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, 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 with\n> difference equations which approximated the differential equations of\n> GR, and used them to demonstrate the behave of GR.\n\nHmm... I\'ve seen many books on gtr, but this doesn\'t ring a bell. Does\nanyone have a full citation?\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 have\n> the exponential form, tending to the Schwarzchild metric as the step\n> size tended to zero.\n\nYes, there is certainly something to explain there.\n\nWhat happens when you try to obtain the ordinary 2 dimensional Laplace\nequation as the continuum limit of the obvious choice for "the"\nanalogous difference equation? The 3 dimensional axisymmetric Laplace\nequation?\n\n> There was a footnote to the effect that, this is only intended to be an\n> approximation, and there\'s no evidence otherwise but maybe, 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\nThe "Regge calculus" is not the same thing, but does offer a\n"discretization" of the EFE. In this approach, we start with a simplicial\napproximation to the underlying manifold M, and add geometric structure by\na four dimensional analog of assigning "curvature deficits" to each vertex\nin a two dimensional triangulation of a manifold like the ordinary sphere\nS^2. We then reformulate the EFE in terms of this combinatorial data.\nThis is an old idea, but with digital computers becoming more powerful,\nthere has recently been renewed interest in exploring its virtues/defects.\nSee this review paper:\n\nauthor = {Adrian P. Gentle},\ntitle = {{R}egge calculus: a unique tool for numerical relativity},\njournal = {Gen. Rel. Grav.},\nvolume = 34,\nyear = 2002,\npages = {1701--1718},\nnote = {gr-qc/0408006}}\n\nI didn\'t understand "maybe, just maybe" though--- maybe I don\'t understand\ncorrectly your goals here?\n\n> What are the biggest problems with such a theory?\n\n[A difference equation analog of the EFE]\n\nAssuming your only goal is to facilitate numerical simulations of -gtr-?\nIf so, getting the wrong metric in the continuum limit would obviously be\na major problem, especially if the modification drastically changes the\ncausal structure! But this might only mean that the authors of the book\nyou saw took the limit incorrectly, or were using an incorrect choice for\n"the" analogous difference equation, as in the example from soliton theory\nwhich I mentioned recently (c.f. the derivation of the KdV from the\nwell-known Fermi-Pasta-Ulam lattice).\n\nAs you may know, "numerical relativity" is a major industry. Many\ncompeting formalisms have been introduced (most of these are variants of\nthe well-known ADM initial value formulation of the EFE). On the ArXiV,\nyou can find several papers reviewing recent progress in this field.\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, 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 behave of GR.
Hmm... I've seen many books on gtr, but this doesn't ring a bell. Does
anyone have a full citation?
> 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?
> 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?
The "Regge calculus" is not the same thing, but does offer a
"discretization" of the EFE. In this approach, we start with a simplicial
approximation to the underlying manifold M, and add geometric structure by
a four dimensional analog of assigning "curvature deficits" to each vertex
in a two dimensional triangulation of a manifold like the ordinary sphere
S^2. We then reformulate the EFE in terms of this combinatorial data.
This is an old idea, but with digital computers becoming more powerful,
there has recently been renewed interest in exploring its virtues/defects.
See this review paper:
author = {Adrian P. Gentle},
title = {{R}egge calculus: a unique tool for numerical relativity},
journal = {Gen. Rel. Grav.},
volume = 34,
year = 2002,
pages = {1701--1718},
note = {http://www.arxiv.org/abs/gr-qc/0408006}}
I didn't understand "maybe, just maybe" though--- maybe I don't understand
correctly your goals here?
> What are the biggest problems with such a theory?
[A difference equation analog of the EFE]
Assuming your only goal is to facilitate numerical simulations of -gtr-?
If so, getting the wrong metric in the continuum limit would obviously be
a major problem, especially if the modification drastically changes the
causal structure! 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).
As you may know, "numerical relativity" is a major industry. Many
competing formalisms have been introduced (most of these are variants of
the well-known ADM initial value formulation of the EFE). On the ArXiV,
you can find several papers reviewing recent progress in this field.
"T. Essel" (hiding somewhere in cyberspace)
> 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.
Hmm... I've seen many books on gtr, but this doesn't ring a bell. Does
anyone have a full citation?
> 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?
> 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?
The "Regge calculus" is not the same thing, but does offer a
"discretization" of the EFE. In this approach, we start with a simplicial
approximation to the underlying manifold M, and add geometric structure by
a four dimensional analog of assigning "curvature deficits" to each vertex
in a two dimensional triangulation of a manifold like the ordinary sphere
S^2. We then reformulate the EFE in terms of this combinatorial data.
This is an old idea, but with digital computers becoming more powerful,
there has recently been renewed interest in exploring its virtues/defects.
See this review paper:
author = {Adrian P. Gentle},
title = {{R}egge calculus: a unique tool for numerical relativity},
journal = {Gen. Rel. Grav.},
volume = 34,
year = 2002,
pages = {1701--1718},
note = {http://www.arxiv.org/abs/gr-qc/0408006}}
I didn't understand "maybe, just maybe" though--- maybe I don't understand
correctly your goals here?
> What are the biggest problems with such a theory?
[A difference equation analog of the EFE]
Assuming your only goal is to facilitate numerical simulations of -gtr-?
If so, getting the wrong metric in the continuum limit would obviously be
a major problem, especially if the modification drastically changes the
causal structure! 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).
As you may know, "numerical relativity" is a major industry. Many
competing formalisms have been introduced (most of these are variants of
the well-known ADM initial value formulation of the EFE). On the ArXiV,
you can find several papers reviewing recent progress in this field.
"T. Essel" (hiding somewhere in cyberspace)