short GR question [Archive]

Blake Winter

Oct22-04, 12:31 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>\n\nI was considering some curious possibilities in GR in which\ncommunication happens "apparantly" faster than light. Most of these\n(when you spin around, for example) are easily solved. But I had one\nsituation which I can\'t quite figure out:\n\nSay that you have a bunch of very small, but massive, little objects.\nVery near them, time will run more slowly than far away from them.\nAssume that their masses are small enough that one must be very, very\nclose to them in order to have a large effect on time, but that near\nthem time runs at some very small percentage of what it runs far away\nfrom them (from the perspective of an ideal observer at infinity).\nNow imagine two very small observers situated far away from each other\nalong this rod-like massive object, such that they are both inside the\nhighly time-dilated part of spacetime, maintaining their distance from\nthe rod with rockets. They will, of course, not be accelerating along\nthe rod, only perpendicular to it. Now, using fiber optics (or some\nother method: perhaps a set of mirrors, which amounts to basically the\nsame thing), one can send a light signal along the rod from one\nobserver to the other or one can send it outside of the slow time area\ninto the fast time area, then down the rod, then back to the observer.\nBecause they are seperated by light years, sending it outside the\narea will, it seems to me, take less time than sending it within the\nslow-time area, without requiring time inside to be very much slower\nthan outside. Therefore one can, it seems, propagate information\nfaster than one would expect using just the area with slower light.\nTherefore, an observer moving at a high velocity (near c) from one\nobserver to the other would encounter the tachyon paradox (although\ninterestingly his hypersurface of simultaneity would be _timelike_\noutside the slow time, if I understand the situation correctly!). So\nwhat is the answer to this seeming paradox?\n\nThe solution I came up with was that one must always consider the\nsmallest s^2 when doing time-ordering problems, and the fact that the\nmoving observer who sees the tachyon effect is using a timelike\nsurface of simultaneity in some regions invalidates his conclusions.\nIs this correct?\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 was considering some curious possibilities in GR in which
communication happens "apparantly" faster than light. Most of these
(when you spin around, for example) are easily solved. But I had one
situation which I can't quite figure out:

Say that you have a bunch of very small, but massive, little objects.
Very near them, time will run more slowly than far away from them.
Assume that their masses are small enough that one must be very, very
close to them in order to have a large effect on time, but that near
them time runs at some very small percentage of what it runs far away
from them (from the perspective of an ideal observer at infinity).
Now imagine two very small observers situated far away from each other
along this rod-like massive object, such that they are both inside the
highly time-dilated part of spacetime, maintaining their distance from
the rod with rockets. They will, of course, not be accelerating along
the rod, only perpendicular to it. Now, using fiber optics (or some
other method: perhaps a set of mirrors, which amounts to basically the
same thing), one can send a light signal along the rod from one
observer to the other or one can send it outside of the slow time area
into the fast time area, then down the rod, then back to the observer.
Because they are seperated by light years, sending it outside the
area will, it seems to me, take less time than sending it within the
slow-time area, without requiring time inside to be very much slower
than outside. Therefore one can, it seems, propagate information
faster than one would expect using just the area with slower light.
Therefore, an observer moving at a high velocity (near c) from one
observer to the other would encounter the tachyon paradox (although
interestingly his hypersurface of simultaneity would be _timelike_
outside the slow time, if I understand the situation correctly!). So
what is the answer to this seeming paradox?

The solution I came up with was that one must always consider the
smallest s^2 when doing time-ordering problems, and the fact that the
moving observer who sees the tachyon effect is using a timelike
surface of simultaneity in some regions invalidates his conclusions.
Is this correct?

tessel@tum.bot

Nov9-04, 03:44 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 Fri, 22 Oct 2004, Blake Winter wrote:\n\n> Say that you have a bunch of very small, but massive, little objects.\n> Very near them, time will run more slowly than far away from them.\n\nNo!-- this is -not- what gtr says at all. I don\'t have time to repeat\nsomething I and others have explained many times in the past, but you can\nlook for past posts on "gravitational time-dilation".\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, 22 Oct 2004, Blake Winter wrote:

> Say that you have a bunch of very small, but massive, little objects.
> Very near them, time will run more slowly than far away from them.

No!-- this is -not- what gtr says at all. I don't have time to repeat
something I and others have explained many times in the past, but you can
look for past posts on "gravitational time-dilation".

"T. Essel" (hiding somewhere in cyberspace)

Ben Rudiak-Gould

Jun21-05, 02:16 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>[crossposted to s.p.research]\n\nBill Hobba wrote:\n> My favorite theory, STM, is a theory\n> of 4D hpersurfaces in flat 5D space - the exact nature of the imbedding\n> leading to the observed properties of matter -\n\nThis general idea has always interested me. The 90+ figure I mentioned\n(based on previous posts to s.p.research) is an upper bound for an isometric\nembedding of an /arbitrary/ GR spacetime in a flat background. In some ways\na more interesting question is how many dimensions are needed to embed the\nobserved geometry of the visible universe. It would be interesting if the\nanswer turned out to be 4+1 instead of 87+3. If it\'s not 4+1, then STM would\npresumably be falsified right off the bat.\n\nWhat are the consequences of starting with GR and adding an embedding\nconstraint of this sort? Obviously it rules out closed timelike curves if\nthere\'s only one timelike dimension. Does it rule out exotic matter?\n\n-- Ben\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>[crossposted to s.p.research]

Bill Hobba wrote:
> My favorite theory, STM, is a theory
> of 4D hpersurfaces in flat 5D space - the exact nature of the imbedding
> leading to the observed properties of matter -

This general idea has always interested me. The 90+ figure I mentioned
(based on previous posts to s.p.research) is an upper bound for an isometric
embedding of an /arbitrary/ GR spacetime in a flat background. In some ways
a more interesting question is how many dimensions are needed to embed the
observed geometry of the visible universe. It would be interesting if the
answer turned out to be 4+1 instead of 87+3. If it's not 4+1, then STM would
presumably be falsified right off the bat.

What are the consequences of starting with GR and adding an embedding
constraint of this sort? Obviously it rules out closed timelike curves if
there's only one timelike dimension. Does it rule out exotic matter?

-- Ben

tessel@tum.bot

Jun24-05, 01:55 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>Bill Hobba remarked:\n\n>> My favorite theory, STM, is a theory of 4D hpersurfaces in flat 5D\n>> space - the exact nature of the imbedding leading to the observed\n>> properties of matter -\n\nSo your "hypersurfaces" are Lorentzian spacetimes isometrically embedded\nin E^(1,5), right? Locally or globally?\n\nBen Rudiak-Gould commented:\n\n> This general idea has always interested me. The 90+ figure I mentioned\n> (based on previous posts to s.p.research) is an upper bound for an\n> isometric embedding of an /arbitrary/ GR spacetime in a flat background.\n> In some ways a more interesting question is how many dimensions are\n> needed to embed the observed geometry of the visible universe.\n^^^^^^^^^^^^^^^^\n\nYou mean some gtr model more realistic than an FRW model, right?\n\nI hope you do -not- mean "the genuine universe in which we live". I am\nnot sure how much sense this phrase means in this context. After all, gtr\nis not a quantum theory, so ultimately cannot be "right", and "the genuine\nuniverse" might very well not be a Lorentzian spacetime at all, or even a\nsmooth manifold-- in fact, this is very unlikely, if you admit Wheeler\'s\nspeculations concerning "spacetime foam". Such objections arise even\n-before- we start to question the wisdom of actually -identifying- "the\nuniverse" with -any- mathematical model. (I prefer to leave it to\nphilosophers to fight over -that- issue!)\n\n> It would be interesting if the answer turned out to be 4+1 instead of\n> 87+3. If it\'s not 4+1, then STM would presumably be falsified right off\n> the bat.\n\nI seem to have missed the beginning of this thread, so forgive me if\nanyone already mentioned the following results.\n\nOn the local side, we have\n\nTheorem [Collison 1968]. There are Lorentzian manifolds which cannot be\neven -locally- isometrically embedded using fewer than two extra\ndimensions.\n\nThese include all nonnull electrovacuums, such as the Reissner/Nordstrom\nsolution. See section 37.4 of\n\nHans Stephani, Dietrich Kramer, Macolm MacCallum, Cornelius Hoensalaers,\nand Eduard Herlt,\nExact Solutions of Einstein\'s Field Equations. 2nd Ed.\nCambridge University Press, 2003.\n\nOn the global side, any spacetime with closed timelike curves, such as\nKerr vacuum, cannot be globally embedded in any E^(1,n).\n\nA famous result of Penrose says that the "Penrose pulse", a kind of\nlimiting case of certain "sandwich waves"--- which are gravitational plane\nwaves and thus exact vacuum solutions of the EFE--- cannot be globally\nisometrically embedded in any E^(1,n). The point here is that this is\ntrue even though pp waves do -not- contain any closed timelike curves:\n\nRoger Penrose,\n"A Remarkable Property of Plane Waves in General Relativity",\nRev. Mod. Phy 11 (1965): 215\n\nThe Penrose pulse is not quite a smooth manifold, but there are exact\ngravitational plane waves which have the same property. Basically, in the\nharmonic chart (aka Brinkmann chart), the "light cones" for the basic plus\npolarized sinusoidal gravitational plane wave look like something like\nthis\n\n...\n____|_____\n****|\n____|_____\n |*****\n____|_____\n****|\n____|_____\n|*****\n... \n\nThis diagram is periodic in the vertical direction. I should probably add\nthat if you squint your eyes, the usual light cone in E^(1,3) looks like\n\n|\n____|_____\n|*****\n|\n...\n\nDo you see it now? :-)\n\nWe have an infinite sequence of expansion and recollapse cycles for the\nforward null geodesics issuing from a given event. Penrose\'s point is that\nthis kind of reconvergence is incompatible with the possibility of global\nisometric embedding in any E^(1,n).\n\n> What are the consequences of starting with GR and adding an embedding\n> constraint of this sort? Obviously it rules out closed timelike curves\n> if there\'s only one timelike dimension.\n\nObviously :-)\n\nThe section of SKMHH cited above concerns the relationship between\nembedding and EFE, so this should be a good start. I think it is fair to\nsay that the short answer to your question is that nothing really decisive\n(or stunningly memorable) seems to be known yet, but you will find here\nsome theorems on certain types of fluid solutions which are known have\nsmall "embedding class".\n\n"T. Essel"\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>Bill Hobba remarked:

>> My favorite theory, STM, is a theory of 4D hpersurfaces in flat 5D
>> space - the exact nature of the imbedding leading to the observed
>> properties of matter -

So your "hypersurfaces" are Lorentzian spacetimes isometrically embedded
in E^(1,5), right? Locally or globally?

Ben Rudiak-Gould commented:

> This general idea has always interested me. The 90+ figure I mentioned
> (based on previous posts to s.p.research) is an upper bound for an
> isometric embedding of an /arbitrary/ GR spacetime in a flat background.
> In some ways a more interesting question is how many dimensions are
> needed to embed the observed geometry of the visible universe.
^^^^^^^^^^^^^^^^

You mean some gtr model more realistic than an FRW model, right?

I hope you do -not- mean "the genuine universe in which we live". I am
not sure how much sense this phrase means in this context. After all, gtr
is not a quantum theory, so ultimately cannot be "right", and "the genuine
universe" might very well not be a Lorentzian spacetime at all, or even a
smooth manifold-- in fact, this is very unlikely, if you admit Wheeler's
speculations concerning "spacetime foam". Such objections arise even
-before- we start to question the wisdom of actually -identifying- "the
universe" with -any- mathematical model. (I prefer to leave it to
philosophers to fight over -that- issue!)

> It would be interesting if the answer turned out to be 4+1 instead of
> 87+3. If it's not 4+1, then STM would presumably be falsified right off
> the bat.

I seem to have missed the beginning of this thread, so forgive me if
anyone already mentioned the following results.

On the local side, we have

Theorem [Collison 1968]. There are Lorentzian manifolds which cannot be
even -locally- isometrically embedded using fewer than two extra
dimensions.

These include all nonnull electrovacuums, such as the Reissner/Nordstrom
solution. See section 37.4 of

Hans Stephani, Dietrich Kramer, Macolm MacCallum, Cornelius Hoensalaers,
and Eduard Herlt,
Exact Solutions of Einstein's Field Equations. 2nd Ed.
Cambridge University Press, 2003.

On the global side, any spacetime with closed timelike curves, such as
Kerr vacuum, cannot be globally embedded in any E^(1,n).

A famous result of Penrose says that the "Penrose pulse", a kind of
limiting case of certain "sandwich waves"--- which are gravitational plane
waves and thus exact vacuum solutions of the EFE--- cannot be globally
isometrically embedded in any E^(1,n). The point here is that this is
true even though pp waves do -not- contain any closed timelike curves:

Roger Penrose,
"A Remarkable Property of Plane Waves in General Relativity",
Rev. Mod. Phy 11 (1965): 215

The Penrose pulse is not quite a smooth manifold, but there are exact
gravitational plane waves which have the same property. Basically, in the
harmonic chart (aka Brinkmann chart), the "light cones" for the basic plus
polarized sinusoidal gravitational plane wave look like something like
this

...
__{__}|__{___}
****|
__{__}|__{___}
|*****
__{__}|__{___}
****|
__{__}|__{___}
|*****
...

This diagram is periodic in the vertical direction. I should probably add
that if you squint your eyes, the usual light cone in E^(1,3) looks like

|
__{__}|__{___}
|*****
|
...

Do you see it now? :-)

We have an infinite sequence of expansion and recollapse cycles for the
forward null geodesics issuing from a given event. Penrose's point is that
this kind of reconvergence is incompatible with the possibility of global
isometric embedding in any E^(1,n).

> What are the consequences of starting with GR and adding an embedding
> constraint of this sort? Obviously it rules out closed timelike curves
> if there's only one timelike dimension.

Obviously :-)

The section of SKMHH cited above concerns the relationship between
embedding and EFE, so this should be a good start. I think it is fair to
say that the short answer to your question is that nothing really decisive
(or stunningly memorable) seems to be known yet, but you will find here
some theorems on certain types of fluid solutions which are known have
small "embedding class".

"T. Essel"