# Is there a Birkhoff-like theorem for axially symmetric gravity fields?

As we know, to derive the Schwarzschild solution in GR we want to determine the unknown functions $a,b$ (sometimes $c$) of $$(ds)^2 = a(r,t)\;dt^2 - b(r,t)\; dr^2 - r^2d\Omega^2$$ or $$(ds)^2 = a(r,t)\;dt^2 + 2b(r,t)\;drdt - c(r,t)\; dr^2 - r^2d\Omega^2$$ where $d\Omega^2 = d\theta^2 + \sin^2 \theta \; d\phi^2 \;$ is the usual line element of the 2-sphere. When applied to the empty space field equations where $G_{ab} \equiv 0$, Birkhoff's theorem says any static, spherically symmetric solution to the vacuum equations of general relativity is necessarily the Schwarzschild solution up to a coordinate transformation. That basically wraps up looking for exterior spherically symmetric solutions in GR. The next logical step is then to look for interior solutions which is a whole different ball game.

I'm interested in considering axially symmetric spacetimes in GR. The usual approach is to look at Weyl metrics (as mentioned in the comments below) and go for broke.

What I am confused about is trying to consider one of the following line element $$(ds)^2 = a(r,\theta)\;dt^2 + 2b(r,\theta)\;drdt - c(r,\theta)\; dr^2 - r^2d\Omega^2.$$

The above line element is axially symmetric also correct? In terms of polar spherical coordinates it is symmetric about the polar angle coordinate.

In terms of a spacetime that models a non-spherically symmetric body for e.g. a planet, I would have thought that this line element is more suitable than the general Weyl metric. However, I have never seen a method beginning with the above line element.

Question Is the Weyl solution to axially symmetric gravitational fields as Schwarzschild solutions are to spherically symmetric gravitational fields? In other words, is there a Birkhoff-type theorem for axially symmetric solutions to the field equations of GR?

• See Weyl metrics: en.wikipedia.org/wiki/Weyl_metrics – Void Jan 5 '17 at 14:37
• @Void: aren't the Weyl metrics are axisymmetric? – Kyle Kanos Jan 5 '17 at 17:22
• @KyleKanos Yes, it is true that Weyl metrics address only to the "angular dependency" (in $\theta$) but not specifically the $\phi$-dependency as proposed here. – Void Jan 5 '17 at 17:59
• That was just an example, the dependency ins't set in stone for e.g. it could be $(r,\theta)$,$(r,\phi)$ or any variation of the coordinates. What I'm interested in is stepping away from the standard Schwarzschild solution and RN solution and that pesky Birkhoff's theorem! – Rumplestillskin Jan 5 '17 at 22:38

## 1 Answer

Many exact solutions to EFE (Einistein's field equations) with varying degrees of symmetry have been found.

This Wikipedia article and references therein are excellent introduction to this field.

As a book reference, the standard one is: H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, “Exact Solutions of Einstein’s Field Equations: 2nd Edition”, (2003), Cambridge University Press.

• Okay, I'll be sure to check them out but more specifically to what I asked have you any experience with solutions to the EFE of this type? – Rumplestillskin Jan 5 '17 at 22:35