View Full Version : General relativity versus black holes
Nick Maclaren
Aug4-04, 01:23 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I think that I know the answer to this, but it may interest\nsome other people. Let\'s consider the Schwarzschild solution.\nIn Newtonian theory, the metric is:\n\nds^2 = dt^2 + dr^2 + r^2 dw^2\n\nIn Einsteinian theory, the metric is:\n\nds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2\n\nNow, let us speculate a unification of quantum mechanics and general\nrelativity that produced the following metric:\n\nds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2\n\nI am, of course, not saying that there is a scrap of evidence for\nsuch a theory. But let us assume one, and assume that it makes\nsimilar changes to other solutions of Einstein\'s equations.\n\nMy question is whether we have any CURRENT data that would enable\nus to distinguish these?\n\nThe relevance is that the speculated formulae would give many of\nthe properties of general relativity, but without black holes and\nother singularities. This, in turns, disproves the claims that\nthe observed binary star results are PROOFS of the existence of\nblack holes.\n\nIf we have any current data that DOES disprove such a theory, then\nthe evidence for the existence of black holes is rather stronger\nthan I think that it is.\n\n\nRegards,\nNick Maclaren.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I think that I know the answer to this, but it may interest
some other people. Let's consider the Schwarzschild solution.
In Newtonian theory, the metric is:
ds^2 = dt^2 + dr^2 + r^2 dw^2
In Einsteinian theory, the metric is:
ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2
Now, let us speculate a unification of quantum mechanics and general
relativity that produced the following metric:
ds^2 = \exp(-2GM/r) dt^2 + \exp(2GM/r) dr^2 + r^2 dw^2
I am, of course, not saying that there is a scrap of evidence for
such a theory. But let us assume one, and assume that it makes
similar changes to other solutions of Einstein's equations.
My question is whether we have any CURRENT data that would enable
us to distinguish these?
The relevance is that the speculated formulae would give many of
the properties of general relativity, but without black holes and
other singularities. This, in turns, disproves the claims that
the observed binary star results are PROOFS of the existence of
black holes.
If we have any current data that DOES disprove such a theory, then
the evidence for the existence of black holes is rather stronger
than I think that it is.
Regards,
Nick Maclaren.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>nmm1@cus.cam.ac.uk (Nick Maclaren) wrote in message news:<ceo9sl\\$d79\\$1@pegasus.csx.cam.ac.uk>...\n > I think that I know the answer to this, but it may interest\n> some other people. Let\'s consider the Schwarzschild solution.\n> In Newtonian theory, the metric is:\n>\n> ds^2 = dt^2 + dr^2 + r^2 dw^2\n>\n> In Einsteinian theory, the metric is:\n>\n> ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2\n>\n> Now, let us speculate a unification of quantum mechanics and general\n> relativity that produced the following metric:\n>\n> ds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2\n>\n> I am, of course, not saying that there is a scrap of evidence for\n> such a theory. But let us assume one, and assume that it makes\n> similar changes to other solutions of Einstein\'s equations.\n>\n> My question is whether we have any CURRENT data that would enable\n> us to distinguish these?\n>\n> The relevance is that the speculated formulae would give many of\n> the properties of general relativity, but without black holes and\n> other singularities. This, in turns, disproves the claims that\n> the observed binary star results are PROOFS of the existence of\n> black holes.\n>\n> If we have any current data that DOES disprove such a theory, then\n> the evidence for the existence of black holes is rather stronger\n> than I think that it is.\n>\n>\n> Regards,\n> Nick Maclaren.\n\n\nI\'m sure many people have noticed that those first two terms in the\nSchwarzschild metric appear to be actually first order terms in a\nTaylor expansion. I know I have. The main problem is that when you\ngo to the equivalent exponential metric form, the metric no longer\nfollows from a vanishing Ricci tensor, as it does in the original\nanalysis. While it appears more elegent with the exponentials, the\nstandard GR analysis would appear to require a very special\ndistribution of mass M to accomodate it.\n\nI think the only way out would be an appropriate modification of the\nEinstein equations to allow this to be derivable from a vanishing\nRicci. The main problem, as I see it, is that any modifications\nwould have to be at the level of derivation of Riemann curvature.\nUnfortunately, this modified geometry would no longer be Riemannian.\nFor that reason I have no idea where you would start with this at all.\nGood luck.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>nmm1@cus.cam.ac.uk (Nick Maclaren) wrote in message news:<ceo9sl$d79$1@pegasus.csx.cam.ac.uk>...
> I think that I know the answer to this, but it may interest
> some other people. Let's consider the Schwarzschild solution.
> In Newtonian theory, the metric is:
>
> ds^2 = dt^2 + dr^2 + r^2 dw^2
>
> In Einsteinian theory, the metric is:
>
> ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2
>
> Now, let us speculate a unification of quantum mechanics and general
> relativity that produced the following metric:
>
> ds^2 = \exp(-2GM/r) dt^2 + \exp(2GM/r) dr^2 + r^2 dw^2
>
> I am, of course, not saying that there is a scrap of evidence for
> such a theory. But let us assume one, and assume that it makes
> similar changes to other solutions of Einstein's equations.
>
> My question is whether we have any CURRENT data that would enable
> us to distinguish these?
>
> The relevance is that the speculated formulae would give many of
> the properties of general relativity, but without black holes and
> other singularities. This, in turns, disproves the claims that
> the observed binary star results are PROOFS of the existence of
> black holes.
>
> If we have any current data that DOES disprove such a theory, then
> the evidence for the existence of black holes is rather stronger
> than I think that it is.
>
>
> Regards,
> Nick Maclaren.
I'm sure many people have noticed that those first two terms in the
Schwarzschild metric appear to be actually first order terms in a
Taylor expansion. I know I have. The main problem is that when you
go to the equivalent exponential metric form, the metric no longer
follows from a vanishing Ricci tensor, as it does in the original
analysis. While it appears more elegent with the exponentials, the
standard GR analysis would appear to require a very special
distribution of mass M to accomodate it.
I think the only way out would be an appropriate modification of the
Einstein equations to allow this to be derivable from a vanishing
Ricci. The main problem, as I see it, is that any modifications
would have to be at the level of derivation of Riemann curvature.
Unfortunately, this modified geometry would no longer be Riemannian.
For that reason I have no idea where you would start with this at all.
Good luck.
Doug Sweetser
Aug5-04, 03:23 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hello Nick:\n\nA few technical corrections before I address your question. Newtonian\nphysics does not have an invariant interval, ds^2. You were thinking\nof flat spacetime versus curved spacetime. The signs for the change in\ntime must be different than the signs for the change in space. For\nds^2, the usual convention is to make the dt^2 negative.\n\nThe coordinate system one uses affect the coefficients. Most books on\ngeneral relativity use the Schwarzschild coordinate system, which look\nlike what you wrote:\n\n> ds^2 = -(1-2GM/c^2 r) c^2 dt^2 + (1-2GM/c^2 r)^-1 dr^2 + r^2 dw^2\n\nThese coordinates are designed to make the resulting metric look\nsimpler. Going around a circle is no longer 2 pi r. I suspect that\nyour proposal is written in isotropic coordinates (all directions are\ntreated equally, a circle is still 2 pi r). These are the coordinates\nused by experimentalists. If one write the Schwarzschild solution in\nisotropic coordinates so apples are compared with apples, then:\n\nds^2 = -((1 - GM/2 c^2 r)/(1 + GM/2 c^2 r))^2 c^2 dt^2\n\n+ (1 + GM/2 c^2 r)^4 dr^2 + r^2 dw^2 (MTW 31.7)\n\nNot as pretty :-( Same information, just different coordinates.\n\n> Now, let us speculate a unification of quantum mechanics and general\n> relativity that produced the following metric [in isotropic\n> coordinates]:\n>\n> ds^2 = -exp(-2GM/c^2 r) c^2 dt^2 + exp(2GM/c^2 r) dr^2 + r^2 dw^2\n>\n> I am, of course, not saying that there is a scrap of evidence for\n> such a theory.\n\nThere is a small body of literature devoted to modification of the\nfield equations of general relativity that results in an exponential\nmetric (authors include Rosen, Yilmaz, Kaniel and Itin, and Watt and\nMisner).\n\nOnly recently did I come across a paper which allows one to predict\nexactly how different light will bend around the Sun from the\nSchwarzschild metric of general relativity ("Post-post-Newtonian\ndeflection of light by the Sun", Epstein and Shapiro, Phys. Rev. D,\n22:2947-2949, 1980). Here is the difference:\n\nSchwarzschild metric: 4 pi (GM/c^2 R) + 3 3/4 pi (GM/c^2 R)^2\nExponential metric: 4 pi (GM/c^2 R) + 4 pi (GM/c^2 R)^2\n-------------------------------------------------------------\nDifference: 1/4 pi (GM/c^2 R)^2\n\nBoth have the same first order in GM/c^2 R bending, 1.76 arcseconds.\nThe exponential metric at second order predicts an additional bending\nof 11.7x10^-6 arcseconds, while the Schwarzschild metric predicts 10.9\nmicroarcseconds, a difference of 0.8 microarcseconds.\n\nAt this level of required sensitivity, other effects come into\nplay. Epstein and Shapiro estimate contributions of 0.2 microseconds\nfrom the quadrapole of the Sun, and 0.7 microarcseconds from the Sun\'s\nrotation.\n\nWhen this paper was written in 1980, the authors did not think in the\nyears time there would be enough of a gain in sensitivity at the\nmicroarcsecond level to use these analytic results. From my reading\nof Clifford Will\'s review article, section 3.4.1, it looks like they\nuncertainty today is around 150 microarcseconds. Therefore it is not\npossible today to accept or reject the exponential metric as a more\naccurate description of a spherical gravitational mass than the\nSchwarzschild metric.\n\nOn aesthetic grounds, the exponential metric looks much prettier than\nthe Schwarzschild metric. Exponential appear in many important\nequations in physics. They play a key role in Lie algebras. For a\nsmall exponent, it means the metric is almost flat. What the algebraic\nfragments that represent the coefficients of the Schwarzschild metric\nmeans I do not grasp.\n\nThere is an experimental reason to reject Rosen\'s proposal. The data\nfor gravity waves indicates that a viable proposal should only have\nquadrapole not dipoles as a generator of waves (a dipole would radiate\nenergy away more quickly than is seen). Energy and momentum\nconservation in general relativity make forming a dipole impossible.\nFor Rosen\'s bi-metric theory, the fixed background metric provides a\nmeans to have a dipole mode of gravity wave generation.\n\nAn often repeated lament for both string theory and loop quantum\ngravity is that they have no testable predictions. Special attention\nshould be focused for work that can be accepted or rejected\non experimental grounds.\n\nThere is no reason to bring in quantum mechanics at this point. One\nwould need a Lagrange density or Hamiltonian that were somehow related\nto this metric before one starts to raise that topic. The exponential\nmetric is a classical animal.\n\n\ndoug\nquaternions.com\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello Nick:
A few technical corrections before I address your question. Newtonian
physics does not have an invariant interval, ds^2. You were thinking
of flat spacetime versus curved spacetime. The signs for the change in
time must be different than the signs for the change in space. For
ds^2, the usual convention is to make the dt^2 negative.
The coordinate system one uses affect the coefficients. Most books on
general relativity use the Schwarzschild coordinate system, which look
like what you wrote:
> ds^2 = -(1-2GM/c^2 r) c^2 dt^2 + (1-2GM/c^2 r)^-1 dr^2 + r^2 dw^2
These coordinates are designed to make the resulting metric look
simpler. Going around a circle is no longer 2 \pi r. I suspect that
your proposal is written in isotropic coordinates (all directions are
treated equally, a circle is still 2 \pi r). These are the coordinates
used by experimentalists. If one write the Schwarzschild solution in
isotropic coordinates so apples are compared with apples, then:
ds^2 = -((1 - GM/2 c^2 r)/(1 + GM/2 c^2 r))^2 c^2 dt^2+ (1 + GM/2 c^2 r)^4 dr^2 + r^2 dw^2[/itex] (MTW 31.7)
Not as pretty :-( Same information, just different coordinates.
> Now, let us speculate a unification of quantum mechanics and general
> relativity that produced the following metric [in isotropic
> coordinates]:
>
> ds^2 = -\exp(-2GM/c^2 r) c^2 dt^2 + \exp(2GM/c^2 r) dr^2 + r^2 dw^2
>
> I am, of course, not saying that there is a scrap of evidence for
> such a theory.
There is a small body of literature devoted to modification of the
field equations of general relativity that results in an exponential
metric (authors include Rosen, Yilmaz, Kaniel and Itin, and Watt and
Misner).
Only recently did I come across a paper which allows one to predict
exactly how different light will bend around the Sun from the
Schwarzschild metric of general relativity ("Post-post-Newtonian
deflection of light by the Sun", Epstein and Shapiro, Phys. Rev. D,
22:2947-2949, 1980). Here is the difference:
Schwarzschild metric: 4 \pi (GM/c^2 R) + 3 3/4 \pi (GM/c^2 R)^2
Exponential metric: 4 \pi (GM/c^2 R) + 4 \pi (GM/c^2 R)^2
-------------------------------------------------------------
Difference: [itex]1/4 \pi (GM/c^2 R)^2
Both have the same first order in GM/c^2 R bending, 1.76 arcseconds.
The exponential metric at second order predicts an additional bending
of 11.7x10^-6 arcseconds, while the Schwarzschild metric predicts 10.9
microarcseconds, a difference of .8 microarcseconds.
At this level of required sensitivity, other effects come into
play. Epstein and Shapiro estimate contributions of .2 microseconds
from the quadrapole of the Sun, and .7 microarcseconds from the Sun's
rotation.
When this paper was written in 1980, the authors did not think in the
years time there would be enough of a gain in sensitivity at the
microarcsecond level to use these analytic results. From my reading
of Clifford Will's review article, section 3.4.1, it looks like they
uncertainty today is around 150 microarcseconds. Therefore it is not
possible today to accept or reject the exponential metric as a more
accurate description of a spherical gravitational mass than the
Schwarzschild metric.
On aesthetic grounds, the exponential metric looks much prettier than
the Schwarzschild metric. Exponential appear in many important
equations in physics. They play a key role in Lie algebras. For a
small exponent, it means the metric is almost flat. What the algebraic
fragments that represent the coefficients of the Schwarzschild metric
means I do not grasp.
There is an experimental reason to reject Rosen's proposal. The data
for gravity waves indicates that a viable proposal should only have
quadrapole not dipoles as a generator of waves (a dipole would radiate
energy away more quickly than is seen). Energy and momentum
conservation in general relativity make forming a dipole impossible.
For Rosen's bi-metric theory, the fixed background metric provides a
means to have a dipole mode of gravity wave generation.
An often repeated lament for both string theory and loop quantum
gravity is that they have no testable predictions. Special attention
should be focused for work that can be accepted or rejected
on experimental grounds.
There is no reason to bring in quantum mechanics at this point. One
would need a Lagrange density or Hamiltonian that were somehow related
to this metric before one starts to raise that topic. The exponential
metric is a classical animal.
doug
quaternions.com
Dallas Kennedy
Aug5-04, 03:24 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Nick,\n\nIf the alternative formula is supposed to reflect a unification of quantum\nmechanics and GR, where are the factors of h? Your alternative looks like a\nmodified classical theory, not a quantum theory.\n\nDallas\n\n"Nick Maclaren" <nmm1@cus.cam.ac.uk> wrote in message\nnews:ceo9sl\\$d79\\$1@pegasus.csx.cam.ac. uk...\n> I think that I know the answer to this, but it may interest\n> some other people. Let\'s consider the Schwarzschild solution.\n> In Newtonian theory, the metric is:\n>\n> ds^2 = dt^2 + dr^2 + r^2 dw^2\n>\n> In Einsteinian theory, the metric is:\n>\n> ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2\n>\n> Now, let us speculate a unification of quantum mechanics and general\n> relativity that produced the following metric:\n>\n> ds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2\n>\n> I am, of course, not saying that there is a scrap of evidence for\n> such a theory. But let us assume one, and assume that it makes\n> similar changes to other solutions of Einstein\'s equations.\n>\n> My question is whether we have any CURRENT data that would enable\n> us to distinguish these?\n>\n> The relevance is that the speculated formulae would give many of\n> the properties of general relativity, but without black holes and\n> other singularities. This, in turns, disproves the claims that\n> the observed binary star results are PROOFS of the existence of\n> black holes.\n>\n> If we have any current data that DOES disprove such a theory, then\n> the evidence for the existence of black holes is rather stronger\n> than I think that it is.\n>\n> Regards,\n> Nick Maclaren.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Nick,
If the alternative formula is supposed to reflect a unification of quantum
mechanics and GR, where are the factors of h? Your alternative looks like a
modified classical theory, not a quantum theory.
Dallas
"Nick Maclaren" <nmm1@cus.cam.ac.uk> wrote in message
news:ceo9sl$d79$1@pegasus.csx.cam.ac.uk...
> I think that I know the answer to this, but it may interest
> some other people. Let's consider the Schwarzschild solution.
> In Newtonian theory, the metric is:
>
> ds^2 = dt^2 + dr^2 + r^2 dw^2
>
> In Einsteinian theory, the metric is:
>
> ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2
>
> Now, let us speculate a unification of quantum mechanics and general
> relativity that produced the following metric:
>
> ds^2 = \exp(-2GM/r) dt^2 + \exp(2GM/r) dr^2 + r^2 dw^2
>
> I am, of course, not saying that there is a scrap of evidence for
> such a theory. But let us assume one, and assume that it makes
> similar changes to other solutions of Einstein's equations.
>
> My question is whether we have any CURRENT data that would enable
> us to distinguish these?
>
> The relevance is that the speculated formulae would give many of
> the properties of general relativity, but without black holes and
> other singularities. This, in turns, disproves the claims that
> the observed binary star results are PROOFS of the existence of
> black holes.
>
> If we have any current data that DOES disprove such a theory, then
> the evidence for the existence of black holes is rather stronger
> than I think that it is.
>
> Regards,
> Nick Maclaren.
tessel@tum.bot
Aug5-04, 03:25 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Wed, 4 Aug 2004, Nick Maclaren wrote:\n\n> I think that I know the answer to this, but it may interest\n> some other people. Let\'s consider the Schwarzschild solution.\n> In Newtonian theory, the metric is:\n>\n> ds^2 = dt^2 + dr^2 + r^2 dw^2\n^\nminus (very important!)\n\nwhere\n\n-infty < t < infty, 0 < r < infty, 0 < u < pi, -pi < v < pi\n\n> In Einsteinian theory, the metric is:\n>\n> ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2\n^\nminus (very important!)\n\nwhere\n\ndw^2 = du^2 + sin(u)^2 dv^2\n\n-infty < t < infty, 2m < r < infty, 0 < u < pi, -pi < v < pi\n\nSigh...\n\nIn the interest of attempting to retard the inexorable decline in the\nintellectual quality of postings to this group, from time to time I try to\nremind contributors that standards of clarity in postings here are higher\nthan in the unmoderated groups. On behalf of all lurkers, as well as\nother posters, I request that all posters here make a serious effort to\nwrite clearly and unambiguously.\n\nNick, I don\'t want to pick nits, but unfortunately you have just given an\nexample of very bad writing which could -easily- have been improved before\nsubmitting your post! IMO this is deplorable, even though in this\nparticular case, probably everyone knows what you presumably -meant- to\nsay.\n\nIn case anyone doesn\'t see what I am complaining about:\n\n"The Schwarzschild vacuum" (or "solution") means a spacetime belonging to\na certain one-parameter family of static spherically symmetric exact\nvacuum solutions to the EFE. This means that it is a Lorentzian manifold\n(M,g), where the metric tensor g can be defined, on "the" exterior region,\nwrt a "polar spherical Schwarzschild coordinate chart", by the second line\nelement above. Strictly speaking, to fully define the underlying smooth\nmanifold M we need to give some additional coordinate charts, defined on\noverlapping domains, together with "transition maps" (diffeomorphisms\ndefined on the overlaps). Such a transition map gives the requisite\n"concordance" between two competing coordinate charts, wherever their\ndomains of definition overlap. Physicists often omit to say all this, but\nsuch laziness can terribly confuse beginners.\n\nCompare Minkowski spacetime E^(1,3), a different (nonisometric) Lorentzian\nmanifold (M,g) whose metric tensor g is defined, on a suitable domain, wrt\na polar spherical coordinate chart, by the first line element above.\nAgain, strictly speaking, to fully define the underlying smooth manifold M\n(usually called R^4), we need to give at least one more chart covering the\nomitted line ("the world line of the observer at r = 0"), plus a\ntransition map. In the case of R^4, we can get away with a single global\nchart (e.g. a Cartesian chart), but this is not possible for many curved\nspacetimes.\n\nBTW, a "Schwarzschild chart" is just any polar spherical chart, on some\nspacetime, in which the "radial coordinate" has the obvious interpretation\nin terms of the surface area of "round spheres". For example, Nick\'s\n"exponential metric" below is a Schwarzschild chart, but the well-known\n"spatially isotropic" polar spherical chart for the Schwarzschild vacuum,\nand the Watt-Misner metric mentioned below, are -not- Schwarzschild\ncharts. So don\'t confuse "a Schwarzschild chart" with "the Schwarzschild\nvacuum"--- they are completely different concepts!\n\nNow, the problem with what Nick said is that writing down the E^(1,3)\nmetric in a polar spherical chart and calling this "the Newtonian analog\nof the Schwarzschild solution", or something like that, is -very-\nmisleading, for a dozen reasons. To name just two:\n\n1. In the "Newtonian" field theory of gravitation, the "field" is a\nscalar function u(x,y,z), namely the gravitational potential, and the\n"field equation" is just Poisson\'s equation\n\nLap(u) = u_(xx) + u_(yy) + u_(zz) = 4 pi mu\n\n(I wrote this in a Cartesian chart, but of course it can be written in any\nother chart defined on some region of R^3.) Thus, a "vacuum solution", in\nthis theory, is simply any harmonic function defined on R^3 (or on some\nopen neighborhood). The validity of euclidean geometry on "space" is an\n-underlying assumption-, but no assumptions about the metric on R^3 are\nneeded to write down the Laplace equation! (It is true that some geometry\nis secretly "preferred" here, since the symmetry group of the 3D Laplace\nequation is the conformal group on E^3.)\n\n2. OTOH, Minkowski geometry is -incompatible- with the Newtonian field\ntheory of gravitation; an easy way to see this is to note that in\nNewtonian theory, the potential responds -instantaneously- everywhere on\nR^3 to changes in the distribution of matter; this obviously violates the\nprinciple that in Minkowski spacetime, information (such as "we just\nchanged the distribution of matter over here!") cannot travel faster than\nthe speed of light. So, Newtonian gravitation is definitely not a\n-relativistic- classical field theory of gravitation! (Compare the\ndiscussion of Cartan\'s notion of "Newtonian spacetime" in MTW.)\n\nOK, as I said, in this -particular- case of very bad writing, probably no\nserious harm was done, but in many other cases, bandwidth is wasted\nbecause respondents misunderstand what the original poster had in mind, or\nhave to request clarification, or because the question never made sense at\nall, etc. So please, -everyone-, let\'s all try to remember that standards\nhave not really been lowered in this n.g., and the charter has not been\nabandoned. Please bear in mind that\n\n(1) the moderators are overworked volunteers (BTW, thanks, guys!),\n\n(2) traffic here is exponentially increasing while the number of\nmoderators remains constant.\n\nSince the moderators are far too busy to continue to suggest felicitous\nimprovements in wording (as John Baez often did in happier days), IMO\n\n-All posters here need to try harder to write well in order to maintain\nour standards-\n\nTIA!\n\nOK, enough of that, onto the rest of Nick\'s post:\n\n> Now, let us speculate a unification of quantum mechanics and general\n> relativity that produced the following metric:\n>\n> ds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2\n>\n> I am, of course, not saying that there is a scrap of evidence for\n> such a theory. But let us assume one,\n\nNo doubt one can come up with some well-defined gravitation theory other\nthan gtr in which this arises as a static spherically symmetric solution,\nat least if you are willing to violate various principles such as Lorentz\ninvariance. As for "uniting QM and gtr", I don\'t think this additional\ndemand is needed to make the point you are trying to make here, so you\nshould be much less ambitious, in the interests of being much more\nspecific about what theory you have in mind!\n\nWarning: in the past, "exponential metrics" such as the one you propose\nhave been put forth by various posters here, in the absence of any\nunderlying theory in which it has the status of a static spherically\nsymmetric solution (but possibly not the -only- such solution). Such\nproposals have, at best, the status of an "Ansatz" rather than a\n"competing theory". Unfortunately, some past posters here could not grasp\nthis elementary distinction, and things got worse from there. This is\nquite irrespective from the not inconsiderable issue of proposing a\ngravitation theory which agrees with available evidence and which doesn\'t\nviolate too many cherished theoretical principles. But I certainly don\'t\nwant to reslay the slain!\n\n> and assume that it makes similar changes to other solutions of\n> Einstein\'s equations.\n\nUnless you can be more specific (or can at least give some more examples),\n"assume similar changes" is too vague to mean anything to me.\n\n> My question is whether we have any CURRENT data that would enable\n> us to distinguish these?\n\nWell, of course this kind of question is the bread and butter of physics.\nWe have some theories, and we want to know: how do they compare with the\ndata? Ideally, we\'d like to test a whole bunch of theories at one fell\nswoop, and pick out the one which best explains all the available\nevidence. In the context of relativistic classical field theories of\ngravitation, PPN formalism provides a handy way to do this for a large\nclass of possible theories. (Of course, you can invent ones which don\'t\nfit into this framework without further work; it is presumably impossible\nto eliminate all competitors without making -some- assumptions about what\nkind of theory you are considering.)\n\nHere is a paper you should read:\n\nauthor = {Keith Watt and Charles W. Misner},\ntitle = {Relativistic Scalar Gravity: A Laboratory for Numerical\nRelativity},\nnote = {gr-qc/9910032}}\n\nThis paper discusses a simple classical gravitation theory (a "stratified\nconformally flat scalar theory of gravitation"). This theory requires a\npreferred frame, and the authors employ a "stratified conformal chart"\n\nds^2 = -f(phi) dt^2 + g(phi) (dx^2 + dy^2 + dz^2),\n\nphi(t,x,y,z) = a scalar field describing gravitation\n\nf,g = functions to be determined\n\nin which the distinguished family of spatial hyperslices are all\nconformally flat. The exact form of phi, f, and g are to be determined by\nsolving their field equation. As you might expect, this problem is\nsimplified if you assume a lot of symmetry, e.g. you can look for static\nspherically symmetric solutions.\n\nNote well: the assumption of a preferred frame drastically violates the\nspirit of gtr, and as the authors note, this assumption implies that the\nWatt-Misner theory cannot exhibit gravitomagnetism. This phenomenon, or\nrather class of phenomena, is predicted by many gravitation theories,\nincluding gtr, but details differ, which is the point--- we can tell\ntheories apart by comparing their predictions in sufficient detail. Even\nas we write, a dedicated satellite experiment is in progress which should\nprovide the first solid test of predicted gravitomagnetic effects near the\nEarth.\n\nAs Watt and Misner point out, the motivation for introducing a stratified\nconformally flat theory is that conformally flat gravitation theories\ncannot exhibit light bending and thus are hopelessly inadequate. OTOH,\nstratified conformally flat theories can exhibit light bending--- but only\nat the cost of breaking global Lorentz invariance. Compare the discussion\nof scalar theories vis a vis vector and tensor theories in MTW (where\nLorentz invariance is assumed).\n\nNext, note that the static spherically symmetric solution in the\nWatt-Misner theory turns out to be given (in a "polar spherical spatially\nisotropic chart" or "stratified conformally flat chart") by:\n\nds^2 = -exp(-2m/r) dt^2 + exp(2m/r) [dr^2 + r^2 (du^2 + sin(u)^2 dv^2)],\n\n-infty < t < infty, 0 < r < infty, 0 < u < pi, -pi < v < pi\n\nExercise: Try to transform your "exponential metric" above into a similar\n"spatially isotropic polar spherical chart" (proceed the same way that you\ntransform the polar spherical Schwarzschild chart for the exterior\nSchwarzschild vacuum to the well-known spatially isotropic polar spherical\nchart for the exterior Schwarzschild vacuum).\n\nFinally, to address your question, note that Watt and Misner give a very\nclear discussion of how to compute the PPN parameters of their theory.\nCompare this with a similar (but less realistic and more complicated!)\nexample in MTW. If you work through these examples, you should get a good\nidea of how we can compare and test gravitation theories.\n\n> The relevance is that the speculated formulae would give many of\n> the properties of general relativity, but without black holes and\n> other singularities.\n\nOh dear, another "obfuscation alert": you must always distinguish clearly\nbetween event horizons (-global- causal features; in the Schwarzschild\nchart the event horizon of the Schwarzschild vacuum happens to form a\nboundary where the coordinate system breaks down, i.e. a "coordinate\nsingularity", but in other coordinate charts the horizon is locally\nunremarkable) and geometric singularities (-local- features) such as\nstrong scalar curvature singularities.\n\nI think you mean here that your spacetime, an "exponential analog" of the\nSchwarzschild vacuum, appears from casual inspection of the form of the\nmetric to lack an -event horizon-. But you should always be very careful\nbefore jumping to any such conclusion. For example, in the Weyl canonical\nchart for "the" exterior of the Schwarzschild vacuum, the horizon appears\nas a -line segment- lying on the axis of axial symmetry used in writing\ndown the chart. (From the form of the chart, it is not evident that in\nfact the spacetime is spherically symmetric.) Point being that we should\nexpect to work through some careful -global analysis- of the causal\nstructure before we can rigorously rule out the existence of an event\nhorizon in a given spacetime model. For, if one exists, it is a -global-\nfeature of the spacetime.\n\n> This, in turns, disproves the claims that the observed binary star\n> results are PROOFS of the existence of black holes.\n\nWhoa, I think you just skipped over a whole lotta highly relevant\nobservations! Such as the orbital decay of certain binary pulsars, which\nis in excellent agreement (so far) with the predictions of gtr, in which\ntheory the decay is attributed to gravitational radiation carrying off\nenergy from the binary system.\n\n> If we have any current data that DOES disprove such a theory, then the\n> evidence for the existence of black holes is rather stronger than I\n> think that it is.\n\nSteve Carlip can probably answer this much better than I can, so I hope he\nwill speak up.\n\n"T. Essel" (hiding somewhere in cyberspace)\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 4 Aug 2004, Nick Maclaren wrote:
> I think that I know the answer to this, but it may interest
> some other people. Let's consider the Schwarzschild solution.
> In Newtonian theory, the metric is:
>
> ds^2 = dt^2 + dr^2 + r^2 dw^2^
minus (very important!)
where
-\infty < t < \infty,< r < \infty,< u < \pi, -\pi < v < \pi
> In Einsteinian theory, the metric is:
>
> ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2^
minus (very important!)
where
dw^2 = du^2 + sin(u)^2 dv^2-\infty < t < \infty, 2m < r < \infty,< u < \pi, -\pi < v < \pi
Sigh...
In the interest of attempting to retard the inexorable decline in the
intellectual quality of postings to this group, from time to time I try to
remind contributors that standards of clarity in postings here are higher
than in the unmoderated groups. On behalf of all lurkers, as well as
other posters, I request that all posters here make a serious effort to
write clearly and unambiguously.
Nick, I don't want to pick nits, but unfortunately you have just given an
example of very bad writing which could -easily- have been improved before
submitting your post! IMO this is deplorable, even though in this
particular case, probably everyone knows what you presumably -meant- to
say.
In case anyone doesn't see what I am complaining about:
"The Schwarzschild vacuum" (or "solution") means a spacetime belonging to
a certain one-parameter family of static spherically symmetric exact
vacuum solutions to the EFE. This means that it is a Lorentzian manifold
(M,g), where the metric tensor g can be defined, on "the" exterior region,
wrt a "polar spherical Schwarzschild coordinate chart", by the second line
element above. Strictly speaking, to fully define the underlying smooth
manifold M we need to give some additional coordinate charts, defined on
overlapping domains, together with "transition maps" (diffeomorphisms
defined on the overlaps). Such a transition map gives the requisite
"concordance" between two competing coordinate charts, wherever their
domains of definition overlap. Physicists often omit to say all this, but
such laziness can terribly confuse beginners.
Compare Minkowski spacetime E^(1,3), a different (nonisometric) Lorentzian
manifold (M,g) whose metric tensor g is defined, on a suitable domain, wrt
a polar spherical coordinate chart, by the first line element above.
Again, strictly speaking, to fully define the underlying smooth manifold M
(usually called R^4), we need to give at least one more chart covering the
omitted line ("the world line of the observer at r = "), plus a
transition map. In the case of R^4, we can get away with a single global
chart (e.g. a Cartesian chart), but this is not possible for many curved
spacetimes.
BTW, a "Schwarzschild chart" is just any polar spherical chart, on some
spacetime, in which the "radial coordinate" has the obvious interpretation
in terms of the surface area of "round spheres". For example, Nick's
"exponential metric" below is a Schwarzschild chart, but the well-known
"spatially isotropic" polar spherical chart for the Schwarzschild vacuum,
and the Watt-Misner metric mentioned below, are -not- Schwarzschild
charts. So don't confuse "a Schwarzschild chart" with "the Schwarzschild
vacuum"--- they are completely different concepts!
Now, the problem with what Nick said is that writing down the E^(1,3)
metric in a polar spherical chart and calling this "the Newtonian analog
of the Schwarzschild solution", or something like that, is -very-
misleading, for a dozen reasons. To name just two:
1. In the "Newtonian" field theory of gravitation, the "field" is a
scalar function u(x,y,z), namely the gravitational potential, and the
"field equation" is just Poisson's equation
Lap(u) = u_(xx) + u_(yy) + u_(zz) = 4 \pi \mu
(I wrote this in a Cartesian chart, but of course it can be written in any
other chart defined on some region of R^3.) Thus, a "vacuum solution", in
this theory, is simply any harmonic function defined on R^3 (or on some
open neighborhood). The validity of euclidean geometry on "space" is an
-underlying assumption-, but no assumptions about the metric on R^3 are
needed to write down the Laplace equation! (It is true that some geometry
is secretly "preferred" here, since the symmetry group of the 3D Laplace
equation is the conformal group on E^3.)
2. OTOH, Minkowski geometry is -incompatible- with the Newtonian field
theory of gravitation; an easy way to see this is to note that in
Newtonian theory, the potential responds -instantaneously- everywhere on
R^3 to changes in the distribution of matter; this obviously violates the
principle that in Minkowski spacetime, information (such as "we just
changed the distribution of matter over here!") cannot travel faster than
the speed of light. So, Newtonian gravitation is definitely not a
-relativistic- classical field theory of gravitation! (Compare the
discussion of Cartan's notion of "Newtonian spacetime" in MTW.)
OK, as I said, in this -particular- case of very bad writing, probably no
serious harm was done, but in many other cases, bandwidth is wasted
because respondents misunderstand what the original poster had in mind, or
have to request clarification, or because the question never made sense at
all, etc. So please, -everyone-, let's all try to remember that standards
have not really been lowered in this n.g., and the charter has not been
abandoned. Please bear in mind that
(1) the moderators are overworked volunteers (BTW, thanks, guys!),
(2) traffic here is exponentially increasing while the number of
moderators remains constant.
Since the moderators are far too busy to continue to suggest felicitous
improvements in wording (as John Baez often did in happier days), IMO
-All[/itex] posters here need to try harder to write well in order to maintain
our standards-
TIA!
OK, enough of that, onto the rest of Nick's post:
> Now, let us speculate a unification of quantum mechanics and general
> relativity that produced the following metric:
>
> ds^2 = \exp(-2GM/r) dt^2 + \exp(2GM/r) dr^2 + r^2 dw^2
>
> I am, of course, not saying that there is a scrap of evidence for
> such a theory. But let us assume one,
No doubt one can come up with some well-defined gravitation theory other
than gtr in which this arises as a static spherically symmetric solution,
at least if you are willing to violate various principles such as Lorentz
invariance. As for "uniting QM and gtr", I don't think this additional
demand is needed to make the point you are trying to make here, so you
should be much less ambitious, in the interests of being much more
specific about what theory you have in mind!
Warning: in the past, "exponential metrics" such as the one you propose
have been put forth by various posters here, in the absence of any
underlying theory in which it has the status of a static spherically
symmetric solution (but possibly not the -only- such solution). Such
proposals have, at best, the status of an "Ansatz" rather than a
"competing theory". Unfortunately, some past posters here could not grasp
this elementary distinction, and things got worse from there. This is
quite irrespective from the not inconsiderable issue of proposing a
gravitation theory which agrees with available evidence and which doesn't
violate too many cherished theoretical principles. But I certainly don't
want to reslay the slain!
> and assume that it makes similar changes to other solutions of
> Einstein's equations.
Unless you can be more specific (or can at least give some more examples),
"assume similar changes" is too vague to mean anything to me.
> My question is whether we have any CURRENT data that would enable
> us to distinguish these?
Well, of course this kind of question is the bread and butter of physics.
We have some theories, and we want to know: how do they compare with the
data? Ideally, we'd like to test a whole bunch of theories at one fell
swoop, and pick out the one which best explains all the available
evidence. In the context of relativistic classical field theories of
gravitation, PPN formalism provides a handy way to do this for a large
class of possible theories. (Of course, you can invent ones which don't
fit into this framework without further work; it is presumably impossible
to eliminate all competitors without making -some- assumptions about what
kind of theory you are considering.)
Here is a paper you should read:
author = {Keith Watt and Charles W. Misner},
title = {Relativistic Scalar Gravity: A Laboratory for Numerical
Relativity},
note = {http://www.arxiv.org/abs/gr-qc/9910032}}
This paper discusses a simple classical gravitation theory (a "stratified
conformally flat scalar theory of gravitation"). This theory requires a
preferred frame, and the authors employ a "stratified conformal chart"
ds^2 = -f(\phi) dt^2 + g(\phi) (dx^2 + dy^2 + dz^2),\phi(t,x,y,z) = a scalar field describing gravitation
f,g = functions to be determined
in which the distinguished family of spatial hyperslices are all
conformally flat. The exact form of \phi, f, and g are to be determined by
solving their field equation. As you might expect, this problem is
simplified if you assume a lot of symmetry, e.g. you can look for static
spherically symmetric solutions.
Note well: the assumption of a preferred frame drastically violates the
spirit of gtr, and as the authors note, this assumption implies that the
Watt-Misner theory cannot exhibit gravitomagnetism. This phenomenon, or
rather class of phenomena, is predicted by many gravitation theories,
including gtr, but details differ, which is the point--- we can tell
theories apart by comparing their predictions in sufficient detail. Even
as we write, a dedicated satellite experiment is in progress which should
provide the first solid test of predicted gravitomagnetic effects near the
Earth.
As Watt and Misner point out, the motivation for introducing a stratified
conformally flat theory is that conformally flat gravitation theories
cannot exhibit light bending and thus are hopelessly inadequate. OTOH,
stratified conformally flat theories can exhibit light bending--- but only
at the cost of breaking global Lorentz invariance. Compare the discussion
of scalar theories vis a vis vector and tensor theories in MTW (where
Lorentz invariance is assumed).
Next, note that the static spherically symmetric solution in the
Watt-Misner theory turns out to be given (in a "polar spherical spatially
isotropic chart" or "stratified conformally flat chart") by:
[itex]ds^2 = -\exp(-2m/r) dt^2 + \exp(2m/r) [dr^2 + r^2 (du^2 + sin(u)^2 dv^2)],-\infty < t < \infty,< r < \infty,< u < \pi, -\pi < v < \pi
Exercise: Try to transform your "exponential metric" above into a similar
"spatially isotropic polar spherical chart" (proceed the same way that you
transform the polar spherical Schwarzschild chart for the exterior
Schwarzschild vacuum to the well-known spatially isotropic polar spherical
chart for the exterior Schwarzschild vacuum).
Finally, to address your question, note that Watt and Misner give a very
clear discussion of how to compute the PPN parameters of their theory.
Compare this with a similar (but less realistic and more complicated!)
example in MTW. If you work through these examples, you should get a good
idea of how we can compare and test gravitation theories.
> The relevance is that the speculated formulae would give many of
> the properties of general relativity, but without black holes and
> other singularities.
Oh dear, another "obfuscation alert": you must always distinguish clearly
between event horizons (-global- causal features; in the Schwarzschild
chart the event horizon of the Schwarzschild vacuum happens to form a
boundary where the coordinate system breaks down, i.e. a "coordinate
singularity", but in other coordinate charts the horizon is locally
unremarkable) and geometric singularities (-local- features) such as
strong scalar curvature singularities.
I think you mean here that your spacetime, an "exponential analog" of the
Schwarzschild vacuum, appears from casual inspection of the form of the
metric to lack an -event horizon-. But you should always be very careful
before jumping to any such conclusion. For example, in the Weyl canonical
chart for "the" exterior of the Schwarzschild vacuum, the horizon appears
as a -line segment- lying on the axis of axial symmetry used in writing
down the chart. (From the form of the chart, it is not evident that in
fact the spacetime is spherically symmetric.) Point being that we should
expect to work through some careful -global analysis- of the causal
structure before we can rigorously rule out the existence of an event
horizon in a given spacetime model. For, if one exists, it is a -global-
feature of the spacetime.
> This, in turns, disproves the claims that the observed binary star
> results are PROOFS of the existence of black holes.
Whoa, I think you just skipped over a whole lotta highly relevant
observations! Such as the orbital decay of certain binary pulsars, which
is in excellent agreement (so far) with the predictions of gtr, in which
theory the decay is attributed to gravitational radiation carrying off
energy from the binary system.
> If we have any current data that DOES disprove such a theory, then the
> evidence for the existence of black holes is rather stronger than I
> think that it is.
Steve Carlip can probably answer this much better than I can, so I hope he
will speak up.
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Aug6-04, 03:05 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Nick Maclaren <nmm1@cus.cam.ac.uk> proposed:\n\n> > Now, let us speculate a unification of quantum mechanics and general\n> > relativity that produced the following metric:\n> >\n> > ds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2\n\nIgor <thoovler@excite.com> commented:\n\n> I\'m sure many people have noticed that those first two terms in the\n> Schwarzschild metric appear to be actually first order terms in a Taylor\n> expansion. I know I have.\n\nYes, this is pointed out in several of the standard textbooks and AFAIK\nthis was first pointed out by Karl Schwarzschild himself.\n\n> The main problem is that when you go to the equivalent exponential\n> metric form, the metric no longer follows from a vanishing Ricci tensor,\n> as it does in the original analysis.\n\nI think Nick knows that his spacetime is not a vacuum solution of the EFE;\nrather, he wants to pretend for the sake of argument that one can concoct\nan alternative gravitation theory in which his spacetime becomes a "vacuum\nsolution". He is assuming (carelessly) that his spacetime has no event\nhorizon (I already pointed out that confirming this guess requires a\ncareful global analysis of the causal structure of this spacetime---\nclearly, this analysis can be carried out by computing a conformal chart\nof this asymptotically flat spacetime, using standard methods), and I\nthink his hope is that one can use his spacetime as a model of the\ngravitational field of astrophysical objects like stars and even\nastrophysical black holes, without disagreeing noticeably with any\ncurrently available evidence.\n\n> While it appears more elegent with the exponentials, the standard GR\n> analysis would appear to require a very special distribution of mass M\n> to accomodate it.\n\nLet us ask: what is a "nonvacuum solution of the EFE"? As various posters\nhave frequently pointed out over the years, we can take -any- Lorentzian\nmanifold, compute the mathematically well-defined tensor field G^(ab),\ndivide by 8 pi, and try to interpret the result as T^(ab) for some\ndistribution of ordinary matter and perhaps various "physical fields" such\nas an EM field, a massless minimally coupled scalar field, or more exotic\nthings. Clearly, this procedure would render gtr useless if we accept\n-any- spacetime as "a solution of the EFE"; since an unfalsifiable theory\nis useless in science. So to use gtr, we must be choosy about what we\nadmit as a legal contribution to T^(ab).\n\nFortunately, it turns out that if we stick to T^(ab) terms produced\n(according to gtr plus standard physics) by ordinary matter (e.g. ideal\ngases, dusts) and well-studied fields like EM fields (including incoherent\nradiation, or "null dusts" in the lingo), we rule out almost all\nLorentzian spacetimes as models admitted by gtr. So in fact "gtr plus\nreasonable restrictions on the form of T^(ab)" (which is of course what\nEinstein had in mind!) actually is quite a stringent theory.\n\nGravitation theories are in some ways analogous to thermodynamics. In the\nsubject of thermodynamics, we study general properties of energy transport\nand "degradation of energy" which hold irrespective of our "theory of\nmatter/radiation". In the subject of gravitation theory, we study general\nproperties of gravitational fields which hold irrespective of our "theory\nof matter/radiation". This analogy is not perfect, but as everyone here\nno doubt knows, in the last quarter of the previous century it emerged\nthat there are in fact profound connections between thermodynamical and\ngravitational concepts. Put another way, it may not make much sense to\ndemand that any gravitation theory be easily falsifiable -in isolation\nfrom the rest of physics-; rather, we should demand that such a theory\nmust be easily falsifiable -when we toss in reasonable assumptions about\nwhat constitutes "legal" matter or nongravitational physical fields-.\nBut unforunately for this sensible approach, at present cosmological\nobservations encourage all kinds of wild speculations concerning exotic\nmatter/fields!\n\nBTW, as simple examples of static spherically symmetric -nonvacuum-\nsolutions to the EFE, with T^(ab) contributions everyone would regard as\nlegal, one might mention:\n\n1. the FRW dust (with E^3 hyperslices orthogonal to the world lines of the\ndust particles),\n\nds^2 = -dt^2 + t^(4/3) [dr^2 + r^2 (du^2 + sin(u)^2 dv^2) ],\n\n0 < t,r < infty, 0 < u < pi, -pi < v < pi\n\n2. the Schwarschild incompressible perfect fluid,\n\nds^2 = -[(3*cos(w0)-cos(w))/2 dt]^2\n\n+ A^2 [dw^2 + w^2 (du^2 + sin(u)^2 dv^2) ]\n\nwhere A, w0 are real parameters and\n\n-infty < t < infty, 0 < w < w0, 0 < u < pi, -pi < v < pi\n\nMore speculative examples include:\n\n3. the Janis-Newman-Winacour massless minimally coupled scalar field\nsolution:\n\nds^2 = -(1-2m/r)^(2 cos a) dt^2\n\n+ (1-2m/r)^(-2 cos a) [dr^2 + r^2 (du^2 + sin(u)^2 dv^2)],\n\nphi = sin(a) log(1-2 m/r)/sqrt(2) = the scalar field\n\nwhere m,a are real parameters and\n\n-infty < t < infty, 2 m < r < infty, 0 < u < pi, -pi < v < pi\n\n(see gr-qc/9908029).\n\nIn all of these examples, we can see the metric of the "round unit sphere"\n\n(d sigma)^2 = du^2 + sin(u)^2 dv^2,\n\n0 < u < pi, -pi < v < pi\n\nComputing the Lie algebra of Killing vector fields confirms that these are\nstatic spherically symmetric solutions.\n\n> I think the only way out would be an appropriate modification of the\n> Einstein equations to allow this to be derivable from a vanishing\n> Ricci.\n\nYes, IOW, we agree that what Nick really wants to do is to concoct a\n-classical- relativistic theory of gravitation (other than gtr) with three\nproperties:\n\n1. The vacuum solutions describing the exterior field of compact objects\n(presumably including rotating objects and non spherically symmetric\nobjects) never admit event horizons,\n\n2. The theory is consistent with all currently available experimental and\nobservational evidence.\n\n3. The theory is self-consistent, mathematically speaking.\n\nThis would appear to be a challenging program!\n\n(We have discussed it here, at length, on many previous occasions. Note\nthat I am assuming that the theory is in some sense a metric theory,\npresumambly a tensor theory with some kind of tensorial field equation\nmore complicated than the EFE, and possibly with various additional\nscalar, vector or tensor fields needed to fully define "the gravitational\nfield". One of the main points here is that in order to achieve 1,2,3, it\nseems that you need to sacrifice simplicity! So the real question here\nis: how simple a theory can one find having these properties?)\n\n> The main problem, as I see it, is that any modifications would have to\n> be at the level of derivation of Riemann curvature. Unfortunately, this\n> modified geometry would no longer be Riemannian.\n^\nLorentzian (special case of semi-Riemannian)\n\nHuh?--- I don\'t know why you think that! After all, Nick\'s spacetime is\ncertainly a Lorentzian spacetime, so it is not unreasonable to expect that\nit might arise as a solution to some kind of "geometric" field equation,\nperhaps the Euler-Lagrange equations induced by some Lagrangian. (But for\nreasons discussed in MTW, if we demand a scalar or vector theory, we\'ll\nhave to abandon at least some "cherished principles", e.g. Lorentz\ninvariance. Or else we\'ll need to introduce other kinds of complications,\nsuch as extra physical fields.)\n\nIt is true that various people -have- proposed a number of theories which\ninvolve connections which are not Christoffel connections (aka Levi-Civita\nconnections), etc. In particular, last year we discussed at length a\nclass of theories formulated in terms of a Weitzenboeck connection (zero\ncurvature but nonzero torsion) instead of a Christoffel connection\n(nonzero curvature but zero torsion). And someone else recently mentioned\nthe recent "nonsymmetric gravity" theory (NSG) of Cornish and Moffat.\nSome work has been done by the authors of these theories suggesting that\nthey pass various observational and experimental tests.\n\nHowever, because gtr is the simplest and by far best studied theory which\nis consistent with -all- the evidence so far, it remains our gold standard\ngravitation theory. Even if it eventually turns out to be wrong under\nsome circumstances (as everyone knows, this seems very likely, because gtr\nis not a quantum theory of gravitation), it seems probable that any\nreplacement will be much harder to formulate (but hopefully, in some sense\nnot much more complicated) and probably much harder to use. Therefore we\nshould expect that just as Newtonian gravitation remains viable under many\ncircumstances--- in such cases, it would usually be idiotic to use gtr,\nwhich is usually harder to work with!-- gtr will certainly remain viable\nunder a wide variety of circumstances--- at the very least, all those so\nfar examined by experimentalists and astronomers. Because gtr and\nNewtonian gravitation are presumably the by far the best-studied "pretty\nsimple and pretty darned good gravitation theories", it seems likely that\nthey will both live as long as the pursuit of physics itself continues.\n\n"T. Essel" (hiding somewhere in cyberspace)\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Nick Maclaren <nmm1@cus.cam.ac.uk> proposed:
> > Now, let us speculate a unification of quantum mechanics and general
> > relativity that produced the following metric:
> >
> > ds^2 = \exp(-2GM/r) dt^2 + \exp(2GM/r) dr^2 + r^2 dw^2
Igor <thoovler@excite.com> commented:
> I'm sure many people have noticed that those first two terms in the
> Schwarzschild metric appear to be actually first order terms in a Taylor
> expansion. I know I have.
Yes, this is pointed out in several of the standard textbooks and AFAIK
this was first pointed out by Karl Schwarzschild himself.
> The main problem is that when you go to the equivalent exponential
> metric form, the metric no longer follows from a vanishing Ricci tensor,
> as it does in the original analysis.
I think Nick knows that his spacetime is not a vacuum solution of the EFE;
rather, he wants to pretend for the sake of argument that one can concoct
an alternative gravitation theory in which his spacetime becomes a "vacuum
solution". He is assuming (carelessly) that his spacetime has no event
horizon (I already pointed out that confirming this guess requires a
careful global analysis of the causal structure of this spacetime---
clearly, this analysis can be carried out by computing a conformal chart
of this asymptotically flat spacetime, using standard methods), and I
think his hope is that one can use his spacetime as a model of the
gravitational field of astrophysical objects like stars and even
astrophysical black holes, without disagreeing noticeably with any
currently available evidence.
> While it appears more elegent with the exponentials, the standard GR
> analysis would appear to require a very special distribution of mass M
> to accomodate it.
Let us ask: what is a "nonvacuum solution of the EFE"? As various posters
have frequently pointed out over the years, we can take -any- Lorentzian
manifold, compute the mathematically well-defined tensor field G^(ab),
divide by 8 \pi, and try to interpret the result as T^(ab) for some
distribution of ordinary matter and perhaps various "physical fields" such
as an EM field, a massless minimally coupled scalar field, or more exotic
things. Clearly, this procedure would render gtr useless if we accept
-any- spacetime as "a solution of the EFE"; since an unfalsifiable theory
is useless in science. So to use gtr, we must be choosy about what we
admit as a legal contribution to T^(ab).
Fortunately, it turns out that if we stick to T^(ab) terms produced
(according to gtr plus standard physics) by ordinary matter (e.g. ideal
gases, dusts) and well-studied fields like EM fields (including incoherent
radiation, or "null dusts" in the lingo), we rule out almost all
Lorentzian spacetimes as models admitted by gtr. So in fact "gtr plus
reasonable restrictions on the form of T^(ab)" (which is of course what
Einstein had in mind!) actually is quite a stringent theory.
Gravitation theories are in some ways analogous to thermodynamics. In the
subject of thermodynamics, we study general properties of energy transport
and "degradation of energy" which hold irrespective of our "theory of
matter/radiation". In the subject of gravitation theory, we study general
properties of gravitational fields which hold irrespective of our "theory
of matter/radiation". This analogy is not perfect, but as everyone here
no doubt knows, in the last quarter of the previous century it emerged
that there are in fact profound connections between thermodynamical and
gravitational concepts. Put another way, it may not make much sense to
demand that any gravitation theory be easily falsifiable -in isolation
from the rest of physics-; rather, we should demand that such a theory
must be easily falsifiable -when we toss in reasonable assumptions about
what constitutes "legal" matter or nongravitational physical fields-.
But unforunately for this sensible approach, at present cosmological
observations encourage all kinds of wild speculations concerning exotic
matter/fields!
BTW, as simple examples of static spherically symmetric -nonvacuum-
solutions to the EFE, with T^(ab) contributions everyone would regard as
legal, one might mention:
1. the FRW dust (with E^3 hyperslices orthogonal to the world lines of the
dust particles),
ds^2 = -dt^2 + t^(4/3) [dr^2 + r^2 (du^2 + sin(u)^2 dv^2) ],
< t,r < \infty,< u < \pi, -\pi < v < \pi
2. the Schwarschild incompressible perfect fluid,
ds^2 = -[(3*cos(w0)-cos(w))/2 dt]^2+ A^2 [dw^2 + w^2 (du^2 + sin(u)^2 dv^2) ]
where A, w0 are real parameters and
-\infty < t < \infty,< w <[/itex] w0, < u < \pi, -\pi < v < \pi
More speculative examples include:
3. the Janis-Newman-Winacour massless minimally coupled scalar field
solution:
ds^2 = -(1-2m/r)^(2 cos a) dt^2+ (1-2m/r)^(-2 cos a) [dr^2 + r^2 (du^2 + sin(u)^2 dv^2)],\phi = sin(a) log(1-2 m/r)/\sqrt(2) = the scalar field
where m,a are real parameters and
[itex]-\infty < t < \infty, 2 m < r < \infty,< u < \pi, -\pi < v < \pi
(see http://www.arxiv.org/abs/gr-qc/9908029).
In all of these examples, we can see the metric of the "round unit sphere"
(d \sigma)^2 = du^2 + sin(u)^2 dv^2,
< u < \pi, -\pi < v < \pi
Computing the Lie algebra of Killing vector fields confirms that these are
static spherically symmetric solutions.
> I think the only way out would be an appropriate modification of the
> Einstein equations to allow this to be derivable from a vanishing
> Ricci.
Yes, IOW, we agree that what Nick really wants to do is to concoct a
-classical- relativistic theory of gravitation (other than gtr) with three
properties:
1. The vacuum solutions describing the exterior field of compact objects
(presumably including rotating objects and non spherically symmetric
objects) never admit event horizons,
2. The theory is consistent with all currently available experimental and
observational evidence.
3. The theory is self-consistent, mathematically speaking.
This would appear to be a challenging program!
(We have discussed it here, at length, on many previous occasions. Note
that I am assuming that the theory is in some sense a metric theory,
presumambly a tensor theory with some kind of tensorial field equation
more complicated than the EFE, and possibly with various additional
scalar, vector or tensor fields needed to fully define "the gravitational
field". One of the main points here is that in order to achieve 1,2,3, it
seems that you need to sacrifice simplicity! So the real question here
is: how simple a theory can one find having these properties?)
> The main problem, as I see it, is that any modifications would have to
> be at the level of derivation of Riemann curvature. Unfortunately, this
> modified geometry would no longer be Riemannian.
^
Lorentzian (special case of semi-Riemannian)
Huh?--- I don't know why you think that! After all, Nick's spacetime is
certainly a Lorentzian spacetime, so it is not unreasonable to expect that
it might arise as a solution to some kind of "geometric" field equation,
perhaps the Euler-Lagrange equations induced by some Lagrangian. (But for
reasons discussed in MTW, if we demand a scalar or vector theory, we'll
have to abandon at least some "cherished principles", e.g. Lorentz
invariance. Or else we'll need to introduce other kinds of complications,
such as extra physical fields.)
It is true that various people -have- proposed a number of theories which
involve connections which are not Christoffel connections (aka Levi-Civita
connections), etc. In particular, last year we discussed at length a
class of theories formulated in terms of a Weitzenboeck connection (zero
curvature but nonzero torsion) instead of a Christoffel connection
(nonzero curvature but zero torsion). And someone else recently mentioned
the recent "nonsymmetric gravity" theory (NSG) of Cornish and Moffat.
Some work has been done by the authors of these theories suggesting that
they pass various observational and experimental tests.
However, because gtr is the simplest and by far best studied theory which
is consistent with -all- the evidence so far, it remains our gold standard
gravitation theory. Even if it eventually turns out to be wrong under
some circumstances (as everyone knows, this seems very likely, because gtr
is not a quantum theory of gravitation), it seems probable that any
replacement will be much harder to formulate (but hopefully, in some sense
not much more complicated) and probably much harder to use. Therefore we
should expect that just as Newtonian gravitation remains viable under many
circumstances--- in such cases, it would usually be idiotic to use gtr,
which is usually harder to work with!-- gtr will certainly remain viable
under a wide variety of circumstances--- at the very least, all those so
far examined by experimentalists and astronomers. Because gtr and
Newtonian gravitation are presumably the by far the best-studied "pretty
simple and pretty darned good gravitation theories", it seems likely that
they will both live as long as the pursuit of physics itself continues.
"T. Essel" (hiding somewhere in cyberspace)
carlip@no-physics-spam.ucdavis.edu
Aug6-04, 03:05 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Nick Maclaren <nmm1@cus.cam.ac.uk> wrote:\n\n[...]\n> Now, let us speculate a unification of quantum mechanics and general\n> relativity that produced the following metric:\n\n> ds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2\n\n> I am, of course, not saying that there is a scrap of evidence for\n> such a theory. But let us assume one, and assume that it makes\n> similar changes to other solutions of Einstein\'s equations.\n\n> My question is whether we have any CURRENT data that would enable\n> us to distinguish these?\n\nIf your metric describes the exterior of a collapsing/collapsed\nobject, then presumably it should change at some boundary to an\ninterior metric describing the interior of the remaining matter.\nFor a general relativistic black hole, this doesn\'t matter so\nmuch, because the transition takes place inside the horizon, and\nis unobservable to those outside. Your metric, on the other\nhand, has no horizon, so the "surface" of the collapsed object\nis not invisible. This means that infalling matter should hit\nthis surface.\n\nThere are two pieces of observational evidence against this that\nI know of. Neither is conclusive, but both strongly suggest that\nthe conventional black hole picture is right. (I\'m not an expert\non these; perhaps others can add more.)\n\nThe first of these is apparent observation of "advection-dominated\naccretion flow," or ADAF. As a gas falls into a black hole, it\nreleases a large amount of gravitational potential energy. Under\nmany circumstances, this energy is radiated away; this is what\nastronomers "see" when they talk about observing a black hole. But\nthere is another possible flow, in which the energy is stored as\nheat, with only a small amount of radiation.\n\nUnder such an advection-dominated flow, the gas becomes extremely hot.\nOne can then ask what happens to the energy. If the gas eventually\nhits a surface, the energy will be released; this is observed for\nflows onto neutron stars. If the object is a black hole, on the other\nhand, the energy will be lost behind the horizon and will not come out\nagain. This is also observed, but only for gas flowing onto objects\nthat are predicted from mass observations to be black holes. While\nI think there is still some controversy over details of ADAF, these\nobservations certainly provide some evidence of a horizon. See, for\nexample, http://cfa-www.harvard.edu/blackhole/release.html, or\nNarayan et al., Ap. J. 478 (1997) L79.\n\nA second argument has to do with the observation (and nonobservation)\nof type I X-ray bursts, which are the result of thermonuclear explosions\nwhen gas accretes onto the surface of a compact star and ignites. It\nseems to be systematically true that such bursts are observed from\nobjects whose mass is low enough that they ought to be neutron stars,\nand are *not* observed from objects whose mass is lowhigh enough that\nthey ought to be black holes. This is again evidence that the black\nhole candidates have no visible "surface" on which the gas can collect.\nThere\'s a nice, not-too-technical lecture by Narayan on this on the\narXiv, astro-ph/0310692, which also discusses possible loopholes.\n\nThese observations are certainly not conclusive. Observations of\ngravitational radiation from colliding black holes and from objects\nfalling into black holes is probably not *too* far off, and this will\neventually allow a detailed investigation of the metric. But the existing\nevidence *does* show that there are strong differences between collapsed\nobjects with neutron-star masses and those with black-hole masses, and\nthat these differences have to do with the question of whether infalling\nmatter hits a surface and releases energy that then escapes.\n\nSteve Carlip\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Nick Maclaren <nmm1@cus.cam.ac.uk> wrote:
[...]
> Now, let us speculate a unification of quantum mechanics and general
> relativity that produced the following metric:
> ds^2 = \exp(-2GM/r) dt^2 + \exp(2GM/r) dr^2 + r^2 dw^2
> I am, of course, not saying that there is a scrap of evidence for
> such a theory. But let us assume one, and assume that it makes
> similar changes to other solutions of Einstein's equations.
> My question is whether we have any CURRENT data that would enable
> us to distinguish these?
If your metric describes the exterior of a collapsing/collapsed
object, then presumably it should change at some boundary to an
interior metric describing the interior of the remaining matter.
For a general relativistic black hole, this doesn't matter so
much, because the transition takes place inside the horizon, and
is unobservable to those outside. Your metric, on the other
hand, has no horizon, so the "surface" of the collapsed object
is not invisible. This means that infalling matter should hit
this surface.
There are two pieces of observational evidence against this that
I know of. Neither is conclusive, but both strongly suggest that
the conventional black hole picture is right. (I'm not an expert
on these; perhaps others can add more.)
The first of these is apparent observation of "advection-dominated
accretion flow," or ADAF. As a gas falls into a black hole, it
releases a large amount of gravitational potential energy. Under
many circumstances, this energy is radiated away; this is what
astronomers "see" when they talk about observing a black hole. But
there is another possible flow, in which the energy is stored as
heat, with only a small amount of radiation.
Under such an advection-dominated flow, the gas becomes extremely hot.
One can then ask what happens to the energy. If the gas eventually
hits a surface, the energy will be released; this is observed for
flows onto neutron stars. If the object is a black hole, on the other
hand, the energy will be lost behind the horizon and will not come out
again. This is also observed, but only for gas flowing onto objects
that are predicted from mass observations to be black holes. While
I think there is still some controversy over details of ADAF, these
observations certainly provide some evidence of a horizon. See, for
example, http://cfa-www.harvard.edu/blackhole/release.html, or
Narayan et al., Ap. J. 478 (1997) L79.
A second argument has to do with the observation (and nonobservation)
of type I X-ray bursts, which are the result of thermonuclear explosions
when gas accretes onto the surface of a compact star and ignites. It
seems to be systematically true that such bursts are observed from
objects whose mass is low enough that they ought to be neutron stars,
and are *not* observed from objects whose mass is lowhigh enough that
they ought to be black holes. This is again evidence that the black
hole candidates have no visible "surface" on which the gas can collect.
There's a nice, not-too-technical lecture by Narayan on this on the
arXiv, http://www.arxiv.org/abs/astro-ph/0310692, which also discusses possible loopholes.
These observations are certainly not conclusive. Observations of
gravitational radiation from colliding black holes and from objects
falling into black holes is probably not *too* far off, and this will
eventually allow a detailed investigation of the metric. But the existing
evidence *does* show that there are strong differences between collapsed
objects with neutron-star masses and those with black-hole masses, and
that these differences have to do with the question of whether infalling
matter hits a surface and releases energy that then escapes.
Steve Carlip
Nick Maclaren
Aug7-04, 05:03 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <ceve49\\$pvq\\$1@godfrey.mcc.ac.uk>, <tessel@tum.bot> wrote:\n>\n>I think Nick knows that his spacetime is not a vacuum solution of the EFE;\n>rather, he wants to pretend for the sake of argument that one can concoct\n>an alternative gravitation theory in which his spacetime becomes a "vacuum\n>solution". He is assuming (carelessly) that his spacetime has no event\n>horizon (I already pointed out that confirming this guess requires a\n>careful global analysis of the causal structure of this spacetime---\n>clearly, this analysis can be carried out by computing a conformal chart\n>of this asymptotically flat spacetime, using standard methods), ...\n\nSigh. No. What I am asking is whether currently observed data is\nenough to exclude such models, where I provided a simple and VERY\npartial example of what I meant. I.e. whether the current claims that\nthe radiation from binary stars demonstrates the existence of black\nholes is indeed correct.\n\nI am (still) enough of a mathematician to know that producing an\nalternative formula for one special case does not mean that it extends\nto the whole domain, still less that it maintains properties when doing\nso. Developing such a theory needs a better mathematician than I am.\n\nIn article <ceug0j\\$f76\\$1@woodrow.ucdavis.edu>,\n<carlip@n o-physics-spam.ucdavis.edu> wrote:\n>\n>If your metric describes the exterior of a collapsing/collapsed\n>object, then presumably it should change at some boundary to an\n>interior metric describing the interior of the remaining matter.\n\nNo, I am hypothesising something without an interior metric.\n\n\nRegards,\nNick Maclaren.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <ceve49$pvq$1@godfrey.mcc.ac.uk>, <tessel@tum.bot> wrote:
>
>I think Nick knows that his spacetime is not a vacuum solution of the EFE;
>rather, he wants to pretend for the sake of argument that one can concoct
>an alternative gravitation theory in which his spacetime becomes a "vacuum
>solution". He is assuming (carelessly) that his spacetime has no event
>horizon (I already pointed out that confirming this guess requires a
>careful global analysis of the causal structure of this spacetime---
>clearly, this analysis can be carried out by computing a conformal chart
>of this asymptotically flat spacetime, using standard methods), ...
Sigh. No. What I am asking is whether currently observed data is
enough to exclude such models, where I provided a simple and VERY
partial example of what I meant. I.e. whether the current claims that
the radiation from binary stars demonstrates the existence of black
holes is indeed correct.
I am (still) enough of a mathematician to know that producing an
alternative formula for one special case does not mean that it extends
to the whole domain, still less that it maintains properties when doing
so. Developing such a theory needs a better mathematician than I am.
In article <ceug0j$f76$1@woodrow.ucdavis.edu>,
<carlip@no-physics-spam.ucdavis.edu> wrote:
>
>If your metric describes the exterior of a collapsing/collapsed
>object, then presumably it should change at some boundary to an
>interior metric describing the interior of the remaining matter.
No, I am hypothesising something without an interior metric.
Regards,
Nick Maclaren.
Nick Maclaren
Aug7-04, 05:04 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <ceug0j\\$f76\\$1@woodrow.ucdavis.edu>,\ncarlip@ no-physics-spam.ucdavis.edu writes:\n|>\n|> For a general relativistic black hole, this doesn\'t matter so\n|> much, because the transition takes place inside the horizon, and\n|> is unobservable to those outside. Your metric, on the other\n|> hand, has no horizon, so the "surface" of the collapsed object\n|> is not invisible. This means that infalling matter should hit\n|> this surface.\n\nNow THAT has the potential to be a good test. It is pretty unlikely\nthat ANY "black hole free" theory does not imply that there will be\na more-or-less definite surface to objects. And the effects of\nimpacts on such things are very different from those where there is\nno such discontiguity.\n\nAre the observational effects clear enough to show the difference?\nMost of the papers I have seen have built their deductions on a\nsufficiently high and ricketty tower of hypotheses that they can\'t\nbe regarded as solid. Probable, yes, but that is different.\n\n\nRegards,\nNick Maclaren.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <ceug0j$f76$1@woodrow.ucdavis.edu>,
carlip@no-physics-spam.ucdavis.edu writes:
|>
|> For a general relativistic black hole, this doesn't matter so
|> much, because the transition takes place inside the horizon, and
|> is unobservable to those outside. Your metric, on the other
|> hand, has no horizon, so the "surface" of the collapsed object
|> is not invisible. This means that infalling matter should hit
|> this surface.
Now THAT has the potential to be a good test. It is pretty unlikely
that ANY "black hole free" theory does not imply that there will be
a more-or-less definite surface to objects. And the effects of
impacts on such things are very different from those where there is
no such discontiguity.
Are the observational effects clear enough to show the difference?
Most of the papers I have seen have built their deductions on a
sufficiently high and ricketty tower of hypotheses that they can't
be regarded as solid. Probable, yes, but that is different.
Regards,
Nick Maclaren.
Doug Sweetser
Aug7-04, 05:08 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hidding somewhere, T Essel wrote:\n\n> This is quite irrespective from the not inconsiderable issue of\n> proposing a gravitation theory which agrees with available evidence\n> and which doesn\'t violate too many cherished theoretical principles.\n\nWhat theoretical principle would people in SPR be willing to give up? I\nthink there is consensus that we will not give up the rule that\nexperimental data trumps theory. For that reason, if we ever measure\nbending of light around the Sun to 1 microarcsecond of accuracy, we may\nbe able to either confirm or reject the exponential metric. This is\njust like what happened at the first order PPN level: the data said\nEinstein\'s second rank theory was right, not Newton\'s scalar theory.\nThe exponential metric predicts more bending the the Schwarzschild\nmetric. Let the data rule.\n\nSomethings I would keep: the weak equivalence principle (inertial equals\ngravitational mass), the strong equivalence principle (the active\ngravitational mass is equivalent to the passive gravitational mass),\nthe local nature of the the equations, and local Lorentz invariance.\nOne "cherished" idea I am willing to part with is the nonlinear nature\nof the vacuum equations. There are many adored thought experiments to\nprove the equations of gravity must necessarily be nonlinear. I have\nread a few of these. The thought experiments are usually done with\nelectrically neutral material. Put a charge on the particles in\nquestion. Rest mass and electric charge are both Lorentz invariant\nscalars. The thought experiments involve transforming that rest mess,\ninvariably destroying electric charge in the process. For me, I see a\nconflict between electric charge conservation and thought experiments\nto show the nonlinearity of gravity. With that choice, I go with\nelectric charge conservation. Once you are clear that you are working\nwith linear field equations, professionals don\'t take the work too\nseriously :-) The real answer lies deeper, they say.\n\nIn a different post under this topic, T. Essel appeared to suggest that\nMTW had provided a reason to reject the Goldilocks approach to gravity\nthe vector approach, inbetween Newton\'s scalar theory and the second\nrank field equations of Einstein\'s work. I found one section that\ndeals with the topic, exercise 7.2. The field strength tensor in that\nproblem was antisymmetric. If gravity is a metric theory which all\ndata indicates, then the ultimate cause of the dynamic metric must have\nthe same symmetry, it must be symmetric. If anyone can provide me with\na modern reference to a decent technical rejection of a purely vector\ntheory, I\'d appreciate it (not vector-tensor as Will discusses in his\nliving review article, just plain Jane vector). So far I have about 5\npapers that only go one for a paragraph, and they all look flawed to me\nso far. One person I talked with said a vector approach was simple\nenough to be insulting to the physicists. That was amusing :-)\n\n\ndoug\nquaternions.com\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hidding somewhere, T Essel wrote:
> This is quite irrespective from the not inconsiderable issue of
> proposing a gravitation theory which agrees with available evidence
> and which doesn't violate too many cherished theoretical principles.
What theoretical principle would people in SPR be willing to give up? I
think there is consensus that we will not give up the rule that
experimental data trumps theory. For that reason, if we ever measure
bending of light around the Sun to 1 microarcsecond of accuracy, we may
be able to either confirm or reject the exponential metric. This is
just like what happened at the first order PPN level: the data said
Einstein's second rank theory was right, not Newton's scalar theory.
The exponential metric predicts more bending the the Schwarzschild
metric. Let the data rule.
Somethings I would keep: the weak equivalence principle (inertial equals
gravitational mass), the strong equivalence principle (the active
gravitational mass is equivalent to the passive gravitational mass),
the local nature of the the equations, and local Lorentz invariance.
One "cherished" idea I am willing to part with is the nonlinear nature
of the vacuum equations. There are many adored thought experiments to
prove the equations of gravity must necessarily be nonlinear. I have
read a few of these. The thought experiments are usually done with
electrically neutral material. Put a charge on the particles in
question. Rest mass and electric charge are both Lorentz invariant
scalars. The thought experiments involve transforming that rest mess,
invariably destroying electric charge in the process. For me, I see a
conflict between electric charge conservation and thought experiments
to show the nonlinearity of gravity. With that choice, I go with
electric charge conservation. Once you are clear that you are working
with linear field equations, professionals don't take the work too
seriously :-) The real answer lies deeper, they say.
In a different post under this topic, T. Essel appeared to suggest that
MTW had provided a reason to reject the Goldilocks approach to gravity
the vector approach, inbetween Newton's scalar theory and the second
rank field equations of Einstein's work. I found one section that
deals with the topic, exercise 7.2. The field strength tensor in that
problem was antisymmetric. If gravity is a metric theory which all
data indicates, then the ultimate cause of the dynamic metric must have
the same symmetry, it must be symmetric. If anyone can provide me with
a modern reference to a decent technical rejection of a purely vector
theory, I'd appreciate it (not vector-tensor as Will discusses in his
living review article, just plain Jane vector). So far I have about 5
papers that only go one for a paragraph, and they all look flawed to me
so far. One person I talked with said a vector approach was simple
enough to be insulting to the physicists. That was amusing :-)
doug
quaternions.com
Robert Shaw
Aug12-04, 08:29 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n"Igor" <thoovler@excite.com> wrote\n>...\n> > I think that I know the answer to this, but it may interest\n> > some other people. Let\'s consider the Schwarzschild solution.\n> > In Newtonian theory, the metric is:\n> >\n> > ds^2 = dt^2 + dr^2 + r^2 dw^2\n> >\n> > In Einsteinian theory, the metric is:\n> >\n> > ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2\n> >\n> > Now, let us speculate a unification of quantum mechanics and general\n> > relativity that produced the following metric:\n> >\n> > ds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2\n>\n> I\'m sure many people have noticed that those first two terms in the\n> Schwarzschild metric appear to be actually first order terms in a\n> Taylor expansion. I know I have. The main problem is that when you\n> go to the equivalent exponential metric form, the metric no longer\n> follows from a vanishing Ricci tensor, as it does in the original\n> analysis. While it appears more elegent with the exponentials, the\n> standard GR analysis would appear to require a very special\n> distribution of mass M to accomodate it.\n>\n> I think the only way out would be an appropriate modification of the\n> Einstein equations to allow this to be derivable from a vanishing\n> Ricci. The main problem, as I see it, is that any modifications\n> would have to be at the level of derivation of Riemann curvature.\n> Unfortunately, this modified geometry would no longer be Riemannian.\n\nI recall reading a book on general relativity for people without\ncalculus, about twenty years ago. Essentially, the authors came up\nwith difference equations which approximated the differential equations\nof GR, and used them to demonstrate the behave of GR.\n\nThey were very clearly that these difference equations were only\nintended to be approximations, for teaching purposes, not a credible\nalternative to GR.\n\nHowever, in their approximation, the metric for a black hole did have\nthe exponential form, tending to the Schwarzchild metric as the step\nsize tended to zero.\n\nThere was a footnote to the effect that, this is only intended to be an\napproximation, and there\'s no evidence otherwise but maybe, just maybe\n....\n\nHas there been any serious investigation of difference equations, on\na lattice or a continuum, as possible classical alternatives to GR?\nWhat are the biggest problems with such a theory?\n\n\n--\nMatter is fundamentally lazy:- It always takes the path of least effort\nMatter is fundamentally stupid:- It tries every other path first.\nThat is the heart of physics - The rest is details.- Robert Shaw\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Igor" <thoovler@excite.com> wrote
>...
> > I think that I know the answer to this, but it may interest
> > some other people. Let's consider the Schwarzschild solution.
> > In Newtonian theory, the metric is:
> >
> > ds^2 = dt^2 + dr^2 + r^2 dw^2
> >
> > In Einsteinian theory, the metric is:
> >
> > ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2
> >
> > Now, let us speculate a unification of quantum mechanics and general
> > relativity that produced the following metric:
> >
> > ds^2 = \exp(-2GM/r) dt^2 + \exp(2GM/r) dr^2 + r^2 dw^2
>
> I'm sure many people have noticed that those first two terms in the
> Schwarzschild metric appear to be actually first order terms in a
> Taylor expansion. I know I have. The main problem is that when you
> go to the equivalent exponential metric form, the metric no longer
> follows from a vanishing Ricci tensor, as it does in the original
> analysis. While it appears more elegent with the exponentials, the
> standard GR analysis would appear to require a very special
> distribution of mass M to accomodate it.
>
> I think the only way out would be an appropriate modification of the
> Einstein equations to allow this to be derivable from a vanishing
> Ricci. The main problem, as I see it, is that any modifications
> would have to be at the level of derivation of Riemann curvature.
> Unfortunately, this modified geometry would no longer be Riemannian.
I recall reading a book on general relativity for people without
calculus, about twenty years ago. Essentially, the authors came up
with difference equations which approximated the differential equations
of GR, and used them to demonstrate the behave of GR.
They were very clearly that these difference equations were only
intended to be approximations, for teaching purposes, not a credible
alternative to GR.
However, in their approximation, the metric for a black hole did have
the exponential form, tending to the Schwarzchild metric as the step
size tended to zero.
There was a footnote to the effect that, this is only intended to be an
approximation, and there's no evidence otherwise but maybe, just maybe
....
Has there been any serious investigation of difference equations, on
a lattice or a continuum, as possible classical alternatives to GR?
What are the biggest problems with such a theory?
--
Matter is fundamentally lazy:- It always takes the path of least effort
Matter is fundamentally stupid:- It tries every other path first.
That is the heart of physics - The rest is details.- Robert Shaw
jgraber
Aug12-04, 08:29 AM
<div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <ceug0j\\$f76\\$1@woodrow.ucdavis.edu>,\ncarlip@no-physics-spam.ucdavis.edu writes:\n|>\n|> For a general relativistic black hole, this doesn\'t matter so\n|> much, because the transition takes place inside the horizon, and\n|> is unobservable to those outside. Your metric, on the other\n|> hand, has no horizon, so the "surface" of the collapsed object\n|> is not invisible. This means that infalling matter should hit\n|> this surface.\n\nNow THAT has the potential to be a good test. It is pretty unlikely\nthat ANY "black hole free" theory does not imply that there will be\na more-or-less definite surface to objects. And the effects of\nimpacts on such things are very different from those where there is\nno such discontiguity.\n\nAre the observational effects clear enough to show the difference?\nMost of the papers I have seen have built their deductions on a\nsufficiently high and ricketty tower of hypotheses that they can\'t\nbe regarded as solid. Probable, yes, but that is different.\n\n\nRegards,\nNick Maclaren.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P>In article <ceug0j$f76$1@woodrow.ucdavis.edu>,
carlip@no-physics-spam.ucdavis.edu writes:
|>
|> For a general relativistic black hole, this doesn't matter so
|> much, because the transition takes place inside the horizon, and
|> is unobservable to those outside. Your metric, on the other
|> hand, has no horizon, so the "surface" of the collapsed object
|> is not invisible. This means that infalling matter should hit
|> this surface.
Now THAT has the potential to be a good test. It is pretty unlikely
that ANY "black hole free" theory does not imply that there will be
a more-or-less definite surface to objects. And the effects of
impacts on such things are very different from those where there is
no such discontiguity.
Are the observational effects clear enough to show the difference?
Most of the papers I have seen have built their deductions on a
sufficiently high and ricketty tower of hypotheses that they can't
be regarded as solid. Probable, yes, but that is different.
Regards,
Nick Maclaren.
Hi Nick, Steve, Doug and T. Essel,
As T. Essel noted, there is a difference between the metric suggested by Nick McLaren and the well studied exponential metric looked at by many authors from Yilmaz to Watt and Misner. A more recent paper by Nandi, Zhang and Kumar (arxix:gr-qc/o407032) points out that this exponential metric actually describes a wormhole.
Anyway, a wormhole, just like a blackhole, lacks a hard surface, and hence would pass the tests due to Narayan and others mentioned by Steve Carlip.
So also would a redhole, which you can think of as a wormhole connecting to a static baby universe.
However, even a body possessing a hard surface, i.e. a super small neutron star like object could pass these tests if it were compact enough that the redshift at its surface exceeds a factor of about 100. This would both attenuate and redshift the emerging radiation sufficiently to pass the tests. For this to work, the infalling energy must thermalize or at least scatter on the surface of the supercompact object. (If the surface acts as a perfect mirror, this does not work.) Also it is important that radiation not directed almost exactly vertically can be trapped by the compact object. (The compact object acts almost, but not quite like a black hole, which is how it can pass the tests.) A highly artificial metric which has these properties is a Schwarzschild metric cut off arbitrarily close to the event horizon.
So it is necessary to determine the exact redshift of the hard surface in Nick McLaren’s metric to see if it would pass the Narayan tests or not. This probably depends on the properties of superdense matter, as well as the metric, i.e. the properties of the implied alternate theory of gravitation.
By the way, does anyone know if Nick’s particular metric has been studied before? I am not currently aware of any references. But it does resemble both the Schwarzschild and the exponential metrics, so perhaps it has been looked at before.
Best to all,
Jim Graber
tessel@tum.bot
Aug12-04, 08:29 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sat, 7 Aug 2004, Doug Sweetser wrote:\n\n> > This is quite irrespective from the not inconsiderable issue of\n> > proposing a gravitation theory which agrees with available evidence\n> > and which doesn\'t violate too many cherished theoretical principles.\n>\n> What theoretical principle would people in SPR be willing to give up?\n\nAny principle, of course--- as long as it seems worthwhile!\n\n> I think there is consensus that we will not give up the rule that\n> experimental data trumps theory.\n\nI trust you don\'t mean you think I was proposing any such thing.\n\n"T. Essel" (hiding somewhere in cyberspace)\n^\none d\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sat, 7 Aug 2004, Doug Sweetser wrote:
> > This is quite irrespective from the not inconsiderable issue of
> > proposing a gravitation theory which agrees with available evidence
> > and which doesn't violate too many cherished theoretical principles.
>
> What theoretical principle would people in SPR be willing to give up?
Any principle, of course--- as long as it seems worthwhile!
> I think there is consensus that we will not give up the rule that
> experimental data trumps theory.
I trust you don't mean you think I was proposing any such thing.
"T. Essel" (hiding somewhere in cyberspace)
^
one d
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nNick Maclaren <nmm1@cus.cam.ac.uk> writes\n>I think that I know the answer to this, but it may interest\n>some other people. Let\'s consider the Schwarzschild solution.\n>In Newtonian theory, the metric is:\n>\n> ds^2 = dt^2 + dr^2 + r^2 dw^2\n>\n>In Einsteinian theory, the metric is:\n>\n> ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2\n>\n>Now, let us speculate a unification of quantum mechanics and general\n>relativity that produced the following metric:\n>\n> ds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2\n>\n>I am, of course, not saying that there is a scrap of evidence for\n>such a theory. But let us assume one, and assume that it makes\n>similar changes to other solutions of Einstein\'s equations.\n>\n>My question is whether we have any CURRENT data that would enable\n>us to distinguish these?\n\nData from Oz? - hell no...\n\nHowever I do wonder what effect this will have on SR.\n\nThere is that slightly niggly conundrum about why highly boosted\nobservers don\'t get to see black holes form in perfectly ordinary (in\ntheir rest frame) particles. It strikes me that there are three\nalternatives (probably a fourth which is the standard explanation that I\nhave forgotten).\n\n1) Lorentz is not exact at high boosts.\n2) GR is not exact at extreme curvatures.\n3) QM effects come into play.\n\nOr a combination, of course.\n\nIn passing I don\'t entirely think simple neat mathematical expressions\nare on the universes agenda, although I think simple processes are. The\ntrouble is that simple mutually interacting non-linear processes have a\ntendency to be mathematically \'interesting\' quite quickly. GR and even\nSR are very significantly more complex than newton, although the basic\nprocess is very simple in both cases. Handling any (should it even\nexist) more \'accurate\' model is likely to be mathematically \'pretty\nchallenging\'.\n\nAlso, in passing, we have two simplified formulations that are very\nsimilar in form: EM and newton. Newton has been extended to GR, whilst\nEM remains as it was in maxwell\'s time. I\'m faintly surprised that\nnobody has considered treating EM in an analogous way to SR_.GR. It\nwould have little practical significance, I admit, due to the non-\naccumulation of same charges, but .....\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Nick Maclaren <nmm1@cus.cam.ac.uk> writes
>I think that I know the answer to this, but it may interest
>some other people. Let's consider the Schwarzschild solution.
>In Newtonian theory, the metric is:
>
> ds^2 = dt^2 + dr^2 + r^2 dw^2
>
>In Einsteinian theory, the metric is:
>
> ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2
>
>Now, let us speculate a unification of quantum mechanics and general
>relativity that produced the following metric:
>
> ds^2 = \exp(-2GM/r) dt^2 + \exp(2GM/r) dr^2 + r^2 dw^2
>
>I am, of course, not saying that there is a scrap of evidence for
>such a theory. But let us assume one, and assume that it makes
>similar changes to other solutions of Einstein's equations.
>
>My question is whether we have any CURRENT data that would enable
>us to distinguish these?
Data from Oz? - hell no...
However I do wonder what effect this will have on SR.
There is that slightly niggly conundrum about why highly boosted
observers don't get to see black holes form in perfectly ordinary (in
their rest frame) particles. It strikes me that there are three
alternatives (probably a fourth which is the standard explanation that I
have forgotten).
1) Lorentz is not exact at high boosts.
2) GR is not exact at extreme curvatures.
3) QM effects come into play.
Or a combination, of course.
In passing I don't entirely think simple neat mathematical expressions
are on the universes agenda, although I think simple processes are. The
trouble is that simple mutually interacting non-linear processes have a
tendency to be mathematically 'interesting' quite quickly. GR and even
SR are very significantly more complex than newton, although the basic
process is very simple in both cases. Handling any (should it even
exist) more 'accurate' model is likely to be mathematically 'pretty
challenging'.
Also, in passing, we have two simplified formulations that are very
similar in form: EM and newton. Newton has been extended to GR, whilst
EM remains as it was in maxwell's time. I'm faintly surprised that
nobody has considered treating EM in an analogous way to SR_.GR. It
would have little practical significance, I admit, due to the non-
accumulation of same charges, but .....
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
carlip@no-physics-spam.ucdavis.edu
Aug12-04, 12:29 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\njgraber <jgra@loc.gov> wrote:\n\n[...]\n> However, even a body possessing a hard surface, i.e. a super small\n> neutron star like object could pass these tests if it were compact\n> enough that the redshift at its surface exceeds a factor of about 100.\n> This would both attenuate and redshift the emerging radiation\n> sufficiently to pass the tests.\n\nThis may be true, but it\'s not obvious. First of all, while outgoing\nradiation will certainly be red-shifted, this shouldn\'t affect the\nenergy balance, since the red shift should essentially compensate for\nthe blue shift of the matter falling in. If you start with matter\nfairly far from the surface and look at what comes out at that same\ndistance, the net effect should be simply that a considerable fraction\nof the mass has been converted to energy, with no net red shift. You\nmight, I suppose, argue that the radiation is shifted to a part of\nthe spectrum where we haven\'t looked or can\'t see very well; I\'d want\nto see some detail, though.\n\nAs for attenuation, I\'m guessing that your argument is that a short\nburst will be stretched out by time dilation, reducing the power.\nThat might make sense, but again, I\'d like to see details. I\'m not\nan expert, but glancing at a few references, it seems that Type I\nX-ray bursts have rise times of about a second and peak counts in the\nthousands per second. Is the technology really bad enough that these\nwouldn\'t be seen if stretched out by a factot of 100? (Again, I guess\nthis might depend on where in the spectrum they\'re red-shifted to. Do\nyou know of a careful analysis?)\n\nSteve Carlip\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>jgraber <jgra@loc.gov> wrote:
[...]
> However, even a body possessing a hard surface, i.e. a super small
> neutron star like object could pass these tests if it were compact
> enough that the redshift at its surface exceeds a factor of about 100.
> This would both attenuate and redshift the emerging radiation
> sufficiently to pass the tests.
This may be true, but it's not obvious. First of all, while outgoing
radiation will certainly be red-shifted, this shouldn't affect the
energy balance, since the red shift should essentially compensate for
the blue shift of the matter falling in. If you start with matter
fairly far from the surface and look at what comes out at that same
distance, the net effect should be simply that a considerable fraction
of the mass has been converted to energy, with no net red shift. You
might, I suppose, argue that the radiation is shifted to a part of
the spectrum where we haven't looked or can't see very well; I'd want
to see some detail, though.
As for attenuation, I'm guessing that your argument is that a short
burst will be stretched out by time dilation, reducing the power.
That might make sense, but again, I'd like to see details. I'm not
an expert, but glancing at a few references, it seems that Type I
X-ray bursts have rise times of about a second and peak counts in the
thousands per second. Is the technology really bad enough that these
wouldn't be seen if stretched out by a factot of 100? (Again, I guess
this might depend on where in the spectrum they're red-shifted to. Do
you know of a careful analysis?)
Steve Carlip
tessel@tum.bot
Aug13-04, 05:41 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 12 Aug 2004, jgraber wrote:\n\n> Nick Maclaren Wrote:\n>\n> > <div class="vbmenu_control"><a href="jabberwocky:;"\n\n[snip further junk]\n\nA simple request (one which, in happier days, the moderators used to make\nin cases like this):\n\n1. please do not quote massive amounts of irrelevant text; take the time\nto trim out the irrelevant stuff,\n\n2. please avoid html code.\n\nThese are among the basic ground rules mentioned in the welcome message.\nPlease, everyone, let\'s all make an effort to observe these basic ground\nrules, and whenever possible, to improve quality of submitted posts by\ntaking the time to organize your thoughts, etc.\n\n> A more recent paper by Nandi, Zhang and Kumar (arxix:gr-qc/o407032)\n^^^^^^^^^^^^^^^^^^^\n????\n\n> points out that this exponential metric actually describes a wormhole.\n\nCitations are good, but they are worthless if you fail to exercise simple\nproofreading skills, with the result (as happened here) that the\n"citation" is worthless! As a matter of fact, whenever possible it is a\ngood idea to give some redundancy in your citations, e.g.\n\nauthor = {Ramesh Narayan},\ntitle = {Evidence for the Black Hole Event Horizon},\nnote = {astro-ph/0310692}}\n\nsince this may enable a reader to find the paper in question even if you\nmade a mistake in giving the citation.\n\nThanks!\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 12 Aug 2004, jgraber wrote:
> Nick Maclaren Wrote:
>
> > <div class="vbmenu_control"><a href="jabberwocky:;"
[snip further junk]
A simple request (one which, in happier days, the moderators used to make
in cases like this):
1. please do not quote massive amounts of irrelevant text; take the time
to trim out the irrelevant stuff,
2. please avoid html code.
These are among the basic ground rules mentioned in the welcome message.
Please, everyone, let's all make an effort to observe these basic ground
rules, and whenever possible, to improve quality of submitted posts by
taking the time to organize your thoughts, etc.
> A more recent paper by Nandi, Zhang and Kumar (arxix:gr-qc/o407032)
^^^^^^^^^^^^^^^^^^^
????
> points out that this exponential metric actually describes a wormhole.
Citations are good, but they are worthless if you fail to exercise simple
proofreading skills, with the result (as happened here) that the
"citation" is worthless! As a matter of fact, whenever possible it is a
good idea to give some redundancy in your citations, e.g.
author = {Ramesh Narayan},
title = {Evidence for the Black Hole Event Horizon},
note = {http://www.arxiv.org/abs/astro-ph/0310692}}
since this may enable a reader to find the paper in question even if you
made a mistake in giving the citation.
Thanks!
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Aug13-04, 05:41 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 12 Aug 2004, Oz wrote:\n\n> There is that slightly niggly conundrum about why highly boosted\n> observers don\'t get to see black holes form in perfectly ordinary (in\n> their rest frame) particles. It strikes me that there are three\n> alternatives (probably a fourth which is the standard explanation that I\n> have forgotten).\n>\n> 1) Lorentz is not exact at high boosts.\n> 2) GR is not exact at extreme curvatures.\n> 3) QM effects come into play.\n>\n> Or a combination, of course.\n\nWe have very recently discussed this very point (and not for the first\ntime, or even the second time!) in this forum. Maybe I misunderstood you,\nOz, but there is no mystery whatever regarding what ultrarelativistic\nobservers experience in a static spherically symmetric gravitational\nfield* -according to gtr-. They experience a certain "distribution\nvalued" vacuum solution to the EFE, a simple pp wave called the\nAichelburg-Sexl solution, which you should think of as "Schwarzschild in\ndisguise".\n\n[*Think of our "observer" as a test particle moving very rapidly past an\nordinary nonrotating spherically symmetric object, i.e. an idealized\nstar.]\n\nThis has -nothing whatever- to do with any "quantum effects" or with\npossible failures of Lorentz symmetry, still less with possible failures\nof gtr--- this is purely classical relativistic field theory in the\ncontext of standard gtr!\n\n(Well, "almost standard", if you like, since to write down the\nAichelburg-Sexl "generalized pp wave", you need to be willing to write a\nDirac delta in the metric. But physicists are very accustomed to such\n"generalized functions" or "distributions" [in the sense of Schwartz], and\nmathematicians have made their theory sufficiently rigorous to handle the\nkind of simple-minded "distribution valued" spacetimes which arise in this\ncontext.)\n\nYou may also find the FAQ helpful:\n\nhttp://www2.corepower.com:8080/~relfaq/black_fast.html\n\n> In passing I don\'t entirely think simple neat mathematical expressions\n> are on the universes agenda, although I think simple processes are.\n\nSome of us would claim that neat mathematical expressions provide some of\nthe best examples of "simple processes".\n\n> The trouble is that simple mutually interacting non-linear processes\n> have a tendency to be mathematically \'interesting\' quite quickly. GR and\n> even SR are very significantly more complex than newton, although the\n> basic process is very simple in both cases. Handling any (should it even\n> exist) more \'accurate\' model is likely to be mathematically \'pretty\n> challenging\'.\n\nThe derivation of the AS solution is fairly straightforward; see\n\nauthor = {Valeri P. Frolov and Igor D. Novikov},\ntitle = {Black Hole Physics: Basic Concepts and New Developments},\npublisher = {Kluwer},\nseries = {Fundamental Theories of Physics},\nvolume = 96,\nyear = 1998}\n\nThere are also dozens of papers in the ArXiV, such as\n\nauthor = {Aichelburg, Peter C. and Belasin, Herbert},\ntitle = {Generalized asymptotic structure of the ultrarelativistic\n{S}chwarzschild black hole},\njournal = {Class. Quant. Grav.},\nvolume = 17,\nyear = 2000,\npages = {3645--3662},\nnote = {gr-qc/9912122}}\n\n>From these, an advanced reader will see that surmountable difficulties do\narise in working with ultrarelativistic boosts of more general solutions,\ne.g. Kerr-Newman and general Ernst-Maxwell electrovacuums (the general\nstationary axisymmetric asympotically flat electrovacuum solution of the\nEFE). But the original AS solution (ultrarelativistic boost of\nSchwarzschild vacuum) is fairly easy to obtain.\n\n> Also, in passing, we have two simplified formulations that are very\n> similar in form: EM and newton. Newton has been extended to GR, whilst\n> EM remains as it was in maxwell\'s time. I\'m faintly surprised that\n> nobody has considered treating EM in an analogous way to SR_.GR.\n\nI am not sure I follow you here, but remember that Newtonian gravity is\nnonrelativistic (the "field equation" can be taken to be the\nthree-dimensional Laplace equation, whose solutions are the so-called\n"harmonic functions"), whereas Maxwell\'s theory of EM is the sine qua non\nof a classical relativistic field theory. So, we should not be too\nsurprised (although we can be grateful, I guess!) that it requires little\nmodification to treat EM in the context of gtr.\n\nOz, are you still interested in continuing the soliton thread?\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 12 Aug 2004, Oz wrote:
> There is that slightly niggly conundrum about why highly boosted
> observers don't get to see black holes form in perfectly ordinary (in
> their rest frame) particles. It strikes me that there are three
> alternatives (probably a fourth which is the standard explanation that I
> have forgotten).
>
> 1) Lorentz is not exact at high boosts.
> 2) GR is not exact at extreme curvatures.
> 3) QM effects come into play.
>
> Or a combination, of course.
We have very recently discussed this very point (and not for the first
time, or even the second time!) in this forum. Maybe I misunderstood you,
Oz, but there is no mystery whatever regarding what ultrarelativistic
observers experience in a static spherically symmetric gravitational
field* -according to gtr-. They experience a certain "distribution
valued" vacuum solution to the EFE, a simple pp wave called the
Aichelburg-Sexl solution, which you should think of as "Schwarzschild in
disguise".
[*Think of our "observer" as a test particle moving very rapidly past an
ordinary nonrotating spherically symmetric object, i.e. an idealized
star.]
This has -nothing whatever- to do with any "quantum effects" or with
possible failures of Lorentz symmetry, still less with possible failures
of gtr--- this is purely classical relativistic field theory in the
context of standard gtr!
(Well, "almost standard", if you like, since to write down the
Aichelburg-Sexl "generalized pp wave", you need to be willing to write a
Dirac \delta in the metric. But physicists are very accustomed to such
"generalized functions" or "distributions" [in the sense of Schwartz], and
mathematicians have made their theory sufficiently rigorous to handle the
kind of simple-minded "distribution valued" spacetimes which arise in this
context.)
You may also find the FAQ helpful:
http://www2.corepower.com:8080/~relfaq/black_fast.html
> In passing I don't entirely think simple neat mathematical expressions
> are on the universes agenda, although I think simple processes are.
Some of us would claim that neat mathematical expressions provide some of
the best examples of "simple processes".
> The trouble is that simple mutually interacting non-linear processes
> have a tendency to be mathematically 'interesting' quite quickly. GR and
> even SR are very significantly more complex than newton, although the
> basic process is very simple in both cases. Handling any (should it even
> exist) more 'accurate' model is likely to be mathematically 'pretty
> challenging'.
The derivation of the AS solution is fairly straightforward; see
author = {Valeri P. Frolov and Igor D. Novikov},
title = {Black Hole Physics: Basic Concepts and New Developments},
publisher = {Kluwer},
series = {Fundamental Theories of Physics},
volume = 96,
year = 1998}
There are also dozens of papers in the ArXiV, such as
author = {Aichelburg, Peter C. and Belasin, Herbert},
title = {Generalized asymptotic structure of the ultrarelativistic
{S}chwarzschild black hole},
journal = {Class. Quant. Grav.},
volume = 17,
year = 2000,
pages = {3645--3662},
note = {http://www.arxiv.org/abs/gr-qc/9912122}}
>From these, an advanced reader will see that surmountable difficulties do
arise in working with ultrarelativistic boosts of more general solutions,
e.g. Kerr-Newman and general Ernst-Maxwell electrovacuums (the general
stationary axisymmetric asympotically flat electrovacuum solution of the
EFE). But the original AS solution (ultrarelativistic boost of
Schwarzschild vacuum) is fairly easy to obtain.
> Also, in passing, we have two simplified formulations that are very
> similar in form: EM and newton. Newton has been extended to GR, whilst
> EM remains as it was in maxwell's time. I'm faintly surprised that
> nobody has considered treating EM in an analogous way to SR_.GR.
I am not sure I follow you here, but remember that Newtonian gravity is
nonrelativistic (the "field equation" can be taken to be the
three-dimensional Laplace equation, whose solutions are the so-called
"harmonic functions"), whereas Maxwell's theory of EM is the sine qua non
of a classical relativistic field theory. So, we should not be too
surprised (although we can be grateful, I guess!) that it requires little
modification to treat EM in the context of gtr.
Oz, are you still interested in continuing the soliton thread?
"T. Essel" (hiding somewhere in cyberspace)
Kefka G
Aug13-04, 05:42 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOz writes:\n\n>\n>Data from Oz? - hell no...\n>\n>However I do wonder what effect this will have on SR.\n>\n>There is that slightly niggly conundrum about why highly boosted\n>observers don\'t get to see black holes form in perfectly ordinary (in\n>their rest frame) particles. It strikes me that there are three\n>alternatives (probably a fourth which is the standard explanation that I\n>have forgotten).\n>\n>1) Lorentz is not exact at high boosts.\n>2) GR is not exact at extreme curvatures.\n>3) QM effects come into play.\n>\n>Or a combination, of course.\n\nWe actually talked about this recently on another thread in the context of\nphotons (short version: a black hole can form from two photons, but never from\none). Actually you don\'t need 2) or 3) (or 1), really, although I\'ll grant you\nthat there is no such thing as a global Lorentz boost in Schwarzschild/Kerr\nspacetime). Recall that a black hole is generally defined by the fact that\nnull geodesics cannot propagate from the interior to infinity. Also recall\nthat this is a coordinate invariant statement - even if we move to some nasty\nnasty frame which is moving very fast relative to the black hole and all skewed\nand stuff, a photon can either get from A (the particle) to B (spatial\ninfinity) or not. Equivalently, we can always conformally rescale the\nspacetime to get the usual conformal diagram of Schwarzschild (or Kerr)\nspacetime. This means, in a vague, loose way, that the frame-dependent kinetic\nenergy of a (set of) particles is not what\'s responsible for gravitation. It\nis only the "rest energy" which has this effect. Note, however, that TWO\nparticles travelling in opposite directions have a well defined energy in the\nCM frame, no matter what speed they move at. This most definitely will\ncontribute to gravitation, and may result in a black hole being formed.\n\nThe only thing that bothers me is that it would appear, then, that a lonely\nphoton (or other massless particle) in empty spacetime cannot gravitate at all,\nbecause if it did, the resulting curvature would have to be a function of its\nenergy. Then there is some frame where it would have enough energy to create a\nblack hole. Or perhaps a photon creates Weyl curvature instead of Ricci, and\nthis somehow allows us to escape that conclusion (can\'t see quite how this\nwould happen, though)? I should just go back and work this out myself using\nthe EFE, but maybe someone knows the answer off the top of their head...\n\n>\n>In passing I don\'t entirely think simple neat mathematical expressions\n>are on the universes agenda, although I think simple processes are. The\n>trouble is that simple mutually interacting non-linear processes have a\n>tendency to be mathematically \'interesting\' quite quickly. GR and even\n>SR are very significantly more complex than newton, although the basic\n>process is very simple in both cases. Handling any (should it even\n>exist) more \'accurate\' model is likely to be mathematically \'pretty\n>challenging\'.\n>\n>Also, in passing, we have two simplified formulations that are very\n>similar in form: EM and newton. Newton has been extended to GR, whilst\n>EM remains as it was in maxwell\'s time. I\'m faintly surprised that\n>nobody has considered treating EM in an analogous way to SR_.GR. It\n>would have little practical significance, I admit, due to the non-\n>accumulation of same charges, but .....\n\nWell, if you want to pull EM into GR, there\'s always Kaluza-Klein theory, where\nwe add an extra periodic spatial dimension to 4-D GR and find that this exactly\ngives us EM gauge invariance if we "factor it away" (Kaluza\'s cylinder\ncondition). It\'s not too satisfying, though, for reasons too detailed to go\ninto here (there\'s a paper at Living Reviews which analyzes a lot of this stuff\nin detail). Maybe you\'re thinking about something else...\n\n-Eric\n\n\n\n>\n>--\n>Oz\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Oz writes:
>
>Data from Oz? - hell no...
>
>However I do wonder what effect this will have on SR.
>
>There is that slightly niggly conundrum about why highly boosted
>observers don't get to see black holes form in perfectly ordinary (in
>their rest frame) particles. It strikes me that there are three
>alternatives (probably a fourth which is the standard explanation that I
>have forgotten).
>
>1) Lorentz is not exact at high boosts.
>2) GR is not exact at extreme curvatures.
>3) QM effects come into play.
>
>Or a combination, of course.
We actually talked about this recently on another thread in the context of
photons (short version: a black hole can form from two photons, but never from
one). Actually you don't need 2) or 3) (or 1), really, although I'll grant you
that there is no such thing as a global Lorentz boost in Schwarzschild/Kerr
spacetime). Recall that a black hole is generally defined by the fact that
null geodesics cannot propagate from the interior to infinity. Also recall
that this is a coordinate invariant statement - even if we move to some nasty
nasty frame which is moving very fast relative to the black hole and all skewed
and stuff, a photon can either get from A (the particle) to B (spatial
infinity) or not. Equivalently, we can always conformally rescale the
spacetime to get the usual conformal diagram of Schwarzschild (or Kerr)
spacetime. This means, in a vague, loose way, that the frame-dependent kinetic
energy of a (set of) particles is not what's responsible for gravitation. It
is only the "rest energy" which has this effect. Note, however, that TWO
particles travelling in opposite directions have a well defined energy in the
CM frame, no matter what speed they move at. This most definitely will
contribute to gravitation, and may result in a black hole being formed.
The only thing that bothers me is that it would appear, then, that a lonely
photon (or other massless particle) in empty spacetime cannot gravitate at all,
because if it did, the resulting curvature would have to be a function of its
energy. Then there is some frame where it would have enough energy to create a
black hole. Or perhaps a photon creates Weyl curvature instead of Ricci, and
this somehow allows us to escape that conclusion (can't see quite how this
would happen, though)? I should just go back and work this out myself using
the EFE, but maybe someone knows the answer off the top of their head...
>
>In passing I don't entirely think simple neat mathematical expressions
>are on the universes agenda, although I think simple processes are. The
>trouble is that simple mutually interacting non-linear processes have a
>tendency to be mathematically 'interesting' quite quickly. GR and even
>SR are very significantly more complex than newton, although the basic
>process is very simple in both cases. Handling any (should it even
>exist) more 'accurate' model is likely to be mathematically 'pretty
>challenging'.
>
>Also, in passing, we have two simplified formulations that are very
>similar in form: EM and newton. Newton has been extended to GR, whilst
>EM remains as it was in maxwell's time. I'm faintly surprised that
>nobody has considered treating EM in an analogous way to SR_.GR. It
>would have little practical significance, I admit, due to the non-
>accumulation of same charges, but .....
Well, if you want to pull EM into GR, there's always Kaluza-Klein theory, where
we add an extra periodic spatial dimension to 4-D GR and find that this exactly
gives us EM gauge invariance if we "factor it away" (Kaluza's cylinder
condition). It's not too satisfying, though, for reasons too detailed to go
into here (there's a paper at Living Reviews which analyzes a lot of this stuff
in detail). Maybe you're thinking about something else...
-Eric
>
>--
>Oz
tessel@tum.bot
Aug13-04, 02:08 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Fri, 13 Aug 2004, Kefka G wrote:\n\n> We actually talked about this recently on another thread in the context\n> of photons (short version: a black hole can form from two photons, but\n> never from one). Actually you don\'t need 2) or 3) (or 1), really,\n> although I\'ll grant you that there is no such thing as a global Lorentz\n> boost in Schwarzschild/Kerr spacetime). Recall that a black hole is\n> generally defined by the fact that null geodesics cannot propagate from\n> the interior to infinity. Also recall that this is a coordinate\n> invariant statement - even if we move to some nasty nasty frame which is\n> moving very fast relative to the black hole and all skewed and stuff,\n\nThis gives the Aichelburg-Sexl ultrarelativistic boost, which is\nessentially a limiting case of a "Gaussian pulse" vacuum pp wave.\n\n> a photon can either get from A (the particle) to B (spatial infinity) or\n> not. Equivalently, we can always conformally rescale the spacetime to\n> get the usual conformal diagram of Schwarzschild (or Kerr) spacetime.\n\n[snip stuff I agree with, except for what I regard as an obligatory\ncomment: "Since gtr is a -classical- relativistic field theory, instead of\n`photons`--- a quantum concept--- let us discuss `Gaussian pulses of\nincoherent EM radiation\'--- which is a purely classical concept", and\nexcept for my strong preference to distinguish conceptually and\nterminologically between -frames- and -charts-.]\n\n> This means, in a vague, loose way, that the frame-dependent kinetic\n> energy of a (set of) particles is not what\'s responsible for\n> gravitation. It is only the "rest energy" which has this effect.\n\nI don\'t think this is a good way to think about it. In gtr, the source of\nthe gravitational field (curvature tensor) is the stress-momentum-energy\ntensor T^(ab). See John Baez\'s exposition of the EFE, for example (you\ncan find it on the ArXiV or on his home page).\n\n> Note, however, that TWO particles travelling in opposite directions have\n> a well defined energy in the CM frame,\n> no matter what speed they move at.\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\nAny speed less than the speed of light?\n\n> Or perhaps a photon creates Weyl curvature instead of Ricci, and this\n> somehow allows us to escape that conclusion (can\'t see quite how this\n> would happen, though)? I should just go back and work this out myself\n> using the EFE, but maybe someone knows the answer off the top of their\n> head...\n\nTwo simple examples of exact solutions of the EFE should be helpful here.\n\nThe well known "Bonnor beam" is an exact time independent "null dust"\nsolution, which consists of an "interior solution", representing a steady\nstate "light beam" as a solid cylindrical tube of incoherent EM radiation,\nwhich is matched to an "exterior solution", which is a vacuum solution\nrepresenting the gravitational field outside the beam.\n\nThe interior region is locally isometric to the type SG17 plane wave:\n\nds^2 = -8 pi m R^2 dU^2 - 2 dU dV + dR^2 + R^2 dTheta^2,\n\n-infty < U,V < infty, 0 < R < R0, -pi < Theta < Pi\n\nThis region happens to be -conformally flat-, so we have vanishing Weyl\ncurvature but nonzero Ricci curvature R^(ab) = G^(ab). IOW, this region\ncontains a -homogeneous- null dust; indeed, wrt an appropriate coframe (an\northonormal basis of covector fields or one-forms, dual to an orthonormal\nbasis of vector fields), such as\n\no^1 = (dU + dV)/sqrt(2) + 4 pi m R^2/sqrt(2) dU\n\no^2 = (-dV + dU)/sqrt(2) + 4 pi m R^2/sqrt(2) dU\n\no^3 = dR\n\no^4 = R dTheta\n\nthe Einstein tensor is just\n\n[ 1 1 0 0 ]\nG^(ab) = 8 pi m [ 1 1 0 0 ]\n[ 0 0 0 0 ]\n[ 0 0 0 0 ]\n\nThis exhibits the -uniform- EM field energy density m. The exterior\nregion is locally isometric to the type EK6 vacuum pp wave:\n\nds^2 = -8 Pi m R0^2 (1 + 2 log(R/R0)) dU^2\n\n-2 dU dV + dR^2 + R^2 dTheta^2,\n\n-infty < U,V < infty, R0 < R < infty, -pi < Theta < Pi\n\nSince this is a vacuum region, it has nonzero Weyl curvature but vanishing\nRicci curvature--- so we have a nice contrast with the interior region!\nThe gravitational acceleration of a test particle (measured in our frame)\nis of course radially inward--- toward the mass-energy concentration---\nand scales like 1/R; the curvature is transverse and scales like 1/R^2.\nThroughout the Bonnor beam spacetime, in the Bel decomposition of the\nRiemann tensor wrt our frame (this is fully analogous to the standard\ndecomposition of the EM field tensor, wrt some frame, into electric and\nmagnetic vectors) the spatial curvature tensors are transverse and have\nthe expected plane wave form. The two regions match across the surface R\n= R0 according to standard matching conditions. (In particular, the\ncurvature of course jumps here, but the metric is continuous across this\nsurface.)\n\nThe Bonnor pulse is similar, but models a Gaussian pulse of incoherent EM\nradiation instead of a steady-state light beam. The interior region is\nlocally isometric to the type SG2 null dust pp wave:\n\nds^2 = - 8 sqrt(pi) m/a R^2 exp(-U^2/a^2) dU^2\n\n-2 dU dV + dR^2 + R^2 dTheta^2,\n\n-infty < U,V < infty, 0 < R < R0, -pi < Theta < Pi\n\nIn an appropriate frame, the Einstein tensor is\n\n[ 1 1 0 0 ]\nG^(ab) = 8 sqrt(pi) m/a exp(-U^2/a^2) [ 1 1 0 0 ]\n[ 0 0 0 0 ]\n[ 0 0 0 0 ]\n\nshowing EM field energy density which is uniform in space, but has\nGaussian behavior wrt time. The exterior region is locally isometric to\nthe type EK4 vacuum pp wave:\n\nds^2 = - 8 sqrt(pi) m R0^2 exp(-U^2/a^2) (1 + 2 log(R/R0))/a dU^2\n\n- 2 dU dV + dR^2 + R^2 dTheta^2,\n\n-infty < U,V < infty, R0 < R < infinity, -pi < Theta < Pi\n\nIn an appropriate frame, the gravitational acceleration is radially inward\nand scales like m/a R0^2/R exp(-U^2/a^2); the curvature is transverse and\nscales like m/a^2 R0^2/R^2 exp(-U^2/a^2). I stress that this falls off\nlike 1/R^2. Again, throughout, in the Bel decomposition of the Riemann\ntensor wrt our frame,the spatial curvature tensors are transverse and have\nthe expected plane wave form.\n\nIn both cases, the exterior solutions are -vacuum- pp waves, i.e. exact\ngravitational wave solutions. In each region, in both cases, we are using\na "harmonic coordinate chart" in the "cylindrical coordinate" form\nappropriate for axial symmetry.\n\nNote too that a familiar limiting case of the line element given above for\nthe -vacuum- Gaussian pulse wave (a type EK4 vacuum pp wave) immediately\ngives the line element for the Aichelburg-Sexl vacuum spacetime (the\n"ultrarelativistic boost of the Schwarzschild vacuum"). This expression\ncontains a "Dirac delta function", so it is not quite a smooth spacetime.\nBTW, the perhaps surprising fact that a static vacuum spacetime yields,\nupon "ultraboosting", a curved spacetime analog of a gravitational plane\nwave (namely, a certain "vacuum pp wave"), represents a general phenomenon\n(this is sometimes called the "Penrose limit"; compare the rate at which\nthe curvature decays as R -> infty as measured by static versus\n"ultraboosted" observers).\n\nIf you use Cartan\'s formalism (curvature two-forms and all that), it is\nnot -terribly- tedious to verify the above claims "by hand". However, in\npast posts to this newsgroup, I have computed and extensively discussed\nthe geodesics, Killing vectors, expansion tensor, curvature tensors,\nconserved quantities, optical scalars, and various other interesting\nphysical/geometric properties of these examples, as well as giving\nalternative frames and alternative coordinate charts. I also explained\nthe Ehlers-Kundt classification of vacuum pp waves by their isometries\n(Killing vector fields) and the generalization by Sippel and Goenner to\ngeneral pp waves.\n\nA few print references:\n\nMaartens & Maharaj, Class. Quant. Grav. 8 (1991): 503--514\n\nSippel & Goenner, Gen. Rel. Grav. 18 (1986): 1229--1243\n\nBonnor: Comm. Math. Phys. 13 (1969): 163--174\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 13 Aug 2004, Kefka G wrote:
> We actually talked about this recently on another thread in the context
> of photons (short version: a black hole can form from two photons, but
> never from one). Actually you don't need 2) or 3) (or 1), really,
> although I'll grant you that there is no such thing as a global Lorentz
> boost in Schwarzschild/Kerr spacetime). Recall that a black hole is
> generally defined by the fact that null geodesics cannot propagate from
> the interior to infinity. Also recall that this is a coordinate
> invariant statement - even if we move to some nasty nasty frame which is
> moving very fast relative to the black hole and all skewed and stuff,
This gives the Aichelburg-Sexl ultrarelativistic boost, which is
essentially a limiting case of a "Gaussian pulse" vacuum pp wave.
> a photon can either get from A (the particle) to B (spatial infinity) or
> not. Equivalently, we can always conformally rescale the spacetime to
> get the usual conformal diagram of Schwarzschild (or Kerr) spacetime.
[snip stuff I agree with, except for what I regard as an obligatory
comment: "Since gtr is a -classical- relativistic field theory, instead of
`photons`--- a quantum concept--- let us discuss `Gaussian pulses of
incoherent EM radiation'--- which is a purely classical concept", and
except for my strong preference to distinguish conceptually and
terminologically between -frames- and -charts-.]
> This means, in a vague, loose way, that the frame-dependent kinetic
> energy of a (set of) particles is not what's responsible for
> gravitation. It is only the "rest energy" which has this effect.
I don't think this is a good way to think about it. In gtr, the source of
the gravitational field (curvature tensor) is the stress-momentum-energy
tensor T^(ab). See John Baez's exposition of the EFE, for example (you
can find it on the ArXiV or on his home page).
> Note, however, that TWO particles travelling in opposite directions have
> a well defined energy in the CM frame,
> no matter what speed they move at.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Any speed less than the speed of light?
> Or perhaps a photon creates Weyl curvature instead of Ricci, and this
> somehow allows us to escape that conclusion (can't see quite how this
> would happen, though)? I should just go back and work this out myself
> using the EFE, but maybe someone knows the answer off the top of their
> head...
Two simple examples of exact solutions of the EFE should be helpful here.
The well known "Bonnor beam" is an exact time independent "null dust"
solution, which consists of an "interior solution", representing a steady
state "light beam" as a solid cylindrical tube of incoherent EM radiation,
which is matched to an "exterior solution", which is a vacuum solution
representing the gravitational field outside the beam.
The interior region is locally isometric to the type SG17 plane wave:
ds^2 = -8 \pi m R^2 dU^2 - 2 dU dV + dR^2 + R^2 dTheta^2,-\infty <[/itex] U,V < \infty,< R < R0, -\pi < \Theta < \Pi
This region happens to be -conformally flat-, so we have vanishing Weyl
curvature but nonzero Ricci curvature R^(ab) = G^(ab). IOW, this region
contains a -homogeneous- null dust; indeed, wrt an appropriate coframe (an
orthonormal basis of covector fields or one-forms, dual to an orthonormal
basis of vector fields), such as
o^1 = (dU + dV)/\sqrt(2) + 4 \pi m R^2/\sqrt(2) dUo^2 = (-dV + dU)/\sqrt(2) + 4 \pi m R^2/\sqrt(2) dUo^3 = dR
o^4 = R dTheta
the Einstein tensor is just
[ 1 1 ]
G^(ab) = 8 \pi m [ 1 1 ]
[ ]
[ ]
This exhibits the -uniform- EM field energy density m. The exterior
region is locally isometric to the type EK6 vacuum pp wave:
ds^2 = -8 \Pi m R0^2 (1 + 2 log(R/R0)) dU^2-2 dU dV + dR^2 + R^2 dTheta^2,-\infty < U,V < \infty, R0 < R < \infty, -\pi < \Theta < \Pi
Since this is a vacuum region, it has nonzero Weyl curvature but vanishing
Ricci curvature--- so we have a nice contrast with the interior region!
The gravitational acceleration of a test particle (measured in our frame)
is of course radially inward--- toward the mass-energy concentration---
and scales like 1/R; the curvature is transverse and scales like 1/R^2.
Throughout the Bonnor beam spacetime, in the Bel decomposition of the
Riemann tensor wrt our frame (this is fully analogous to the standard
decomposition of the EM field tensor, wrt some frame, into electric and
magnetic vectors) the spatial curvature tensors are transverse and have
the expected plane wave form. The two regions match across the surface R
= R0 according to standard matching conditions. (In particular, the
curvature of course jumps here, but the metric is continuous across this
surface.)
The Bonnor pulse is similar, but models a Gaussian pulse of incoherent EM
radiation instead of a steady-state light beam. The interior region is
locally isometric to the type SG2 null dust pp wave:
ds^2 = - 8 \sqrt(\pi) m/a R^2 \exp(-U^2/a^2) dU^2-2 dU dV + dR^2 + R^2 dTheta^2,-\infty < U,V < \infty,< R < R0, -\pi < \Theta < \Pi
In an appropriate frame, the Einstein tensor is
[ 1 1 ]
G^(ab) = 8 \sqrt(\pi) m/a \exp(-U^2/a^2) [ 1 1 ]
[ ]
[ ]
showing EM field energy density which is uniform in space, but has
Gaussian behavior wrt time. The exterior region is locally isometric to
the type EK4 vacuum pp wave:
ds^2 = - 8 \sqrt(\pi) m R0^2 \exp(-U^2/a^2) (1 + 2 log(R/R0))/a dU^2- 2 dU dV + dR^2 + R^2 dTheta^2,-\infty < U,V < \infty, R0 < R < infinity, [itex]-\pi < \Theta < \Pi
In an appropriate frame, the gravitational acceleration is radially inward
and scales like m/a R0^2/R \exp(-U^2/a^2); the curvature is transverse and
scales like m/a^2 R0^2/R^2 \exp(-U^2/a^2). I stress that this falls off
like 1/R^2. Again, throughout, in the Bel decomposition of the Riemann
tensor wrt our frame,the spatial curvature tensors are transverse and have
the expected plane wave form.
In both cases, the exterior solutions are -vacuum- pp waves, i.e. exact
gravitational wave solutions. In each region, in both cases, we are using
a "harmonic coordinate chart" in the "cylindrical coordinate" form
appropriate for axial symmetry.
Note too that a familiar limiting case of the line element given above for
the -vacuum- Gaussian pulse wave (a type EK4 vacuum pp wave) immediately
gives the line element for the Aichelburg-Sexl vacuum spacetime (the
"ultrarelativistic boost of the Schwarzschild vacuum"). This expression
contains a "Dirac \delta function", so it is not quite a smooth spacetime.
BTW, the perhaps surprising fact that a static vacuum spacetime yields,
upon "ultraboosting", a curved spacetime analog of a gravitational plane
wave (namely, a certain "vacuum pp wave"), represents a general phenomenon
(this is sometimes called the "Penrose limit"; compare the rate at which
the curvature decays as R -> \infty as measured by static versus
"ultraboosted" observers).
If you use Cartan's formalism (curvature two-forms and all that), it is
not -terribly- tedious to verify the above claims "by hand". However, in
past posts to this newsgroup, I have computed and extensively discussed
the geodesics, Killing vectors, expansion tensor, curvature tensors,
conserved quantities, optical scalars, and various other interesting
physical/geometric properties of these examples, as well as giving
alternative frames and alternative coordinate charts. I also explained
the Ehlers-Kundt classification of vacuum pp waves by their isometries
(Killing vector fields) and the generalization by Sippel and Goenner to
general pp waves.
A few print references:
Maartens & Maharaj, Class. Quant. Grav. 8 (1991): 503--514
Sippel & Goenner, Gen. Rel. Grav. 18 (1986): 1229--1243
Bonnor: Comm. Math. Phys. 13 (1969): 163--174
"T. Essel" (hiding somewhere in cyberspace)
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\ntessel@tum.bot writes\n>On Thu, 12 Aug 2004, Oz wrote:\n>\n>> There is that slightly niggly conundrum about why highly boosted\n>> observers don\'t get to see black holes form in perfectly ordinary (in\n>> their rest frame) particles. It strikes me that there are three\n>> alternatives (probably a fourth which is the standard explanation that I\n>> have forgotten).\n>>\n>> 1) Lorentz is not exact at high boosts.\n>> 2) GR is not exact at extreme curvatures.\n>> 3) QM effects come into play.\n>>\n>> Or a combination, of course.\n>\n>We have very recently discussed this very point (and not for the first\n>time, or even the second time!) in this forum.\n\n<sorry>\n\nTemporary lapse of brain cell....\n\n<grovel...>\n\n>> In passing I don\'t entirely think simple neat mathematical expressions\n>> are on the universes agenda, although I think simple processes are.\n>\n>Some of us would claim that neat mathematical expressions provide some of\n>the best examples of "simple processes".\n\nThey do, but some quickly become pathological.\nConsider the three body newtonian gravity example.\n\n>> Also, in passing, we have two simplified formulations that are very\n>> similar in form: EM and newton. Newton has been extended to GR, whilst\n>> EM remains as it was in maxwell\'s time. I\'m faintly surprised that\n>> nobody has considered treating EM in an analogous way to SR_.GR.\n>\n>I am not sure I follow you here, but remember that Newtonian gravity is\n>nonrelativistic (the "field equation" can be taken to be the\n>three-dimensional Laplace equation, whose solutions are the so-called\n>"harmonic functions"), whereas Maxwell\'s theory of EM is the sine qua non\n>of a classical relativistic field theory. So, we should not be too\n>surprised (although we can be grateful, I guess!) that it requires little\n>modification to treat EM in the context of gtr.\n>\n>Oz, are you still interested in continuing the soliton thread?\n\nAbsolutely. As deep as I can manage given my highly limited ability.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot writes
>On Thu, 12 Aug 2004, Oz wrote:
>
>> There is that slightly niggly conundrum about why highly boosted
>> observers don't get to see black holes form in perfectly ordinary (in
>> their rest frame) particles. It strikes me that there are three
>> alternatives (probably a fourth which is the standard explanation that I
>> have forgotten).
>>
>> 1) Lorentz is not exact at high boosts.
>> 2) GR is not exact at extreme curvatures.
>> 3) QM effects come into play.
>>
>> Or a combination, of course.
>
>We have very recently discussed this very point (and not for the first
>time, or even the second time!) in this forum.
<sorry>
Temporary lapse of brain cell....
<grovel...>
>> In passing I don't entirely think simple neat mathematical expressions
>> are on the universes agenda, although I think simple processes are.
>
>Some of us would claim that neat mathematical expressions provide some of
>the best examples of "simple processes".
They do, but some quickly become pathological.
Consider the three body newtonian gravity example.
>> Also, in passing, we have two simplified formulations that are very
>> similar in form: EM and newton. Newton has been extended to GR, whilst
>> EM remains as it was in maxwell's time. I'm faintly surprised that
>> nobody has considered treating EM in an analogous way to SR_.GR.
>
>I am not sure I follow you here, but remember that Newtonian gravity is
>nonrelativistic (the "field equation" can be taken to be the
>three-dimensional Laplace equation, whose solutions are the so-called
>"harmonic functions"), whereas Maxwell's theory of EM is the sine qua non
>of a classical relativistic field theory. So, we should not be too
>surprised (although we can be grateful, I guess!) that it requires little
>modification to treat EM in the context of gtr.
>
>Oz, are you still interested in continuing the soliton thread?
Absolutely. As deep as I can manage given my highly limited ability.
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nKefka G <kefkag@aol.com> writes\n>\n>\n>Oz writes:\n>\n>>\n>>Data from Oz? - hell no...\n>>\n>>However I do wonder what effect this will have on SR.\n>>\n>>There is that slightly niggly conundrum about why highly boosted\n>>observers don\'t get to see black holes form in perfectly ordinary (in\n>>their rest frame) particles.\n\n>Recall that a black hole is generally defined by the fact that\n>null geodesics cannot propagate from the interior to infinity. Also recall\n>that this is a coordinate invariant statement\n\nAaarrrgghh!\n[Oz wishes he sometimes thought before posting]\nOf course,\n<weakly> silly me....\n<ingratiating but unconvincing smile>\n\n>The only thing that bothers me is that it would appear, then, that a lonely\n>photon (or other massless particle) in empty spacetime cannot gravitate at all,\n>because if it did, the resulting curvature would have to be a function of its\n>energy. Then there is some frame where it would have enough energy to create a\n>black hole. Or perhaps a photon creates Weyl curvature instead of Ricci, and\n>this somehow allows us to escape that conclusion (can\'t see quite how this\n>would happen, though)? I should just go back and work this out myself using\n>the EFE, but maybe someone knows the answer off the top of their head...\n\nOoops, sorry about that ....\n\n\n>Well, if you want to pull EM into GR, there\'s always Kaluza-Klein theory, where\n>we add an extra periodic spatial dimension to 4-D GR and find that this exactly\n>gives us EM gauge invariance if we "factor it away" (Kaluza\'s cylinder\n>condition). It\'s not too satisfying, though, for reasons too detailed to go\n>into here (there\'s a paper at Living Reviews which analyzes a lot of this stuff\n>in detail). Maybe you\'re thinking about something else...\n\nProbably I was, but I expect KK were closer.\n\nWhy do we have to \'factor it away\'? Is it because we don\'t see this\nextra dimension, but then given we are electrically neutral then I\nwouldn\'t expect us to see it directly. Perhaps I am missing something\nhere.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Kefka G <kefkag@aol.com> writes
>
>
>Oz writes:
>
>>
>>Data from Oz? - hell no...
>>
>>However I do wonder what effect this will have on SR.
>>
>>There is that slightly niggly conundrum about why highly boosted
>>observers don't get to see black holes form in perfectly ordinary (in
>>their rest frame) particles.
>Recall that a black hole is generally defined by the fact that
>null geodesics cannot propagate from the interior to infinity. Also recall
>that this is a coordinate invariant statement
Aaarrrgghh!
[Oz wishes he sometimes thought before posting]
Of course,
<weakly> silly me....
<ingratiating but unconvincing smile>
>The only thing that bothers me is that it would appear, then, that a lonely
>photon (or other massless particle) in empty spacetime cannot gravitate at all,
>because if it did, the resulting curvature would have to be a function of its
>energy. Then there is some frame where it would have enough energy to create a
>black hole. Or perhaps a photon creates Weyl curvature instead of Ricci, and
>this somehow allows us to escape that conclusion (can't see quite how this
>would happen, though)? I should just go back and work this out myself using
>the EFE, but maybe someone knows the answer off the top of their head...
Ooops, sorry about that ....
>Well, if you want to pull EM into GR, there's always Kaluza-Klein theory, where
>we add an extra periodic spatial dimension to 4-D GR and find that this exactly
>gives us EM gauge invariance if we "factor it away" (Kaluza's cylinder
>condition). It's not too satisfying, though, for reasons too detailed to go
>into here (there's a paper at Living Reviews which analyzes a lot of this stuff
>in detail). Maybe you're thinking about something else...
Probably I was, but I expect KK were closer.
Why do we have to 'factor it away'? Is it because we don't see this
extra dimension, but then given we are electrically neutral then I
wouldn't expect us to see it directly. Perhaps I am missing something
here.
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
Nick Maclaren
Aug15-04, 12:57 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>[[Mod. note -- I apolgise to the poster and to readers for the\nseveral-day delay in posting this article -- I mistakenly buried it\nin an overflowing mailbox and only just now saw it.\n-- Jonathan Thornburg, sci.astro.research deupty moderator]]\n\n\nIn article <cf340m\\$tah\\$1@pcls4.std.com>,\nDoug Sweetser <sweetser@alum.mit.edu> wrote:\n>\n>What theoretical principle would people in SPR be willing to give up? I\n>think there is consensus that we will not give up the rule that\n>experimental data trumps theory. For that reason, if we ever measure\n>bending of light around the Sun to 1 microarcsecond of accuracy, we may\n>be able to either confirm or reject the exponential metric. This is\n>just like what happened at the first order PPN level: the data said\n>Einstein\'s second rank theory was right, not Newton\'s scalar theory.\n>The exponential metric predicts more bending the the Schwarzschild\n>metric. Let the data rule.\n\nThanks. Yes, that is what I was trying to ask - i.e. exactly what\nfixed ground do the current data give us.\n\n>Somethings I would keep: the weak equivalence principle (inertial equals\n>gravitational mass), the strong equivalence principle (the active\n>gravitational mass is equivalent to the passive gravitational mass),\n>the local nature of the the equations, and local Lorentz invariance.\n\nDo you mean that you would want them to hold for short-range interactions\nunder conditions of extreme stresses (gravitational, temperature, etc.)?\nGiven quantum mechanics\' slightly bizarre concept of physical measurement,\nit isn\'t clear that they could simply \'hold\' in a combined model.\n\n\nRegards,\nNick Maclaren.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>[[Mod. note -- I apolgise to the poster and to readers for the
several-day delay in posting this article -- I mistakenly buried it
in an overflowing mailbox and only just now saw it.
-- Jonathan Thornburg, sci.astro.research deupty moderator]]
In article <cf340m$tah$1@pcls4.std.com>,
Doug Sweetser <sweetser@alum.mit.edu> wrote:
>
>What theoretical principle would people in SPR be willing to give up? I
>think there is consensus that we will not give up the rule that
>experimental data trumps theory. For that reason, if we ever measure
>bending of light around the Sun to 1 microarcsecond of accuracy, we may
>be able to either confirm or reject the exponential metric. This is
>just like what happened at the first order PPN level: the data said
>Einstein's second rank theory was right, not Newton's scalar theory.
>The exponential metric predicts more bending the the Schwarzschild
>metric. Let the data rule.
Thanks. Yes, that is what I was trying to ask - i.e. exactly what
fixed ground do the current data give us.
>Somethings I would keep: the weak equivalence principle (inertial equals
>gravitational mass), the strong equivalence principle (the active
>gravitational mass is equivalent to the passive gravitational mass),
>the local nature of the the equations, and local Lorentz invariance.
Do you mean that you would want them to hold for short-range interactions
under conditions of extreme stresses (gravitational, temperature, etc.)?
Given quantum mechanics' slightly bizarre concept of physical measurement,
it isn't clear that they could simply 'hold' in a combined model.
Regards,
Nick Maclaren.
Kefka G
Aug16-04, 12:55 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOz writes:\n\n>KefkaG <kefkag@aol.com> writes:\n>>Well, if you want to pull EM into GR, there\'s always Kaluza-Klein theor=\ny,\n>where\n>>we add an extra periodic spatial dimension to 4-D GR and find that this\n>exactly\n>>gives us EM gauge invariance if we "factor it away" (Kaluza\'s cylinder\n>>condition). It\'s not too satisfying, though, for reasons too detailed =\nto go\n>>into here (there\'s a paper at Living Reviews which analyzes a lot of th=\nis\n>stuff\n>>in detail). Maybe you\'re thinking about something else...\n>\n>Probably I was, but I expect KK were closer.\n>\n>Why do we have to \'factor it away\'? Is it because we don\'t see this\n>extra dimension, but then given we are electrically neutral then I\n>wouldn\'t expect us to see it directly. Perhaps I am missing something\n>here.=20\n>\n\nHere are the basic assumptions of Kaluza-Klein theory in its original for=\nm:\n\n1) The laws of general relativity hold fast\n2) We have five dimensions - one time, four space\n3) The fourth spatial dimension is periodic, and somehow so small that "n=\nothing\ndepends" on it - i.e. we drop all derivatives in this direction, and assu=\nme\nthat we can\'t measure any distances in that direction.\n4) Gravity and matter is all that we have\n\nTo start out, following Kaluza we take the line element to look like this=\n:\n\nds^2 =3D g*_uv (x,y) dx*^u dx*^v\n=3D g_uv (x) dx^u dx^v + ( dy + k A_u (x) dx^u )^2\n\nwhere the *\'d variables refer to the bulk 5-d space, and the unstarred on=\nes\nrefer to 4-d space (in which case the fourth spatial dimension is called =\ny).=20\nNotice that A_u does NOT depend on y here, so we\'ve already essentially t=\naken\nthe Kaluza cylinder hypothesis 3). I\'ll let you work out that in order t=\no have\ninvariance of this line element under transformations y -> y\' =3D f(x^u) =\nwe must\nhave the following:\n\nA_=B5 -> A\'_=B5 =3D A_u - (df/dx^=B5) / k where the d\'s should be parti=\nals.\n\nNote that f does not depend on y - convince yourself that this is necessa=\nry for\nthe line element to be preserved.\n\nAlso note that we can take this last equation as the gauge transformation=\nof\nthe scalar potential, so this gives us hope that the EFE will be equivale=\nnt to\ngravity coupled to e/m. I\'ll let you work out the curvatures and whatnot=\n-\nit\'s not that tough if you use a non-orthonormal basis. Do remember to d=\nrop\nall y-derivatives by the cylinder condition! But in the end, we find a s=\nimple\nrelation between the 5-d and 4-d curvatures:\n\nR* =3D R - (k^2 / 4) F^uv F_uv\n\nwhere F_uv is the usual Maxwell tensor. Note that this equation is invar=\niant\nunder choice of metric signature (being the scalar contractions of two\neven-rank tensors), which is nice if you\'re like me and use different\nconventions depending on your whim. Now just plug this into the Hilbert\naction, integrate out the fifth coordinate (trivially since nothing depen=\nds on\nit) and you\'ll get a 4-d action integral describing (for the appropriate =\nchoice\nof k) 4-d gravity coupled to Maxwell\'s equations.\n\nSo in short, the reason we need to factor out the fifth dimension is that\notherwise, we\'re just doing relativity in 5-d. We\'ll have waves in that\ndirection, particles moving in that direction, etc. In order to get anyt=\nhing\nelse, we need to pretend that we\'re in 4-d, and a good way to do this is =\nto\npretend that the fifth dimension is just too small to see. Keep in mind t=\nhat so\nfar, all of this is in empty (5-dimensional) space, too, so in 4-d, at mo=\nst\nwe\'re covering e/m radiation coupled to gravity with no matter.\n\nIn particular, one thing that sucks about this theory is the following. =\nIn\nfive dimensions everything is sourceless, so the scalar R* =3D 0. Conseq=\nuently,\nthanks to the 5-d EFE, R*_ab =3D 0 for all a, b. I haven\'t shown this, b=\nut as it\nturns out,\n\nR*_44 =3D - (1/4) F_ab F^ab =3D 0\n\nwhich in turn means that F_ab F^ab =3D 0 everywhere, so all of these equa=\ntions\nare trivial! As it turns out, we\'d need to have included a scalar field =\nin the\noriginal metric hypothesis in order to avoid this problem. It\'s much mor=\ne\ncomplicated to get anything interesting out, so I\'ll leave it to you to f=\nind\npapers where this is discussed - there are plenty. But I think this disc=\nussion\nat least illustrates the basic idea quite nicely - if you tack on an extr=\na set\nof dimensions invariant under some isometry group, then the coordinate\ninvariance of GR will automatically turn that isometry group into a group=\nof\ngauge transformations in higher dimensional gravity, at least if we "fact=\nor\nthem away," i.e. just don\'t pay attention to them. This creates problems=\nof\nits own, however, such as the fact that the resulting equations may turn =\nout\ntrivial if we\'re not careful (here the problem arises from assuming that =\ng_44\nis always equal to 1(indices from 0-4), which we\'re not free to do - if w=\ne\nallow this to "flap in the wind", we can obtain non-trivial equations whi=\nch are\nno longer so simple as 4-d gravity + e/m).\n\nIt\'s an interesting exercise to try this out with linear fields as well -=\nin\nparticular, try this whole "factoring out a dimension" thing on a scalar =\nfield\n(where as it turns out there are no observable consequences - a 5-d scala=\nr\ntheory compactified to 4-d is exactly equivalent to a 4-d scalar theory) =\nand on\nthe Maxwell field (using the obvious generalization of the Maxwell equati=\nons).=20\nAs it turns out, it is NOT true that a fifth dimension would be unobserva=\nble if\nit were really small under Maxwell\'s equations - we end up with an extra =\nscalar\nfield which has a non-trivial effect on 4-dimensional matter, i.e. 5-d Ma=\nxwell\ncompactified to 4-d =3D 4-d Maxwell + 4-d scalar. Of course, to produce =\nthis\nextra field requires current flowing in the extra dimension, which maybe =\nwe\'d\nprefer to "factor out" - in any case, these are interesting matters, and\nthere\'s a million fun things to play with if you\'re interested.\n\nAnd that\'s not even starting to talk about what happens when we add in qu=\nantum\nmechanics, higher gauge groups, projective spaces, stability issues, sour=\nces,\nblack holes, Casimir forces, etc (much of which I don\'t understand very w=\nell\nyet). Actually, all that I\'ve talked about above was really Kaluza\'s the=\nory -\nKlein made separate contributions which I haven\'t talked about yet. Anyh=\now,\nI\'m hungry so I need to go out now. Sorry if all this was a little vague=\n-\ne-mail me if you\'d like a longer, more cogent explanation.\n\n-Eric\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Oz writes:
>KefkaG <kefkag@aol.com> writes:
>>Well, if you want to pull EM into GR, there's always Kaluza-Klein theor=
y,
>where
>>we add an extra periodic spatial dimension to 4-D GR and find that this
>exactly
>>gives us EM gauge invariance if we "factor it away" (Kaluza's cylinder
>>condition). It's not too satisfying, though, for reasons too detailed =
to go
>>into here (there's a paper at Living Reviews which analyzes a lot of th=
is
>stuff
>>in detail). Maybe you're thinking about something else...
>
>Probably I was, but I expect KK were closer.
>
>Why do we have to 'factor it away'? Is it because we don't see this
>extra dimension, but then given we are electrically neutral then I
>wouldn't expect us to see it directly. Perhaps I am missing something
>here.=20
>
Here are the basic assumptions of Kaluza-Klein theory in its original for=
m:
1) The laws of general relativity hold fast
2) We have five dimensions - one time, four space
3) The fourth spatial dimension is periodic, and somehow so small that "n=
othing
depends" on it - i.e. we drop all derivatives in this direction, and assu=
me
that we can't measure any distances in that direction.
4) Gravity and matter is all that we have
To start out, following Kaluza we take the line element to look like this=
:
ds^2 =3D g*_uv (x,y) dx*^u dx*^v=3D g_{uv} (x) dx^u dx^v + ( dy + k A_u (x) dx^u )^2
where the *'d variables refer to the bulk 5-d space, and the unstarred on=
es
refer to 4-d space (in which case the fourth spatial dimension is called =
y).=20
Notice that A_u does NOT depend on y here, so we've already essentially t=
aken
the Kaluza cylinder hypothesis 3). I'll let you work out that in order t=
o have
invariance of this line element under transformations y -> y' =3D f(x^u) =
we must
have the following:
A_=B5 -> A'_=B5 =3D A_u - (df/dx^=B5) / k[/itex] where the d's should be parti=
als.
Note that f does not depend on y - convince yourself that this is necessa=
ry for
the line element to be preserved.
Also note that we can take this last equation as the gauge transformation=
of
the scalar potential, so this gives us hope that the EFE will be equivale=
nt to
gravity coupled to e/m. I'll let you work out the curvatures and whatnot=
-
it's not that tough if you use a non-orthonormal basis. Do remember to d=
rop
all y-derivatives by the cylinder condition! But in the end, we find a s=
imple
relation between the 5-d and 4-d curvatures:
[itex]R* =3D R - (k^2 / 4) F^{uv} F_{uv}
where F_{uv} is the usual Maxwell tensor. Note that this equation is invar=
iant
under choice of metric signature (being the scalar contractions of two
even-rank tensors), which is nice if you're like me and use different
conventions depending on your whim. Now just plug this into the Hilbert
action, integrate out the fifth coordinate (trivially since nothing depen=
ds on
it) and you'll get a 4-d action integral describing (for the appropriate =
choice
of k) 4-d gravity coupled to Maxwell's equations.
So in short, the reason we need to factor out the fifth dimension is that
otherwise, we're just doing relativity in 5-d. We'll have waves in that
direction, particles moving in that direction, etc. In order to get anyt=
hing
else, we need to pretend that we're in 4-d, and a good way to do this is =
to
pretend that the fifth dimension is just too small to see. Keep in mind t=
hat so
far, all of this is in empty (5-dimensional) space, too, so in 4-d, at mo=
st
we're covering e/m radiation coupled to gravity with no matter.
In particular, one thing that sucks about this theory is the following. =
In
five dimensions everything is sourceless, so the scalar R* =3D . Conseq=
uently,
thanks to the 5-d EFE, R*_ab =3D for all a, b. I haven't shown this, b=
ut as it
turns out,
R*_44 =3D - (1/4) F_{ab} F^{ab} =3D
which in turn means that F_{ab} F^{ab} =3D everywhere, so all of these equa=
tions
are trivial! As it turns out, we'd need to have included a scalar field =
in the
original metric hypothesis in order to avoid this problem. It's much mor=
e
complicated to get anything interesting out, so I'll leave it to you to f=
ind
papers where this is discussed - there are plenty. But I think this disc=
ussion
at least illustrates the basic idea quite nicely - if you tack on an extr=
a set
of dimensions invariant under some isometry group, then the coordinate
invariance of GR will automatically turn that isometry group into a group=
of
gauge transformations in higher dimensional gravity, at least if we "fact=
or
them away," i.e. just don't pay attention to them. This creates problems=
of
its own, however, such as the fact that the resulting equations may turn =
out
trivial if we're not careful (here the problem arises from assuming that =
g_{44}
is always equal to 1(indices from 0-4), which we're not free to do - if w=
e
allow this to "flap in the wind", we can obtain non-trivial equations whi=
ch are
no longer so simple as 4-d gravity + e/m).
It's an interesting exercise to try this out with linear fields as well -=
in
particular, try this whole "factoring out a dimension" thing on a scalar =
field
(where as it turns out there are no observable consequences - a 5-d scala=
r
theory compactified to 4-d is exactly equivalent to a 4-d scalar theory) =
and on
the Maxwell field (using the obvious generalization of the Maxwell equati=
ons).=20
As it turns out, it is NOT true that a fifth dimension would be unobserva=
ble if
it were really small under Maxwell's equations - we end up with an extra =
scalar
field which has a non-trivial effect on 4-dimensional matter, i.e. 5-d Ma=
xwell
compactified to 4-d =3D 4-d Maxwell + 4-d scalar. Of course, to produce =
this
extra field requires current flowing in the extra dimension, which maybe =
we'd
prefer to "factor out" - in any case, these are interesting matters, and
there's a million fun things to play with if you're interested.
And that's not even starting to talk about what happens when we add in qu=
antum
mechanics, higher gauge groups, projective spaces, stability issues, sour=
ces,
black holes, Casimir forces, etc (much of which I don't understand very w=
ell
yet). Actually, all that I've talked about above was really Kaluza's the=
ory -
Klein made separate contributions which I haven't talked about yet. Anyh=
ow,
I'm hungry so I need to go out now. Sorry if all this was a little vague=
-
e-mail me if you'd like a longer, more cogent explanation.
-Eric
jgraber
Aug16-04, 12:55 PM
Hi Steve,
No, I don’t know of a careful published analysis. Obviously, if the only effect is redshift out, that is totally compensated for by blueshift in. That was point of my comment that the energy must scatter or thermalize. As I said, if the surface acts like a perfect mirror, the energy will emerge promptly and in the same spectral range from a surface at any finite redshift and hence such a surface would not pass the Narayan tests. However, if the energy themalizes to the same spectral range, it will be redshifted by roughly the depth of the surface. The attenuation factor is however much more than merely that due to the time dilation of the escaping radiation, i.e. much more than a factor of one hundred if the redshift is one hundred. (Actually, this statement is metric dependent, but it holds for most of the metrics I’ve thought about, including exponential and truncated Schwazschild. I’m not sure it would apply to Nick’s metric, but my first guess is yes for that one, too.) The reason is due to the “loss cone” or escape cone as illustrated on the bottom of page 675 of MTW. This loss cone also gets narrower and narrower as the redshift gets deeper. (And more than linearly so in the cases I have considered.) Accordingly an even larger fraction of the radiation does not escape, but merely bounces onto another section of the compact objects surface. When it does finally escape, a much longer time has elapsed and the energy has rethermalized many times and hence probably also moved to a far lower energy band than would result from only a single redshift. Perhaps I can try to do a toy model calculation in the next few days.
Best, Jim
Robert Shaw
Aug17-04, 11:27 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n<tessel@tum.bot> wrote in message news:411c9ae9\\$1@news.sentex.net...\n> On Thu, 12 Aug 2004, Robert Shaw wrote:\n>\n> > I recall reading a book on general relativity for people without\n> > calculus, about twenty years ago. Essentially, the authors came up\n> > with difference equations which approximated the differential\n> > equations of GR, and used them to demonstrate the behaviour\n> > of GR.\n>\n> Hmm... I\'ve seen many books on gtr, but this doesn\'t ring a bell.\n> Does anyone have a full citation?\n>\nIt was published in the UK. I don\'t remember anyhting else.\n\n> > They were very clearly that these difference equations were only\n> > intended to be approximations, for teaching purposes, not a credible\n> > alternative to GR.\n> >\n> > However, in their approximation, the metric for a black hole did\n> > have the exponential form, tending to the Schwarzchild metric\n> > as the step size tended to zero.\n>\n> Yes, there is certainly something to explain there.\n>\n> What happens when you try to obtain the ordinary 2 dimensional Laplace\n> equation as the continuum limit of the obvious choice for "the"\n> analogous difference equation? The 3 dimensional axisymmetric Laplace\n> equation?\n\nIn general you can get many problems, including failure to converge\nand spurious solutions.\n\n>\n> > There was a footnote to the effect that, this is only intended to be\n> > an approximation, and there\'s no evidence otherwise but maybe,\n> > just maybe\n> > ...\n> >\n> > Has there been any serious investigation of difference equations, on\n> > a lattice or a continuum, as possible classical alternatives to GR?\n>\n\n>\n> I didn\'t understand "maybe, just maybe" though--- maybe I don\'t\n> understand correctly your goals here?\n\nThey were, in tones of wild speculation, saying that general relativity\nmight be the approximation, and a discrete theory the actuality.\n\nA discrete theory would be equivalent to general relativity plus\nan infinite series of higher order derivatives terms.\n\nI understand that adding one such higher order term can give a\ntheory whose quantum version doesn\'t even need renormalising\n(but isn\'t unitary). Adding an infinite set of such terms might well\ncancel out all the infinities, and a discrete theory is a natural way\nto specify the infinite series without needing an infinite set of\nconstants.\n\nHowever, it\'s at least as plausible that any classical discrete theory\nto which general relativity is a reasonable approximation would\nhave insuperable problems.\n\nWhat I\'m wondering is how insuperable those problems are.\n\n>\n> > What are the biggest problems with such a theory?\n>\n> [A difference equation analog of the EFE]\n>\n> But this might only mean that the authors of the book\n> you saw took the limit incorrectly, or were using an incorrect choice\n> for "the" analogous difference equation, as in the example from\nsoliton\n> theory which I mentioned recently (c.f. the derivation of the KdV from\n> the well-known Fermi-Pasta-Ulam lattice).\n\nQuite possibly. I was in no position to judge at the time.\n\n--\nMatter is fundamentally lazy:- It always takes the path of least effort\nMatter is fundamentally stupid:- It tries every other path first.\nThat is the heart of physics - The rest is details.- Robert Shaw\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky><tessel@tum.bot> wrote in message news:411c9ae9$1@news.sentex.net...
> On Thu, 12 Aug 2004, Robert Shaw wrote:
>
> > I recall reading a book on general relativity for people without
> > calculus, about twenty years ago. Essentially, the authors came up
> > with difference equations which approximated the differential
> > equations of GR, and used them to demonstrate the behaviour
> > of GR.
>
> Hmm... I've seen many books on gtr, but this doesn't ring a bell.
> Does anyone have a full citation?
>
It was published in the UK. I don't remember anyhting else.
> > They were very clearly that these difference equations were only
> > intended to be approximations, for teaching purposes, not a credible
> > alternative to GR.
> >
> > However, in their approximation, the metric for a black hole did
> > have the exponential form, tending to the Schwarzchild metric
> > as the step size tended to zero.
>
> Yes, there is certainly something to explain there.
>
> What happens when you try to obtain the ordinary 2 dimensional Laplace
> equation as the continuum limit of the obvious choice for "the"
> analogous difference equation? The 3 dimensional axisymmetric Laplace
> equation?
In general you can get many problems, including failure to converge
and spurious solutions.
>
> > There was a footnote to the effect that, this is only intended to be
> > an approximation, and there's no evidence otherwise but maybe,
> > just maybe
> > ...
> >
> > Has there been any serious investigation of difference equations, on
> > a lattice or a continuum, as possible classical alternatives to GR?
>
>
> I didn't understand "maybe, just maybe" though--- maybe I don't
> understand correctly your goals here?
They were, in tones of wild speculation, saying that general relativity
might be the approximation, and a discrete theory the actuality.
A discrete theory would be equivalent to general relativity plus
an infinite series of higher order derivatives terms.
I understand that adding one such higher order term can give a
theory whose quantum version doesn't even need renormalising
(but isn't unitary). Adding an infinite set of such terms might well
cancel out all the infinities, and a discrete theory is a natural way
to specify the infinite series without needing an infinite set of
constants.
However, it's at least as plausible that any classical discrete theory
to which general relativity is a reasonable approximation would
have insuperable problems.
What I'm wondering is how insuperable those problems are.
>
> > What are the biggest problems with such a theory?
>
> [A difference equation analog of the EFE]
>
> But this might only mean that the authors of the book
> you saw took the limit incorrectly, or were using an incorrect choice
> for "the" analogous difference equation, as in the example from
soliton
> theory which I mentioned recently (c.f. the derivation of the KdV from
> the well-known Fermi-Pasta-Ulam lattice).
Quite possibly. I was in no position to judge at the time.
--
Matter is fundamentally lazy:- It always takes the path of least effort
Matter is fundamentally stupid:- It tries every other path first.
That is the heart of physics - The rest is details.- Robert Shaw
tessel@tum.bot
Aug18-04, 04:08 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 17 Aug 2004, Robert Shaw wrote:\n\n> <tessel@tum.bot> wrote in message news:411c9ae9\\$1@news.sentex.net...\n>\n> They were, in tones of wild speculation, saying that general relativity\n> might be the approximation, and a discrete theory the actuality.\n>\n> A discrete theory would be equivalent to general relativity plus\n> an infinite series of higher order derivatives terms.\n\nFWIW, on the ArXiV you can find some preprints dealing with "higher order\n[derivative] gravity [theories]". For example, some such theories can be\ndefined via a Lagrangian which resembles the Einstein-Hilbert Lagrangian\nof gtr -plus- some additional quadratic terms, and admit as "vacuum\nsolutions" all vacuum solutions of the EFE -plus- some additional\nsolutions. This is of course a Baconian defect, since such a theory is\nharder to falsify than gtr--- at least if one is studying vacuum regions.\n\n> I understand that adding one such higher order term can give a\n> theory whose quantum version doesn\'t even need renormalising\n> (but isn\'t unitary). Adding an infinite set of such terms might well\n> cancel out all the infinities, and a discrete theory is a natural way\n> to specify the infinite series without needing an infinite set of\n> constants.\n\nSounds like there is general principle here worth explaining briefly.\n\nI understand how a generating function (e.g. a rational function or some\nother easily written function) "can serve as a clothsline on which we hang\nan infinite number of coefficients" (Herbert Wilf writes something like\nthis in the introduction to his delightful book Generatingfunctionology),\ne.g.\n\n1/(1-x-x^2) = 1 + x + 2 x^2 + 3 x^3 + 5 x^4 + 8 x^5 + ...\n\nis a convenient way of compactly representing the Fibonacci series, many\ncombinatorial identities are most easily derived using the machinery of\ngenerating function methods, and so on and so forth. (Readers who have\nnever encountered generating functions will be well rewarded by looking up\nJohn Baez\'s posts in the This Week series on "combinatorial species", a\nfar-reaching "categorifation" of generating functions. Alas, I never did\nget around to explaining a very general and quite delightful connection\nbetween combinatorial species, aka "structors", and the Cameron cycle\nindex/Polya counting/oligomorphic group actions.)\n\nBut I\'m not grokking what you just said. What is a "discrete theory" and\nhow does it enable us "to specify an infinite series without needing an\ninfinite set of coefficients"? Does this claim factor through the idea I\nalready understand? That is, does "a discrete theory" yield a generating\nfunction? If so, how?\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 17 Aug 2004, Robert Shaw wrote:
> <tessel@tum.bot> wrote in message news:411c9ae9$1@news.sentex.net...
>
> They were, in tones of wild speculation, saying that general relativity
> might be the approximation, and a discrete theory the actuality.
>
> A discrete theory would be equivalent to general relativity plus
> an infinite series of higher order derivatives terms.
FWIW, on the ArXiV you can find some preprints dealing with "higher order
[derivative] gravity [theories]". For example, some such theories can be
defined via a Lagrangian which resembles the Einstein-Hilbert Lagrangian
of gtr -plus- some additional quadratic terms, and admit as "vacuum
solutions" all vacuum solutions of the EFE -plus- some additional
solutions. This is of course a Baconian defect, since such a theory is
harder to falsify than gtr--- at least if one is studying vacuum regions.
> I understand that adding one such higher order term can give a
> theory whose quantum version doesn't even need renormalising
> (but isn't unitary). Adding an infinite set of such terms might well
> cancel out all the infinities, and a discrete theory is a natural way
> to specify the infinite series without needing an infinite set of
> constants.
Sounds like there is general principle here worth explaining briefly.
I understand how a generating function (e.g. a rational function or some
other easily written function) "can serve as a clothsline on which we hang
an infinite number of coefficients" (Herbert Wilf writes something like
this in the introduction to his delightful book Generatingfunctionology),
e.g.
1/(1-x-x^2) = 1 + x + 2 x^2 + 3 x^3 + 5 x^4 + 8 x^5 + ...
is a convenient way of compactly representing the Fibonacci series, many
combinatorial identities are most easily derived using the machinery of
generating function methods, and so on and so forth. (Readers who have
never encountered generating functions will be well rewarded by looking up
John Baez's posts in the This Week series on "combinatorial species", a
far-reaching "categorifation" of generating functions. Alas, I never did
get around to explaining a very general and quite delightful connection
between combinatorial species, aka "structors", and the Cameron cycle
index/Polya counting/oligomorphic group actions.)
But I'm not grokking what you just said. What is a "discrete theory" and
how does it enable us "to specify an infinite series without needing an
infinite set of coefficients"? Does this claim factor through the idea I
already understand? That is, does "a discrete theory" yield a generating
function? If so, how?
"T. Essel" (hiding somewhere in cyberspace)
Robert Shaw
Aug18-04, 09:58 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n<tessel@tum.bot> wrote\n> On Tue, 17 Aug 2004, Robert Shaw wrote:\n>\n>\n> I understand how a generating function (e.g. a rational function or\nsome\n> other easily written function) "can serve as a clothsline on which we\nhang\n> an infinite number of coefficients" (Herbert Wilf writes something\nlike\n> this in the introduction to his delightful book\nGeneratingfunctionology),\n> e.g.\n>\n> 1/(1-x-x^2) = 1 + x + 2 x^2 + 3 x^3 + 5 x^4 + 8 x^5 + ...\n>\n> is a convenient way of compactly representing the Fibonacci series,\nmany\n> combinatorial identities are most easily derived using the machinery\nof\n> generating function methods, and so on and so forth.\n\n> But I\'m not grokking what you just said. What is a "discrete theory"\nand\n> how does it enable us "to specify an infinite series without needing\nan\n> infinite set of coefficients"? Does this claim factor through the\nidea I\n> already understand? That is, does "a discrete theory" yield a\ngenerating\n> function? If so, how?\n\nI may have my terminolgy confused, but the principle is the same.\n\nThe difference equation, x_(i+1) + x_(i-1) = 2x_i, which only\ntake values on a lattice of discrete points, spaced h apart,\nhas the same solutions has the infinite order differential equation\n0 = (D^2 + 2h^2D^4 /4! + 2h^4D^6 /6! .... ) x\nat the points where both solutions are defined.\n\nYou\'ve got an infinite series of higher order derivatives specified\nby a single parameter, with generating function 2( cosh hD -1)\n\n\n--\nMatter is fundamentally lazy:- It always takes the path of least effort\nMatter is fundamentally stupid:- It tries every other path first.\nThat is the heart of physics - The rest is details.- Robert Shaw\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky><tessel@tum.bot> wrote
> On Tue, 17 Aug 2004, Robert Shaw wrote:
>
>
> I understand how a generating function (e.g. a rational function or
some
> other easily written function) "can serve as a clothsline on which we
hang
> an infinite number of coefficients" (Herbert Wilf writes something
like
> this in the introduction to his delightful book
Generatingfunctionology),
> e.g.
>
> 1/(1-x-x^2) = 1 + x + 2 x^2 + 3 x^3 + 5 x^4 + 8 x^5 + ...
>
> is a convenient way of compactly representing the Fibonacci series,
many
> combinatorial identities are most easily derived using the machinery
of
> generating function methods, and so on and so forth.
> But I'm not grokking what you just said. What is a "discrete theory"
and
> how does it enable us "to specify an infinite series without needing
an
> infinite set of coefficients"? Does this claim factor through the
idea I
> already understand? That is, does "a discrete theory" yield a
generating
> function? If so, how?
I may have my terminolgy confused, but the principle is the same.
The difference equation, x_(i+1) + x_(i-1) = 2x_i, which only
take values on a lattice of discrete points, spaced h apart,
has the same solutions has the infinite order differential equation
= (D^2 + 2h^2D^4 /4! + 2h^4D^6 /6! .... ) x
at the points where both solutions are defined.
You've got an infinite series of higher order derivatives specified
by a single parameter, with generating function 2( cosh hD -1)
--
Matter is fundamentally lazy:- It always takes the path of least effort
Matter is fundamentally stupid:- It tries every other path first.
That is the heart of physics - The rest is details.- Robert Shaw
jgraber
Aug31-04, 12:41 PM
More on Hard Surface Compact Pseudo Black Holes
Narayan et al analysis based on General Relativity
First of all, let me point out that the Narayan et al papers are all based on General Relativity, and hence assume that the hard surface can not be at a redshift greater than about unity, a result that is true in GR and goes all the wayback to Chandrasekhar and Bondi. In fact, the diagram in the most recent paper, astro-ph/0401549, cuts off at a redshift of z=.5.ie z equal one half. (Neutron star surfaces are usually calculated to be at redshifts of about .2 or in the range of .1 to .3 ie about 10 to 30 percent.)
On the contrary, I assume a surface redshift of z=100, a factor of 200 to 1000 times larger, and hence assume that GR is incorrect, and that some alternate theory of gravity is correct. Hence there is no conflict between my remarks and the conclusions of Narayan et al, we simply start with different hypotheses, and end with different conclusions.
Highly redshifted flat sufaces have more energy to hide.
Since I still assume conservation of energy, the fact that the hard surface is so highly redshifted requires that almost all of the rest mass of the infalling material has been converted into binding energy. For neutron stars, this is sometimes referred to by saying that their gravitational mass is less than their baryonic mass. Presumably this missing mass has escaped as radiation. Hence a z = 100 hard surface object has to dispose of 99 percent of the infalling energy, while a z=.2 object only has todispose of approximately 20%. Hence, the more compact object has approximately five times as much energy to hide.
Nevertheless, the black hole candidate objects are one hundred to one thousand times less luminous in the x-ray band than the neutron stars during quiescent accretion.
(Of couse, during outbursts, the black hole candidates are much more luminus than the neutron stars, a fact that is still not yet well understood.)
How can compact objects hide energy?
There are several possibilities:
1.They can redshift it out of the x-ray band.
2.They can convert it to thermal energy.
3.They can convert it to trapped photons.
4.They can convert it to trapped gravitons.
(Since the black hole candidates have masses larger than the masses of the neutron stars, a substantial fraction of the energy cannot escape, but most be somehow trapped or retained.)
Since I assume that these hypothetical hard surface hypercompact objects are entirely inside the photon orbit, they can trap photons and hence also gravitons. Vertically directed radiation can still escape, and also radiation emitted into a very small vertically directed loss cone, but radiation emitted at any larger angle away from the vertical will be trapped and return to the surface for another bounce, or another chance to be absorbed. If the photon is absorbed, its energy may be thermalized and reradiated as several weaker photons. This absorbtion, thermalization and reradiation process is even more effective than gravitational redshift in converting the energy from x-rays to longer wavelength radiation.
My basic claim is that if the loss cone is small enough to reduce the prompt emission by slightly more than the observed factor in the x-ray band, this reradiation process will hide the rest of it.
(Narayan et al base their argument on luminosity in the .5 to 10 kev band according to astro-ph/0107387)
Since the area of the loss cone tends to be inversely proportional to the square of the redshift z, it is possible that a z of only 10 or 20 would be sufficent to model the Narayan et al observations. If z is 100, it should be more than sufficent.
More later perhaps if anyone is interested.
Best, Jim
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"jgraber" <jgra@loc.gov> wrote in message\nnews:jgraber.1b73lo@physicsforums.com...\ n\n.....\n\n> First of all, let me point out that the Narayan et al papers are all\n> based on General Relativity, and hence assume that the hard surface can\n> not be at a redshift greater than about unity, a result that is true in\n> GR and goes all the wayback to Chandrasekhar and Bondi. In fact, the\n> diagram in the most recent paper, astro-ph/0401549, cuts off at a\n> redshift of z=.5.ie z equal one half. (Neutron star surfaces are\n> usually calculated to be at redshifts of about .2 or in the range of .1\n> to .3 ie about 10 to 30 percent.)\n>\n> On the contrary, I assume a surface redshift of z=100, a factor of 200\n> to 1000 times larger, and hence assume that GR is incorrect, and that\n> some alternate theory of gravity is correct.\n\nDear Jim,\n\nit is an - albeit very common - misperception, that GR does not allow\nredshifts higher than z \\approx 2.\n\nThis is simply not true. All "proofs" for z<=2 require - besides the\nfield equations of general relativity - *additional* assumptions, such\nas isotropic pressure, non-negative pressure etc.\n\nIn fact, one can see this already by a constructive argument:\n\nThere are exact solutions of the Einstein field equations\n(with and without gravitational constant Lambda) for spherically\nsymmetric compact bodies with (hard, material) surfaces , whose\nsurface redshift can become arbitrarily large (depending on the\nmass of the body).\n\nThe holographic solution (see gr-qc/0306063, gr-qc/0306066) is one\nsuch solution. It solves the unmodified field equations of GR (without\ngravitational constant) exactly, but has a negative string-type\ninterior pressure. Its surface redshift scales with \\sqrt{M / M_Pl}.\nA "holostar" with the mass of the sun has a surface redshift of\nroughly 10^20.\n\nThe gravastar - although somwhat awkward, because the "interior\npart" of the solution solves the field equations *with* gravitational\nconstant, whereas the "exterior part" solves the field equations\n*without* gravitational constant - is another such solution. The\ngravastar\'s (matter-filled) interior deSitter core has a negative\nisotropic pressure.\n\nTherefore, if one is willing to accept\n\n- negative pressures (dark energy = negative pressure is\nexperimentally verified by the recent supernova measurements!\nSo there should be no problem to accept the physical relevance\nof negative pressure),\n\n- or even negative *anisotropic* pressures (such as commonly\noccurring in string theory: strings have tension = negative\npressure, but the tension acts in only one direction - namely\nthe direction of the string. The "transverse" pressure of a\nclassical string is zero),\n\ncompact objects with arbitrarily large surface redshifts are fully\ncompatible with GR.\n\nGR in itself does not give any prescription what the matter-state\nor the equation of state of a compact self-gravitating body should\nbe. GR can "live" with zero matter-density and zero pressure (the\nclassical black hole solutions), it can live with an anisotropic\nstring type pressure (such as domain-wall or string-type solutions,\nor the recently discovered holographic solution), as well as it can\nlive with isotropic positive pressure (e.g. the radiation dominated\nFRW-solution), or with an isotropic negative pressure (deSitter\nsolution, gravastars, the Lamdba CDM "standard" model of the\nuniverse).\n\nKeep in mind, that science is a continuous process of refining,\nreviewing and adapting our knowledge to the most recent experimental\nand theoretical findings. The "old" assumptions, which were reasonable\nin the past, must not necessarily remain reasonable forever. The\nassumptions of non-negative and isotropic pressure, leading to the\n"prediction" z<2, were very reasonable assumptions regarding our state\nof knowledge roughly 10 years ago. But recent experimental and\ntheoretical results not only suggest, they actually *require* a\nre-evaluation of these "old" assumptions and therefore a re-evaluation\nof ALL RESULTS BASED ON THESE ASSUMPTIONS.\n\n<sorry for the "screaming", but this important and highly trivial fact\nhas not registered yet in the scientific community, which happily\ncontinues to pour out (or mindlessly repeat) claims, which are based\non these assumptions.\n\nI don\'t have anything against *assuming* positive or isotropic pressure,\nbut in view of the recent experimental and theoretical findings it is\nmandatory to mention *all* assumptions on which a certain claim\nrelies.\n\nMy guess is, that almost everyone will agree that the "old" assumption\nof non-negative pressure has been proven to be experimentally incorrect\nby the recent supernova-measurements. Negative pressure (= tension)\ndominating the large scale structure of the observable universe has\nbecome known to the public under the somewhat obscure term "dark\nenergy". Note however, that negative pressure (tension) was quite\ncommon even before this discovery: Everybody knows e.g. about surface\ntension. Just look at some water-drops.\n\nThe case of isotropic vs. anisotropic pressure is not so clear. What is\nclear however, that there is no general physical principle which requires\nthe pressure to be isotropic. In fact, we know for certain that there are\nseveral real world situations (crystals, electromagnetic fields) where\npressure actually *is* an-isotropic. We know from gravitational theory\n(exterior field of a static or rotating charge distribution - see the\nReissner-Nordstroem and the Kerr-Newman solutions), that anisotropic\npressure is not only possible, but highly probable up to the very largest\nscales: The exterior (electro-magnetic) field of a charged black hole\n(rotating or not) has an anisotropic stress-energy tensor! Furthermore,\nif we believe in what (some) string theorists want to tell us ("all matter\nis\nconstructed out of strings"), one must be prepared to accept that the\nassumption of isotropic pressure might fail already at the microscopic\nlevel (i.e. the level of "magnification", where we start to see the string\nnature of the particles). We might already be seeing this "failure" in\ntoday\'s particle-accelerators: The debris of a high energy-collision\ninvolving quarks or gluons is not distributed isotropically (for a single\nevent). Rather the particles produced in highly energetic collisions\nare collimated in jets.\n\n\n> Since the area of the loss cone tends to be inversely proportional to\n> the square of the redshift z, it is possible that a z of only 10 or 20\n> would be sufficent to model the Narayan et al observations. If z is\n> 100, it should be more than sufficent.\n\nI agree with your argument about the loss-cone.\n\n> More later perhaps if anyone is interested.\n\nLet\'s see what interest you (or I) can arouse.\n\nReevaluation of results based on "crumbling" assumptions is a painful\nprocess, a process most researchers are neither willing (nor capable)\nof performing. Such a process requires as one of its basic prerequisites,\nthat one knows *precisely* on what assumptions certain claims (such\nas "z>>2 is incompatible with GR") are based. Fact is, most of\nthe claims we utter, are not our own claims, but just mere repetitions\nof claims made by other (hopefully more knowledgable) scientists. If\nwe are honest (which we mostly are not, because we like to pretend to know\nmore than what we actually do), for most claims which are not\nour own we *don\'t* really know on what assumptions these claims\nare based. We just trust that the person who tells us (often just repeating\nwhat he has heard from X, who again repeats what he has read from Y etc.\netc.) is correct. I am sure, that Chandrasekhar knows on what\nassumptions his result z< 2 is based. But from personal experience I\nknow, that many others do not..\n\nI don\'t know from whom you got this particular claim that GR is\nincompatible with z >> 2. Most likely it is not your own claim. And\nclearly it is not the claim that Chandrasekhar wanted to make.\nUnfortunately such misleading claims are all too common. Ask e.g. what\nthe basic assumptions of the singularity theorems are and whether there\nis experimental and/or theoretical evidence supporting these basic\nassumptions and whether the situation changed with the recent discoveries\n(hint: "dark energy" does not obey the strong energy condition!)\n\nYou will be surprised at the poor quality of the answers. In fact, it is\nhighly probable that if you ask this question spontaneously, you will not\nget any decent answer at all.\n\nM.P.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"jgraber" <jgra@loc.gov> wrote in message
news:jgraber.1b73lo@physicsforums.com...
.....
> First of all, let me point out that the Narayan et al papers are all
> based on General Relativity, and hence assume that the hard surface can
> not be at a redshift greater than about unity, a result that is true in
> GR and goes all the wayback to Chandrasekhar and Bondi. In fact, the
> diagram in the most recent paper, http://www.arxiv.org/abs/astro-ph/0401549, cuts off at a
> redshift of z=.5.ie z equal one half. (Neutron star surfaces are
> usually calculated to be at redshifts of about .2 or in the range of .1
> to .3 ie about 10 to 30 percent.)
>
> On the contrary, I assume a surface redshift of z=100, a factor of 200
> to 1000 times larger, and hence assume that GR is incorrect, and that
> some alternate theory of gravity is correct.
Dear Jim,
it is an - albeit very common - misperception, that GR does not allow
redshifts higher than z \approx 2.
This is simply not true. All "proofs" for z<=2 require - besides the
field equations of general relativity - *additional* assumptions, such
as isotropic pressure, non-negative pressure etc.
In fact, one can see this already by a constructive argument:
There are exact solutions of the Einstein field equations
(with and without gravitational constant \Lambda) for spherically
symmetric compact bodies with (hard, material) surfaces , whose
surface redshift can become arbitrarily large (depending on the
mass of the body).
The holographic solution (see http://www.arxiv.org/abs/gr-qc/0306063, http://www.arxiv.org/abs/gr-qc/0306066) is one
such solution. It solves the unmodified field equations of GR (without
gravitational constant) exactly, but has a negative string-type
interior pressure. Its surface redshift scales with \sqrt{M / M_{Pl}}.
A "holostar" with the mass of the sun has a surface redshift of
roughly 10^20.
The gravastar - although somwhat awkward, because the "interior
part" of the solution solves the field equations *with* gravitational
constant, whereas the "exterior part" solves the field equations
*without* gravitational constant - is another such solution. The
gravastar's (matter-filled) interior deSitter core has a negative
isotropic pressure.
Therefore, if one is willing to accept
- negative pressures (dark energy = negative pressure is
experimentally verified by the recent supernova measurements!
So there should be no problem to accept the physical relevance
of negative pressure),
- or even negative *anisotropic* pressures (such as commonly
occurring in string theory: strings have tension = negative
pressure, but the tension acts in only one direction - namely
the direction of the string. The "transverse" pressure of a
classical string is zero),
compact objects with arbitrarily large surface redshifts are fully
compatible with GR.
GR in itself does not give any prescription what the matter-state
or the equation of state of a compact self-gravitating body should
be. GR can "live" with zero matter-density and zero pressure (the
classical black hole solutions), it can live with an anisotropic
string type pressure (such as domain-wall or string-type solutions,
or the recently discovered holographic solution), as well as it can
live with isotropic positive pressure (e.g. the radiation dominated
FRW-solution), or with an isotropic negative pressure (deSitter
solution, gravastars, the Lamdba CDM "standard" model of the
universe).
Keep in mind, that science is a continuous process of refining,
reviewing and adapting our knowledge to the most recent experimental
and theoretical findings. The "old" assumptions, which were reasonable
in the past, must not necessarily remain reasonable forever. The
assumptions of non-negative and isotropic pressure, leading to the
"prediction" z<2, were very reasonable assumptions regarding our state
of knowledge roughly 10 years ago. But recent experimental and
theoretical results not only suggest, they actually *require* a
re-evaluation of these "old" assumptions and therefore a re-evaluation
of ALL RESULTS BASED ON THESE ASSUMPTIONS.
<sorry for the "screaming", but this important and highly trivial fact
has not registered yet in the scientific community, which happily
continues to pour out (or mindlessly repeat) claims, which are based
on these assumptions.
I don't have anything against *assuming* positive or isotropic pressure,
but in view of the recent experimental and theoretical findings it is
mandatory to mention *all* assumptions on which a certain claim
relies.
My guess is, that almost everyone will agree that the "old" assumption
of non-negative pressure has been proven to be experimentally incorrect
by the recent supernova-measurements. Negative pressure (= tension)
dominating the large scale structure of the observable universe has
become known to the public under the somewhat obscure term "dark
energy". Note however, that negative pressure (tension) was quite
common even before this discovery: Everybody knows e.g. about surface
tension. Just look at some water-drops.
The case of isotropic vs. anisotropic pressure is not so clear. What is
clear however, that there is no general physical principle which requires
the pressure to be isotropic. In fact, we know for certain that there are
several real world situations (crystals, electromagnetic fields) where
pressure actually *is* an-isotropic. We know from gravitational theory
(exterior field of a static or rotating charge distribution - see the
Reissner-Nordstroem and the Kerr-Newman solutions), that anisotropic
pressure is not only possible, but highly probable up to the very largest
scales: The exterior (electro-magnetic) field of a charged black hole
(rotating or not) has an anisotropic stress-energy tensor! Furthermore,
if we believe in what (some) string theorists want to tell us ("all matter
is
constructed out of strings"), one must be prepared to accept that the
assumption of isotropic pressure might fail already at the microscopic
level (i.e. the level of "magnification", where we start to see the string
nature of the particles). We might already be seeing this "failure" in
today's particle-accelerators: The debris of a high energy-collision
involving quarks or gluons is not distributed isotropically (for a single
event). Rather the particles produced in highly energetic collisions
are collimated in jets.
> Since the area of the loss cone tends to be inversely proportional to
> the square of the redshift z, it is possible that a z of only 10 or 20
> would be sufficent to model the Narayan et al observations. If z is
> 100, it should be more than sufficent.
I agree with your argument about the loss-cone.
> More later perhaps if anyone is interested.
Let's see what interest you (or I) can arouse.
Reevaluation of results based on "crumbling" assumptions is a painful
process, a process most researchers are neither willing (nor capable)
of performing. Such a process requires as one of its basic prerequisites,
that one knows *precisely* on what assumptions certain claims (such
as "z>>2 is incompatible with GR") are based. Fact is, most of
the claims we utter, are not our own claims, but just mere repetitions
of claims made by other (hopefully more knowledgable) scientists. If
we are honest (which we mostly are not, because we like to pretend to know
more than what we actually do), for most claims which are not
our own we *don't* really know on what assumptions these claims
are based. We just trust that the person who tells us (often just repeating
what he has heard from X, who again repeats what he has read from Y etc.
etc.) is correct. I am sure, that Chandrasekhar knows on what
assumptions his result z< 2 is based. But from personal experience I
know, that many others do not..
I don't know from whom you got this particular claim that GR is
incompatible with z >> 2. Most likely it is not your own claim. And
clearly it is not the claim that Chandrasekhar wanted to make.
Unfortunately such misleading claims are all too common. Ask e.g. what
the basic assumptions of the singularity theorems are and whether there
is experimental and/or theoretical evidence supporting these basic
assumptions and whether the situation changed with the recent discoveries
(hint: "dark energy" does not obey the strong energy condition!)
You will be surprised at the poor quality of the answers. In fact, it is
highly probable that if you ask this question spontaneously, you will not
get any decent answer at all.
M.P.
Ilja Schmelzer
Sep6-04, 06:31 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"jgraber" <jgra@loc.gov> schrieb\n> On the contrary, I assume a surface redshift of z=100, a factor of 200\n> to 1000 times larger, and hence assume that GR is incorrect, and that\n> some alternate theory of gravity is correct.\n\nAs an example for such a theory of gravity see gr-qc/0205035.\n\nIlja\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"jgraber" <jgra@loc.gov> schrieb
> On the contrary, I assume a surface redshift of z=100, a factor of 200
> to 1000 times larger, and hence assume that GR is incorrect, and that
> some alternate theory of gravity is correct.
As an example for such a theory of gravity see http://www.arxiv.org/abs/gr-qc/0205035.
Ilja
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