Sixteenth Marcel Grossmann Meeting

Static, equipotential photon surfaces have no hair

6 Jul 2021, 11:10

20m

Invited talk in the parallel session Gravitational Lensing and Shadows Gravitational Lensing and Shadows

Speaker

Carla Cederbaum (Tuebingen University)

Description

The Schwarzschild spacetime of positive mass is well-known to possess a unique “photon sphere” – meaning a cylindrical, timelike hypersurface $P$ such that any null geodesic initially tangent to $P$ remains tangent to $P$ – in all dimensions. We will show that it also possesses a rich family of spatially spherically symmetric “photon surfaces” – general timelike hypersurfaces $P$ such that any null geodesic initially tangent to $P$ remains tangent to $P$. This generalizes a result of Foertsch, Hasse, and Perlick from $2+1$ to higher dimensions. Furthermore, we will discuss how these photon surfaces behave across the black hole horizon and towards infinity in the Kruskal--Szekeres extension.

Similar results can be obtained in a large class of static, spherically symmetric spacetimes, including for example sub-extremal Reissner--Nordström spacetimes, but also other relevant examples. We show that they are (almost) necessarily rotationally symmetric and give concrete ODEs for their radial profile, including a solvability analysis of said ODEs.

We will also present a general theorem that implies that any static, vacuum, asymptotically flat spacetime possessing a so-called “equipotential” photon surface must already be the Schwarzschild spacetime. The proof of the theorem uses and extends Riemannian geometry arguments first introduced by Bunting and Masood-ul-Alam in their proof of static black hole uniqueness. It holds in all dimensions $n+1\ge 3+1$ and naturally generalizes to electro-vacuum.

Parts of this work are joint with Gregory J. Galloway, with Sophia Jahns and Olivia Vicanek Martinez, and with Markus Wolff.

Presentation materials

Marcel Grossmann Meeting 06.07.2021.pdf