Graduate Extra Killing Vector Field in Kerr Spacetime?

[PeterDonis]

In a recent thread, the following was posted regarding the "no hair" theorem for black holes:

Carter and Robinson proved it assuming that the space-time has axial symmetry. Hawking removed that assumption, but had an analyticity, assumption which is unreasonable both mathematically and physically. The problem is still open. In the mathematical literature it is known as the rigidity conjecture.

www.ihes.fr/~vanhove/Slides/Klainerman-ihes-fev2011.pdf

https://arxiv.org/pdf/1501.01587

In the arxiv paper linked to, it says the following (p. 2, after Theorem 1.1):

"Hawking has shown that in addition to the original, stationary, Killing field, which has to be tangent to the event horizon, there must exist, infinitesimally along the horizon and tangent to its generators, an additional Killing vector-field."

This is the first I've heard of this, and it's not elaborated on in the paper. Can anyone explain what this refers to?

 

[Ibix]

A bit above my pay grade, but I think this is relevant: https://arxiv.org/abs/gr-qc/9811021

I got there via https://web.math.princeton.edu/~seri/homepage/papers/LocalH.pdf

 

[George Jones]

"Hawking has shown that in addition to the original, stationary, Killing field, which has to be tangent to the event horizon, there must exist, infinitesimally along the horizon and tangent to its generators, an additional Killing vector-field."

This is the first I've heard of this, and it's not elaborated on in the paper. Can anyone explain what this refers to?

Either I have forgotten completely about this, or I also have never heard about it. Page 323 (attached) of 'General Relativity' by Wald also mentions the result. In particular, see the sentence that starts "Finally, in case (iii) ..."

 

[PeterDonis]

see the sentence that starts "Finally, in case (iii) ..."

Ah, I get it. It's just a different way of organizing the Killing fields. The way I usually think of them is that we have the stationary KVF $\xi^a$, which is timelike at infinity, and the axial KVF $\psi^a$, which has closed orbits. But $\xi^a$ is spacelike on the horizon (it's null at the static limit, not the horizon), and the horizon has to be generated by null geodesics, so there is also a null KVF $\chi^a$ that is tangent to the horizon generators. In my usual way of thinking of these, we have $\chi^a = \xi^a + \Omega \psi^a$, where $\Omega$ is the "angular velocity of the horizon". But Hawking's proof can be viewed as starting with $\xi^a$ (whose existence follows from the spacetime being stationary), proving the existence of $\chi^a$ (i.e., of a KVF linearly independent of $\xi^a$ that is null on the horizon), and then using those two to construct $\psi^a$ (i.e., proving that the spacetime is axially symmetric).

 

[PeterDonis]

Either I have forgotten completely about this, or I also have never heard about it.

Since I've read Wald before, I evidently had seen it before, but had forgotten.