# Expanding Schwarzschild solution using spherical harmonics?

The well known Schwarzschild solution is given by

\begin{equation} (ds)^2 = \left( 1 - \frac{2GM}{c^2r} \right) dt^2 - \left( 1 - \frac{2GM}{c^2r} \right)^{-1} dr^2 - r^2 d\Omega^2. \end{equation} It has been on my mind a lot about attempting to represent an aspherical body in general relativity to model the geometry of the Earth. In Newtonian mechanics a more accurate description of the Earth's oblate geometry can be obtained by expanding the Newtonian potential using spherical harmonics. We can clearly identify the Newtonian potential in the Schwarzschild solution above.

There is obviously a reason stopping the spherical harmonic expansion into the above equation. However, I don't know why? Any suggestions?

In fact there is a lot of work that has been done involving the non spherical perturbations of Schwarzschild spacetimes. It is of great use in estimating the stability and gravitational radiation emitted in those cases and including Black Hole perturbations that then settle down to the symmetric case by emitting gravitational radiation. The same is true for Kerr solutions. One recent summary is at https://arxiv.org/pdf/gr-qc/0502064.pdf.

It's known in General Relativity that for a mass distribution, the first term that radiates gravitationally is the changing quadrupole moment.

So those are equivalent to your Newtonian approximations to a non perfect earth. Although much more complex the same is done in General Relativity.

For two Black Holes merging it is also more complex because in the strong field regime you need to integrate solutions numerically for the two body problem, so more complex than the Newtonian reduction to a one body problem.

• so what about just expanding the Coefficients in both temporal and radial direction? Is that something that is common? I think it would be interesting – Rumplestillskin Jun 3 '17 at 4:44
• There may be different things to try. Usually r and t expansions are done for grav waves, and in fact incorporate the time dependence on the n-pole moments, with the r dependence going as increasing powers of 1/r for higher n-poles, similar to electromagnetic expansions. The other thing done is in terms of t-r and t+r, for light rays or light like conformal infinity. – Bob Bee Jun 5 '17 at 17:19
• that's interesting about $t-r$ and $t+r$ would you have any reference that I could investigate? – Rumplestillskin Jun 6 '17 at 0:21
• There is a lot of work and uses for those, called null coordinates, in both special and general relativity. One introduces coordinates u and v defined from r and t (and how you define it depends on the metric) such that u or v = constant are the light cones in that spacetime. In Minkowski it is t-r and t+r. Also called lightlike coordinates. It's in a lot of places and papers and books. See eg en.m.wikipedia.org/wiki/Eddington–Finkelstein_coordinates for the Schwarzschild spacetime in those coordinates. Penrose used them to define the causal structure of spacetime. Bondi used them. Etc – Bob Bee Jun 6 '17 at 2:03
• That's great! I'll be sure to look into them. Now, my intuition is the look at the $GM/r$ and expand this using spherical harmonics and see what happens. However, I'm not sure if that is entirely "legal" (maybe something to do with the non-linearity). BUT I know Ashby does it as soon as he linearises the Schwarzschild line element. Have you had experience with this before? It seems like an obvious thing to try. – Rumplestillskin Jun 6 '17 at 2:06