Solve Eq. (9.58) on Schutz Page 246 - A First Course in GR

[Jimmy Snyder]

Here is a question from problem 26 part a on page 246 of "A First Course in GR" by Schutz. I doubt that it can be answered by someone without a copy of the book, but I have thought that before and been wrong. If someone without the book wants to help and needs more information then of course I would promptly provide it.

Eq. (9.58) in the vacuum region outside the source - i.e., where $S_{\mu \nu} = 0$ - can be solved by separation of variables.

Eq (9.58) (edited) follows:

$$(\nabla^2 + \Omega^2)(\bar{h}_{\mu \nu}e^{i\Omega t}) = 0$$

Assume a solution for $\bar{h}_{\mu \nu}$ has the form

$$\Sigma_{km}A^{km}_{\mu \nu}f_k(r)Y_{km}(\theta, \phi)/\sqrt{r}$$

where $Y_{km}$ is the spherical harmonic. (The book uses l as does everyone else on the planet, but I changed l to k so that this post would read more easily).

(a) Show that $f_k(r)$ satisfies the equation:

$$\ddot{f}_k + \frac{1}{r}\dot{f}_k + [\Omega^2 - \frac{(k+\frac{1}{2})^2}{r^2}]f_k = 0$$

where dot means differentiation with respect to r. Without even trying to solve this problem, my question is simply this: how can k show up in the differential equation? Neither A nor Y are functions of r, and k is just a subscript on f.

 

[George Jones]

k is (related to) the separation constant. When solving partial differential equations by the technique of separation of variables, the separation constant often appears, after rearrangement, in the resulting ordinary differential equations.

Regards,
George

 

[Jimmy Snyder]

k is (related to) the separation constant.

Thanks George, I got out a book on DiffEQs and looked up separation constant. It looks like I'll be spending some time on this.