ebunn@lfa221051.richmond.edu
Jun12-04, 07:21 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>In article <3E6yc.5344\\$Y3.3907@newsread2.news.atl.earthlink .net>,\nDavid Park <djmp@earthlink.net> wrote:\n>\n>Foster & Nightingale have a problem in their text "A Short Course in General\n>Relativity" that concerns spherical masses that have just attained the\n>radius to become black holes. They then want to compare the densities of the\n>two objects. If the object has the mass m, they say its density is (at least\n>proportional to) m/m^3 = 1/m^2. So the larger mass has the smaller density.\n>\n>But how would one actually calculate the volume of such a spherical region?\n>What metric would be used. Is there actually a finite volume? What justifies\n>simply using r in a flat space metric?\n>\n>Is density of a black hole a meaningful concept?\n\nI haven\'t studied this book in detail, so I don\'t know exactly what\nthey\'re getting at, but I do know that many descriptions of black\nholes contain similar descriptions of the density. They often seem to\nmean the mass divided by the volume, with the volume given by (4/3) pi\n(schwarzschild radius)^3. You\'re quite right that this is not a\nmeaningful thing to do. The interior of a Schwarschild black hole is\nnot a spacelike 3-volume, so it doesn\'t have a volume.\n\nFrom your description, it sounds like Foster & Nightingale are talking\nabout an object that has not yet become a black hole. In that case,\none can imagine that there might be sensible ways to define a\nspacelike 3-volume that you might want to call the interior of the\nobject at an instant. The volume of that thing would be well-defined,\nalthough there\'s no reason to expect it to be (4/3) pi r^3.\n\nIf you had a family of such objects of different sizes, and you caught\nthem all at the same stage of their evolution, so that their\ngeometries were all rescaled versions of each other, then it is true\nthat their volumes (however you defined them) would be proportional to\nthe cubes of their radii (in whatever reasonable coordinates you were\nusing). So the scaling relationship\n\n(Average density of something that\'s about to become a black hole)\nis proportional to 1/m^2\n\ncan be thought of as meaningful and correct in some sense.\n\n-Ted\n\n--\n[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]\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 <3E6yc.5344$Y3.3907@newsread2.news.atl.earthlink.ne t>,
David Park <djmp@earthlink.net> wrote:
>
>Foster & Nightingale have a problem in their text "A Short Course in General
>Relativity" that concerns spherical masses that have just attained the
>radius to become black holes. They then want to compare the densities of the
>two objects. If the object has the mass m, they say its density is (at least
>proportional to) m/m^3 = 1/m^2. So the larger mass has the smaller density.
>
>But how would one actually calculate the volume of such a spherical region?
>What metric would be used. Is there actually a finite volume? What justifies
>simply using r in a flat space metric?
>
>Is density of a black hole a meaningful concept?
I haven't studied this book in detail, so I don't know exactly what
they're getting at, but I do know that many descriptions of black
holes contain similar descriptions of the density. They often seem to
mean the mass divided by the volume, with the volume given by (4/3) \pi
(schwarzschild radius)^3. You're quite right that this is not a
meaningful thing to do. The interior of a Schwarschild black hole is
not a spacelike 3-volume, so it doesn't have a volume.
From your description, it sounds like Foster & Nightingale are talking
about an object that has not yet become a black hole. In that case,
one can imagine that there might be sensible ways to define a
spacelike 3-volume that you might want to call the interior of the
object at an instant. The volume of that thing would be well-defined,
although there's no reason to expect it to be (4/3) \pi r^3.
If you had a family of such objects of different sizes, and you caught
them all at the same stage of their evolution, so that their
geometries were all rescaled versions of each other, then it is true
that their volumes (however you defined them) would be proportional to
the cubes of their radii (in whatever reasonable coordinates you were
using). So the scaling relationship
(Average density of something that's about to become a black hole)
is proportional to 1/m^2
can be thought of as meaningful and correct in some sense.
-Ted
--
[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]
tessel@tum.bot
Jun16-04, 03:42 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, 11 Jun 2004, David Park wrote:\n\n> Foster & Nightingale have a problem in their text "A Short Course in\n> General Relativity" that concerns spherical masses that have just\n> attained the radius to become black holes. They then want to compare the\n> densities of the two objects. If the object has the mass m, they say its\n> density is (at least proportional to) m/m^3 = 1/m^2. So the larger mass\n> has the smaller density.\n\nI don\'t think this problem makes sense out of context---- ideally, you\nshould restate the problem -exactly- as the authors give it!\n(Unfortunately I myself am about to go AFK for a week or more, so I might\nnot see your response.)\n\nBut I can say this: the intent of the problem must be to provide some\nintuition for the well-known fact that the strength of the gravitational\nfield of a hole "at" the horizon, as defined operationally in various\nways, -decreases- as the mass -increases-. There are several other ways\nof getting at this idea, including these:\n\n1. wrt a suitable "frame", compute the acceleration of a static observer\n\nD_X X = M/r^2 1/sqrt(1-2M/r) e_2\n\nevaluate at r = 3M (the radius of the circular orbits of laser pulses) or\nr = 6M (the radius of the minimal possible circular orbit of a test\nparticle), and consider how your results scale with M,\n\n2. compute the "tidal tensor" (aka the "electrogravitic tensor" in the\n"Bel" decomposition of the Riemann tensor)\n\nE(X)_(ab) = R_(ambn) X^m X^n\n\n"=" M/r^3 diag (-2,1,1)\n\non an object falling freely and radially into the hole (say, from rest at\nspatial infinity), with world line given by the vector field X, evaluate\nat r = 2M, and consider how your result scales with M.\n\n3. look up the definition and interpretation of the "surface gravity" of a\nSchwarzschild hole in this textbook:\n\nauthor = {Sean Carroll},\ntitle = {Spacetime and geometry: an introduction to general relativity},\npublisher = {Addison-Wesley},\nyear = 2004}\n\nand consider how this quantity scales with M.\n\nBTW, I have looked at Foster and Nightingale, although not recently, and\nit has some nice features, but for a first look at gtr I\'d recommend\ninstead Carroll\'s book or any of these three:\n\nauthor = {Bernard F. Schutz},\ntitle = {A First Course in General Relativity},\npublisher = {Cambridge University Press},\nyear = 1985}\n\nauthor = {Hans Stephani},\ntitle = {General Relativity: An Introduction of the Theory of the\nGravitational Field},\npublisher = {Cambridge University Press},\nedition = {Second},\nnote = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},\nyear = 1990}\n\nauthor = {Ray D\'Inverno},\ntitle = {Introducing {E}instein\'s Relativity},\npublisher = {Clarendon Press},\nyear = 1995}\n\nDepending on your interests, one of these might stand out; e.g. for\n-linearized- gravitational waves, Schutz or Carroll is probably best, but\n(AFAIK) only Stephani mentions the important -exact- gravitational "plane\nwaves" (or more correctly, "pp waves"). You might try the comparison\nchart on the website Relativity on the World Wide Web, which you can find\nat John Baez\'s website (but Carroll\'s book was published after the last\nupdate of that site, so it isn\'t mentioned there, but see the list of\n"on-line course notes").\n\n> But how would one actually calculate the volume of such a spherical\n> region? What metric would be used. Is there actually a finite volume?\n> What justifies simply using r in a flat space metric?\n\nYou are of course correct that Schwarzschild holes do not have, in any\nobvious sense, a well-defined "volume of space contained inside the\nhorizon". (See the very clear discussion of this point in the classic\ngraduate textbook by Misner, Thorne, and Wheeler, Gravitation, Freeman,\n1973.) Clearly, the intent of the problem cannot involve "volume"!\n\nAlso of course, a black hole cannot coexist with another massive object\nwithout its geometry (in particular, the geometrical "shape" of their\nhorizons) becoming distorted by the gravitational effect of the other\nobject. Clearly, the intent of the problem cannot involve the interaction\nof a black hole with a massive object (such another black hole), either!\n\nBTW, there is another very important point connected with the expression\n\nD_X X = M/r^2 1/sqrt(1-2M/r) e_2 (*)\n\nThis point has nothing to do with black holes per se; rather, it is a\nfundamental point concerning the consequences of the fact that gtr is a\n-nonlinear- theory for the notion of "parameters characterizing a physical\nfield". To wit: in a static spacetime which arises as an exact solution\nof the EFE, say an "electrovacuum solution", if we define with sufficient\ncare what it means for a static observer to have a\ngeometrically/physically meaningful "radius" ("position" wrt the source of\nthe field), then we can contemplate thought experiments in which we\nincrease the mass or charge parameters of the object modelled by the\nsolution, by dropping in more matter/charges, and compare "before" and\n"after" values of the physical fields as measured by our static observer.\nThen, these fields will in general be found to scale -nonlinearly- with\nthe mass or charge parameters: doubling the mass more than doubles the\ngravitational force at a given (geometrically/physically meaningful)\n"radius". Only in the limit r0 -> infty do we recover our usual intuition\nabout gravitostatic or electrostatic fields scaling linearly (m/r^2, q/r^2\nrespectively) with mass or charge, and inverse square wrt distance!\n\nIf you know about\n\n1. Weyl\'s family of exact solutions comprising all static axisymmetric\nvacuum solutions of the EFE, including the Schwarzschild vacuum, to wit:\n\nds^2 = -exp(-2u) dt^2 + exp(2u) [exp(2v) (dz^2+r^2) + r^2 dphi^2],\n\n-infty < t,z < infty, r0 < r < infty, -pi < phi < pi\n\nu, v are functions of the coordinates z, r only\n\nu_(zz) + u_(rr) + u_r/r = 0 (u is axisymmetric harmonic function)\n\nv_r = r (u_r^2 - u_z^2)\n\nv_z = 2 r u_r u_z\n\n2. the symmetry analysis of systems of nonlinear PDEs,\n\nthen a very good exercise is this:\n\n(a) Discuss the physical/geometric meaning of the metric functions u,v,\n\n(b) Compute the point symmetry group of the Weyl system of PDEs, given\nabove, which defines his solutions,\n\n(c) Discuss the physical/geometric meaning of the transformations\ncomprising this point symmetry group.\n\nSimilarly for the Weyl-Gordon massless minimally coupled field solutions\n(including e.g. Winacour solution), the Weyl-Maxwell electrovacuums (all\nstatic axisymmetric electrovacuum solutions, including the\nReissner-Nordstrom solution), the Beck vacuums, etc. and even, with more\nthought, rotating generalizations such as the Ernst vacuums (all\nstationary axisymmetric vacuum solutions, including the Kerr solution).\n\nFor symmetry analysis of DEs, see\n\nauthor = {Brian J. Cantwell},\ntitle = {Introduction to symmetry analysis},\npublisher = {Cambridge University Press},\nyear = 2002}\n\nauthor = {Bluman, George W., and Kumei, Sukeyuki},\ntitle = {Symmetries and Differential Equations},\nseries = {Applied mathematical sciences},\nvolume =81,\npublisher = {Springer-Verlag},\nyear = {1989}}\n\nauthor = {Peter J. Olver},\ntitle = {Applications of {L}ie Groups to Differential Equations},\nseries = {Graduate Texts in Mathematics},\nvolume = 107,\npublisher = {Springer-Verlag},\nyear = 1993}\n\nFor exact solutions of the EFE, including the Weyl and Ernst families, see\n\nauthor = {Jiri Bic\\\'ak},\ntitle = {Selected solutions of {E}instein\'s field equations:\ntheir role in general relativity and astrophysics},\nbooktitle = {{E}instein Field Equations and Their Physical Implications\n(Selected essays in honour of {J}uergen {E}hlers)},\neditor = {Bernd G. Schmidt},\npublisher = {Springer-Verlag},\nyear = 2000,\nseries = {Lecture Notes in Physics},\nvolume = 540,\nnote = {gr-qc/0004016}}\n\nauthor = {J\\"urgen Ehlers and Wolfgang Kundt},\ntitle = {Exact Solutions of the Gravitational Field Equations},\nbooktitle = {Gravitation: an Introduction to Current Research},\neditor = {Louis Witten},\npublisher = {Wiley},\nyear = 1962,\npages = {49--101}}\n\nauthor = {D. Kramer and H. Stephani and E. Herlt and M. MacCallum},\ntitle = {Exact Solutions of {E}instein\'s Field Equations},\npublisher = {Cambridge University Press},\nedition = {second},\nseries = {Cambridge monographs on mathematical physics},\nvolume = 6,\nyear = 2003}\n\n> Is density of a black hole a meaningful concept?\n\nThis is actually discussed in the FAQ; see "is the Universe a Black\nHole?".\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 Fri, 11 Jun 2004, David Park wrote:
> Foster & Nightingale have a problem in their text "A Short Course in
> General Relativity" that concerns spherical masses that have just
> attained the radius to become black holes. They then want to compare the
> densities of the two objects. If the object has the mass m, they say its
> density is (at least proportional to) m/m^3 = 1/m^2. So the larger mass
> has the smaller density.
I don't think this problem makes sense out of context---- ideally, you
should restate the problem -exactly- as the authors give it!
(Unfortunately I myself am about to go AFK for a week or more, so I might
not see your response.)
But I can say this: the intent of the problem must be to provide some
intuition for the well-known fact that the strength of the gravitational
field of a hole "at" the horizon, as defined operationally in various
ways, -decreases- as the mass -increases-. There are several other ways
of getting at this idea, including these:
1. wrt a suitable "frame", compute the acceleration of a static observer
D_X X = M/r^2 1/\sqrt(1-2M/r) e_2
evaluate at r = 3M (the radius of the circular orbits of laser pulses) or
r = 6M (the radius of the minimal possible circular orbit of a test
particle), and consider how your results scale with M,
2. compute the "tidal tensor" (aka the "electrogravitic tensor" in the
"Bel" decomposition of the Riemann tensor)
E(X)_(ab) = R_(ambn) X^m X^n"=" M/r^3[/itex] diag (-2,1,1)
on an object falling freely and radially into the hole (say, from rest at
spatial infinity), with world line given by the vector field X, evaluate
at r = 2M, and consider how your result scales with M.
3. look up the definition and interpretation of the "surface gravity" of a
Schwarzschild hole in this textbook:
author = {Sean Carroll},
title = {Spacetime and geometry: an introduction to general relativity},
publisher = {Addison-Wesley},
year = 2004}
and consider how this quantity scales with M.
BTW, I have looked at Foster and Nightingale, although not recently, and
it has some nice features, but for a first look at gtr I'd recommend
instead Carroll's book or any of these three:
author = {Bernard F. Schutz},
title = {A First Course in General Relativity},
publisher = {Cambridge University Press},
year = 1985}
author = {Hans Stephani},
title = {General Relativity: An Introduction of the Theory of the
Gravitational Field},
publisher = {Cambridge University Press},
edition = {Second},
note = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},
year = 1990}
author = {Ray D'Inverno},
title = {Introducing {E}instein's Relativity},
publisher = {Clarendon Press},
year = 1995}
Depending on your interests, one of these might stand out; e.g. for
-linearized- gravitational waves, Schutz or Carroll is probably best, but
(AFAIK) only Stephani mentions the important -exact- gravitational "plane
waves" (or more correctly, "pp waves"). You might try the comparison
chart on the website Relativity on the World Wide Web, which you can find
at John Baez's website (but Carroll's book was published after the last
update of that site, so it isn't mentioned there, but see the list of
"on-line course notes").
> But how would one actually calculate the volume of such a spherical
> region? What metric would be used. Is there actually a finite volume?
> What justifies simply using r in a flat space metric?
You are of course correct that Schwarzschild holes do not have, in any
obvious sense, a well-defined "volume of space contained inside the
horizon". (See the very clear discussion of this point in the classic
graduate textbook by Misner, Thorne, and Wheeler, Gravitation, Freeman,
1973.) Clearly, the intent of the problem cannot involve "volume"!
Also of course, a black hole cannot coexist with another massive object
without its geometry (in particular, the geometrical "shape" of their
horizons) becoming distorted by the gravitational effect of the other
object. Clearly, the intent of the problem cannot involve the interaction
of a black hole with a massive object (such another black hole), either!
BTW, there is another very important point connected with the expression
D_X X = M/r^2 1/\sqrt(1-2M/r) e_2 (*)
This point has nothing to do with black holes per se; rather, it is a
fundamental point concerning the consequences of the fact that gtr is a
-nonlinear- theory for the notion of "parameters characterizing a physical
field". To wit: in a static spacetime which arises as an exact solution
of the EFE, say an "electrovacuum solution", if we define with sufficient
care what it means for a static observer to have a
geometrically/physically meaningful "radius" ("position" wrt the source of
the field), then we can contemplate thought experiments in which we
increase the mass or charge parameters of the object modelled by the
solution, by dropping in more matter/charges, and compare "before" and
"after" values of the physical fields as measured by our static observer.
Then, these fields will in general be found to scale -nonlinearly- with
the mass or charge parameters: doubling the mass more than doubles the
gravitational force at a given (geometrically/physically meaningful)
"radius". Only in the limit r0 -> \infty do we recover our usual intuition
about gravitostatic or electrostatic fields scaling linearly (m/r^2, q/r^2
respectively) with mass or charge, and inverse square wrt distance!
If you know about
1. Weyl's family of exact solutions comprising all static axisymmetric
vacuum solutions of the EFE, including the Schwarzschild vacuum, to wit:
ds^2 = -\exp(-2u) dt^2 + \exp(2u) [\exp(2v) (dz^2+r^2) + r^2 dphi^2],-\infty < t,z < \infty, r0 < r < \infty, -\pi < \phi < \pi
u, v are functions of the coordinates z, r only
u_(zz) + u_(rr) + u_r/r = (u is axisymmetric harmonic function)
[itex]v_r = r (u_r^2 - u_z^2)v_z = 2 r u_r u_z
2. the symmetry analysis of systems of nonlinear PDEs,
then a very good exercise is this:
(a) Discuss the physical/geometric meaning of the metric functions u,v,
(b) Compute the point symmetry group of the Weyl system of PDEs, given
above, which defines his solutions,
(c) Discuss the physical/geometric meaning of the transformations
comprising this point symmetry group.
Similarly for the Weyl-Gordon massless minimally coupled field solutions
(including e.g. Winacour solution), the Weyl-Maxwell electrovacuums (all
static axisymmetric electrovacuum solutions, including the
Reissner-Nordstrom solution), the Beck vacuums, etc. and even, with more
thought, rotating generalizations such as the Ernst vacuums (all
stationary axisymmetric vacuum solutions, including the Kerr solution).
For symmetry analysis of DEs, see
author = {Brian J. Cantwell},
title = {Introduction to symmetry analysis},
publisher = {Cambridge University Press},
year = 2002}
author = {Bluman, George W., and Kumei, Sukeyuki},
title = {Symmetries and Differential Equations},
series = {Applied mathematical sciences},
volume =81,
publisher = {Springer-Verlag},
year = {1989}}
author = {Peter J. Olver},
title = {Applications of {L}ie Groups to Differential Equations},
series = {Graduate Texts in Mathematics},
volume = 107,
publisher = {Springer-Verlag},
year = 1993}
For exact solutions of the EFE, including the Weyl and Ernst families, see
author = {Jiri Bic\'ak},
title = {Selected solutions of {E}instein's field equations:
their role in general relativity and astrophysics},
booktitle = {{E}instein Field Equations and Their Physical Implications
(Selected essays in honour of {J}uergen {E}hlers)},
editor = {Bernd G. Schmidt},
publisher = {Springer-Verlag},
year = 2000,
series = {Lecture Notes in Physics},
volume = 540,
note = {http://www.arxiv.org/abs/gr-qc/0004016}}
author = {J\"urgen Ehlers and Wolfgang Kundt},
title = {Exact Solutions of the Gravitational Field Equations},
booktitle = {Gravitation: an Introduction to Current Research},
editor = {Louis Witten},
publisher = {Wiley},
year = 1962,
pages = {49--101}}
author = {D. Kramer and H. Stephani and E. Herlt and M. MacCallum},
title = {Exact Solutions of {E}instein's Field Equations},
publisher = {Cambridge University Press},
edition = {second},
series = {Cambridge monographs on mathematical physics},
volume = 6,
year = 2003}
> Is density of a black hole a meaningful concept?
This is actually discussed in the FAQ; see "is the Universe a Black
Hole?".
"T. Essel" (hiding somewhere in cyberspace)
vBulletin® v3.8.7, Copyright ©2000-2012, vBulletin Solutions, Inc.