Understanding Perturbation Theory of GR [Archive]

Bill Kavanagh

Nov3-04, 09:54 AM

tessel@tum.bot

Nov9-04, 03:45 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, 3 Nov 2004, Bill Kavanagh wrote:\n\n> I am currently trying to learn the basics (and beyond) of Perturbation\n> Theory in General Relativity for which there doesn\'t seem to be any\n> really good sources.\n\nI\'ve posted on this very extensively before, so I\'ll be brief.\n\nYou didn\'t specify your mathematical background, so please forgive me if I\nam incorrect in assuming (on the basis of certain small clues) that you do\nnot have a strong modern undergraduate background in what is often called\n"mathematical methods".\n\nOne essential point to understand is that perturbation theory is a body of\nideas which applies not just to the problem of solving the EFE but to\ndifferential equations in general, and indeed to such deceptively simple\ntopics as locating the roots of a univariate polynomial. These ideas are\namong the most "applicable" in all of "applied mathematics", so everyone\nshould know them!\n\nSee for example the chapter on perturbation theory in\n\nauthor = {Richards, Derek},\ntitle = {Advanced mathematical methods with {M}aple},\npublisher = {Cambridge University Press},\nyear = 2002}\n\nCan you see how to apply the root location techniques explained by\nRichards in general relativity?\n\nExercise: review a textbook discussion of the horizons in Schwarzschild\nvacuum, then look up Schwarzschild-de Sitter and solve the analogous\nproblem.\n\nNext, can you see how standard techniques for perturbing solutions of an\nODE, also explained by Richards, are employed in the derivation of the\nextra-Newtonian precession of Mercury?\n\nExercise: review a textbook discussion, then repeat using a suitable chart\nfor the static weak-field monopole solution. (E.g., polar spherical, then\nCartesian.) Now try to extend your result to a static isolated object\nwith nonzero monopole and quadrupole moments. Can you explain the\nprecession of Mercury using hypothetical solar oblateness? If so, can you\nexplain the precession of -both- Mercury and Venus this way? Hint:\nperturbation techniques you have learned above should help in comparing\nhow the monopole and quadrupole effects scale with distance.\n\n(If you get stuck, look for past posts to this group where these two\nexercises are worked in detail. The second exercise is taken from a\nproblem from the problem book by Press et al.)\n\n> My current understanding of perturbation theory\n\nsay rather something like "metric perturbations of a Lorentzian manifold"\n\n> is that we take a metric and add some small terms to the metric\n> components and then put this into Einstein\'s equations and through away\n> higher order terms. I\'ve studied the weak-field (g = \\eta + h ; where\n> h<<1) equations which result from a perturbation off Minkowski. Of the\n> things I don\'t fully understand, the idea that h is a background\n> \'tensor\' and the use of gauge transformations are the most critical.\n\nActually, eta_(ab) is the "background metric" and h_(ab) is the\nperturbation.\n\n> Despite these details though, I am now trying to understand more\n> complicated perturbations. The perturbations of the Schwarzschild\n> Black-Hole in chapter 4 of S. Chandrasekhar\'s "The Mathematical Theory\n> of Black Holes" (1983) is what I have chosen to look at (it is\n> relevant in some way to my research).\n\nYou certainly do not lack in audacity! :-/\n\nBut courage is only one of the prerequisites for this particular book...\n\n> I run into trouble immediately when he talks about modes of\n> perturbations. What is a mode of perturbation? (i.e. a\n> non-axisymmetric mode of perturbation with an e^(im\\phi) dependence)\n> He remarks that restricting to time-dependant axisymmetric modes can be\n> done without loss of generality.\n\nI easily found the paragraph you are looking at, and I believe its meaning\nwill be clear to well prepared students. So you probably need to back off\nand focus on acquiring some of the background needed for understanding\nthis and many other books.\n\nAs a start, I\'d recommend the appropriate chapters in the book cited\nabove, plus\n\nauthor = {Boas, Mary L.},\ntitle = {Mathematical methods in the physical sciences},\npublisher = {Wiley},\nyear = 1983}\n\nand perhaps\n\nauthor = {Folland, Gerald B.},\ntitle = {{F}ourier analysis and its applications},\npublisher = { Wadsworth & Brooks/Cole Advanced Books},\nyear = 1992}\n\nFurther buzzwords include "Sturm-Liouville", "special functions",\n"harmonic analysis", "separation of variables" applied to a "boundary\nvalue problem", and "spherical harmonics".\n\nFor example, I think you\'ll find discussion of a vibrating drumhead (a\nstandard topic in mathematical methods courses--- I don\'t remember if Boas\ndiscusses that but Folland certainly does) useful for understanding the\nnotion of a "mode". Such vibration problems (controlled by the Laplace\nequation) are actually quite relevant to metric perturbations of\nLorentzian manifolds-- you can google for past posts here connected\nharmonic polynomials, multipoles, Laplace-Beltrami equation on S^2, H^2,\netc., and metric perturbations of (semi)-Riemannian manifolds. One quick\nway to see that this claim is plausible is to review the form of the\nweak-field EFE written ---in an appropriate gauge (i.e. restricting to a\nparticular family of local coordinate charts)--- in terms of the "trace\nreversed perturbation" hbar_(ab).\n\nFor "weak-field EFE", "gauge", "trace-reverse" operation, etc., try these\nrecent and very readable UG textbooks:\n\nauthor = {Sean Carroll},\ntitle = {Spacetime and geometry: an introduction to general relativity},\npublisher = {Addison-Wesley},\nyear = 2004}\n\nauthor = {Bernard F. Schutz},\ntitle = {A First Course in General Relativity},\npublisher = {Cambridge University Press},\nyear = 1985}\n\nThere are many other excellent books, but I think you will find these\nparticularly useful for weak-field gtr. For "multipole moments" in\nweak-field gtr, try also\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\nAgain, you can also look for past posts here on these topics.\n\nSorry I don\'t have time to say more! Hope this helps...\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 Wed, 3 Nov 2004, Bill Kavanagh wrote:

> I am currently trying to learn the basics (and beyond) of Perturbation
> Theory in General Relativity for which there doesn't seem to be any
> really good sources.

I've posted on this very extensively before, so I'll be brief.

You didn't specify your mathematical background, so please forgive me if I
am incorrect in assuming (on the basis of certain small clues) that you do
not have a strong modern undergraduate background in what is often called
"mathematical methods".

One essential point to understand is that perturbation theory is a body of
ideas which applies not just to the problem of solving the EFE but to
differential equations in general, and indeed to such deceptively simple
topics as locating the roots of a univariate polynomial. These ideas are
among the most "applicable" in all of "applied mathematics", so everyone
should know them!

See for example the chapter on perturbation theory in

author = {Richards, Derek},
title = {Advanced mathematical methods with {M}aple},
publisher = {Cambridge University Press},
year = 2002}

Can you see how to apply the root location techniques explained by
Richards in general relativity?

Exercise: review a textbook discussion of the horizons in Schwarzschild
vacuum, then look up Schwarzschild-de Sitter and solve the analogous
problem.

Next, can you see how standard techniques for perturbing solutions of an
ODE, also explained by Richards, are employed in the derivation of the
extra-Newtonian precession of Mercury?

Exercise: review a textbook discussion, then repeat using a suitable chart
for the static weak-field monopole solution. (E.g., polar spherical, then
Cartesian.) Now try to extend your result to a static isolated object
with nonzero monopole and quadrupole moments. Can you explain the
precession of Mercury using hypothetical solar oblateness? If so, can you
explain the precession of -both- Mercury and Venus this way? Hint:
perturbation techniques you have learned above should help in comparing
how the monopole and quadrupole effects scale with distance.

(If you get stuck, look for past posts to this group where these two
exercises are worked in detail. The second exercise is taken from a
problem from the problem book by Press et al.)

> My current understanding of perturbation theory

say rather something like "metric perturbations of a Lorentzian manifold"

> is that we take a metric and add some small terms to the metric
> components and then put this into Einstein's equations and through away
> higher order terms. I've studied the weak-field (g = \eta + h ; where
> h<<1) equations which result from a perturbation off Minkowski. Of the
> things I don't fully understand, the idea that h is a background
> 'tensor' and the use of gauge transformations are the most critical.

Actually, \eta_(ab) is the "background metric" and h_(ab) is the
perturbation.

> Despite these details though, I am now trying to understand more
> complicated perturbations. The perturbations of the Schwarzschild
> Black-Hole in chapter 4 of S. Chandrasekhar's "The Mathematical Theory
> of Black Holes" (1983) is what I have chosen to look at (it is
> relevant in some way to my research).

You certainly do not lack in audacity! :-/

But courage is only one of the prerequisites for this particular book...

> I run into trouble immediately when he talks about modes of
> perturbations. What is a mode of perturbation? (i.e. a
> non-axisymmetric mode of perturbation with an e^(im\phi) dependence)
> He remarks that restricting to time-dependant axisymmetric modes can be
> done without loss of generality.

I easily found the paragraph you are looking at, and I believe its meaning
will be clear to well prepared students. So you probably need to back off
and focus on acquiring some of the background needed for understanding
this and many other books.

As a start, I'd recommend the appropriate chapters in the book cited
above, plus

author = {Boas, Mary L.},
title = {Mathematical methods in the physical sciences},
publisher = {Wiley},
year = 1983}

and perhaps

author = {Folland, Gerald B.},
title = {{F}ourier analysis and its applications},
publisher = { Wadsworth & Brooks/Cole Advanced Books},
year = 1992}

Further buzzwords include "Sturm-Liouville", "special functions",
"harmonic analysis", "separation of variables" applied to a "boundary
value problem", and "spherical harmonics".

For example, I think you'll find discussion of a vibrating drumhead (a
standard topic in mathematical methods courses--- I don't remember if Boas
discusses that but Folland certainly does) useful for understanding the
notion of a "mode". Such vibration problems (controlled by the Laplace
equation) are actually quite relevant to metric perturbations of
Lorentzian manifolds-- you can google for past posts here connected
harmonic polynomials, multipoles, Laplace-Beltrami equation on S^2, H^2,
etc., and metric perturbations of (semi)-Riemannian manifolds. One quick
way to see that this claim is plausible is to review the form of the
weak-field EFE written ---in an appropriate gauge (i.e. restricting to a
particular family of local coordinate charts)--- in terms of the "trace
reversed perturbation" \hbar_(ab).

For "weak-field EFE", "gauge", "trace-reverse" operation, etc., try these
recent and very readable UG textbooks:

author = {Sean Carroll},
title = {Spacetime and geometry: an introduction to general relativity},
publisher = {Addison-Wesley},
year = 2004}

author = {Bernard F. Schutz},
title = {A First Course in General Relativity},
publisher = {Cambridge University Press},
year = 1985}

There are many other excellent books, but I think you will find these
particularly useful for weak-field gtr. For "multipole moments" in
weak-field gtr, try also

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}

Again, you can also look for past posts here on these topics.

Sorry I don't have time to say more! Hope this helps...

"T. Essel" (hiding somewhere in cyberspace)