# Is there a Birkhoff-like theorem for stationary axisymmetric metrics?

I know about the theorem by Robinson and Carter about the uniqueness of the Kerr metric in the case of stationary axisymmetric (SA) black holes. Are there any uniqueness theorems like Birkhoff's theorem for stationary axisymmetric metrics?

1. Note first of all that $U(1)$ axial symmetry is much smaller than $SO(3)$ spherical symmetry.

2. (Let us put the cosmological constant $\Lambda=0$ to zero.) Where as spherically symmetric vacuum solutions are static and there are no spherically symmetric gravitational waves, the axisymmetric vacuum solutions are not necessarily stationary and there are axisymmetric gravitational waves. Even if the axisymmetric vacuum solution is additionally assumed to be stationary or static, there is still too much freedom left. Hence there is no axisymmetric version of Birkhoff's theorem.

3. The following electrostatic analogy with 3D Poisson's equation is telling: Spherically symmetric solutions $\phi$ to Laplace equation are restricted to just $\phi= Ar^2+B/r$. On the other hand, for axisymmetric solutions to Laplace equation in cylindrical coordinates, we have no control over the $z$-dependence.

• But what about the stationary axisymmetric metric? – skdatta Apr 11 '17 at 5:12
• Even if the axisymmetric spacetime is additionally assumed to be stationary or static, there is still too much freedom left. – Qmechanic Apr 11 '17 at 8:59
• can you suggest me some literature, from where it will become vivid? – skdatta Apr 11 '17 at 9:07
• Not all spherically symmetric spacetimes are static, only the vacuum ones. – Slereah Apr 11 '17 at 9:35
• $\uparrow$ Right. – Qmechanic Apr 11 '17 at 9:45

There is nothing as strong as the Birkhoff theorem in the case of stationarity and axisymmetry. Note that the Birkhoff theorem in its strongest form can be stated as:

"If even a piece of the space-time is spherically symmetric and a vacuum, then it is a piece of the Schwarzschild space-time."

There are various theorems about axially symmetric and stationary space-times which say that we reduce to Kerr under various conditions such as e.g. regularity outside of and on the horizon, connectedness of the horizon, asymptotic flatness, and global vacuum. I.e., with some physicist-faith leeway, we can construct arguments that as a reasonable, globally strictly vacuum and asymptotically flat space-time, the Kerr space-time is unique.

However, we do not know any reasonable matter solution which would match the Kerr space-time as an "exterior" solution. Geroch has even conjectured that there is no "interior" solution to the Kerr metric, i.e. a non-black-hole star which would reduce to Kerr outside its surface. (I personally believe that Geroch's conjecture is true.)

In practice, when we construct solutions of neutron stars, we find that they always differ from the Kerr case in the quadrupole and higher mass-multipole momenta of the space-time and we have to match them to approximately constructed non-Kerr metrics. I.e., when we are not globally vacuous, the uniqueness of Kerr is broken for sure.

There is even a rather well-known class of solutions derived by Manko & Novikov in 1992 which allow to set all of the infinite asymptotic values of mass-multipole momenta to arbitrary values. This, however, comes at the cost of weird singularities at the horizon and/or singular matter sources outside it. If you want a simpler playground for gaining some intuition on this, you can check out the axisymmetric and static Weyl metrics where you can plug in any Newtonian (axisymmetric and stationary) gravitational potential to generate a new space-time deviating from Kerr in its vacuum regions.

No. The key point is the (in general) non zero value of the multipoles modes.

Physically, it must be emphasised that there is no Birkhoff theorem for rotating spacetimes — it is not true that the spacetime geometry in the vacuum region outside a generic rotating star (or planet) is a part of the Kerr geometry. The best result one can obtain is the much milder statement that outside a rotating star (or planet) the geometry asymptotically approaches the Kerr geometry. The basic problem is that in the Kerr geometry all the multipole moments are very closely related to each other whereas in real physical stars (or planets) the mass quadrupole, octopole, and higher moments of the mass distribution can in principle be independently specified. Of course from electromagnetism you will remember that higher n-pole fields fall off as 1/r^{2+n}, so that far away from the object the lowest multipoles dominate it is in this asymptotic sense that the Kerr geometry is relevant for rotating stars or planets. On the other hand, if the star (or planet) gravitationally collapses then classically a black hole can be formed. In this case there are a number of powerful uniqueness theorems which guarantee the direct physical relevance of the Kerr spacetime, but as the unique exact solution corresponding to stationary rotating black holes, (as opposed to merely being an asymptotic solution to the far field of rotating stars or planets)

Source: Visser (2008)