The Schwarzschild spacetime of positive mass is well known to possess a unique photon sphere – a cylindrical, timelike hypersurface P such that any null geodesic initially tangent to P remains tangent to P. We will show that it also possesses a rich family of spherically symmetric photon surfaces – general timelike hypersurfaces P such that any null geodesic initially tangent to P remains tangent to P --, and prove that these are indeed the only photon surfaces in the Schwarzschild spacetime besides some partial hyperplanes. These results hold in all dimensions $n+1≥ 4$ and generalize to other spacetimes such as sub-extremal Reissner–Nordström and AdS.
Furthermore, we will show that the Schwarzschild spacetime is indeed the only static, vacuum, asymptotically flat spacetime possessing static black hole horizons, photon spheres, and so-called “equipotential” photon surfaces, again in all dimensions. Physically speaking, this asserts that static, vacuum, equipotential photon surfaces have no hair.
The above results are joint work with Galloway and with Jahns and Vičánek Martínez. They build on a rigidity result for asymptotically flat Riemannian manifolds of non-negative scalar curvature with special umbilic, CMC, constant scalar curvature inner boundary that globally support a harmonic function satisfying an overdetermined system of Dirichlet and Neumann conditions on the inner boundary and suitable asymptotic assumptions. The proof of this rigidity result extends Riemannian and conformal geometry arguments first introduced by Bunting and Masood-ul-Alam in their proof of static black hole uniqueness, a higher dimensional analog by Gibbons, Ida, and Shiromizu, and previous joint work with Galloway on the uniqueness of photon spheres. It relies on Schoen and Yau’s higher dimensional positive mass theorem as well as on a result by McFerron and Szekelyhidi. Under somewhat stronger assumptions, this result of the author has since been generalized by Jahns to electro-vacuum in higher dimensions, implying static, electro-vacuum black hole and photon sphere uniqueness and extending previous results of Galloway and the author.