Derivation of the Teukolsky Equation

[geoffc]

Several years ago as an exercise, I was able to work through the derivation of thethe Teukolsky equation but have since lost my notes. Has anyone else gone through the derivation, and if so, could you give me a sort of complete idiot's guide to going through the steps? It's been quite some time since I worked through the details. At some point I'd like to do the separation of variables and look at some of the papers on perturbation theory.

Many thanks in advance.

 

[Chris Hillman]

Good grief! That would take an awful lot of LaTeX, I think! As you recall, the fact that the Teukolsky equation exists at all is something of a miracle--- like a delicate orchid, it requires special care to present. Particularly if you want an "idiots guide", how about backing way up and discussing metric perturbations generally, to try to carry along some of the more ambitious PF seekers.

For those who have no idea what we are talking about, we are discussing a core topic in the theory of wave propagation near a Kerr object.

 

[Chris Hillman]

Suggestion: Work up to it slowly!

Er... I wasn't trying to discourage you, in fact I was encouraging you to try your hand at exposition with some introduction to perturbations generally (in the general math forum, I guess). For example many books on perturbation theory, e.g. Simmonds and Mann, A First Look at Perturbation Theory, Dover reprint of a book originally published in 1986, begin by analyzing the roots of polynomials, and this is directly relevant to a common problem in gtr, in which we have something like the Kottler lambdavacuum (aka Schwarzschild-de Sitter lambdavacuum) and have observed that roots of a fourth order polynomial give the location of two horizons. But the exact expression for these roots are awkward, and it makes good sense to apply perturbation theory to obtain perfectly adequate but much simpler approximations! Then you can discuss application of perturbation theory of ODEs to analyzing some of the classical solar system tests. Continuing, eventually you get to discussing metric perturbations, e.g. linearized gravitational waves propagating on Minkowski or FRW or Schwarzschild backgrounds. The latter brings us to the Regge-Wheeler equation and tensor harmonics. After discussing that we stand at the threshold of generalizing to the Kerr vacuum and deriving the Teukolsky equation.

 

[geoffc]

Thank you very much for the reference. It's one that I haven't had a chance to take a look at, but certainly will when I get a chance. I had been meaning to sit down and work through a few examples in celestial mechanics which seem to be of increasing importance since people are now treating the three body problem in gr which looks like a complete mess.

Many years back I had worked through a few examples of the Regge-Wheeler equation which, from what I remember, was quite reminiscent of scattering problems from quantum mechanics. What seemed to be quite different was the issue of gauge, which was enough of an issue to confuse even a Chandra.

But assuming I meticulously work through all the examples you suggest (a great idea, by the way), all I really wanted to do was write down the Teukolsky equation pretty much the same way a person would tediously write down the Laplacian in spherical coordinates. You have a wave operator...

 

[Chris Hillman]

all I really wanted to do was write down the Teukolsky equation pretty much the same way a person would tediously write down the Laplacian in spherical coordinates

Now I am confused: if you just want to write down the Teukolsky equation without understanding some way to obtain it (or motivate it, or extend it), you can simply look it up in various books.