Stationary Black Holes and Light Rings.

Pedro V P Cunha,Carlos A R Herdeiro

doi:10.1103/physrevlett.124.181101

Pedro V P Cunha, Carlos A R Herdeiro

Open Access

https://doi.org/10.1103/physrevlett.124.181101

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Abstract

The ringdown and shadow of the astrophysically significant Kerr black hole (BH) are both intimately connected to a special set of bound null orbits known as light rings (LRs). Does it hold that a generic equilibrium BH must possess such orbits? In this Letter we prove the following theorem. A stationary, axisymmetric, asymptotically flat black hole spacetime in 1+3 dimensions, with a nonextremal, topologically spherical, Killing horizon admits, at least, one standard LR outside the horizon for each rotation sense. The proof relies on a topological argument and assumes C^{2} smoothness and circularity, but makes no use of the field equations. The argument is also adapted to recover a previous theorem establishing that a horizonless ultracompact object must admit an even number of nondegenerate LRs, one of which is stable.

Highlights

Introduction.—The second decade of the 21st century will be celebrated as the dawn of precision strong gravity
The ringdown and shadow of the astrophysically significant Kerr black hole (BH) are both intimately connected to a special set of bound null orbits known as light rings (LRs)
Does it hold that a generic equilibrium BH must possess such orbits? In this Letter we prove the following theorem

Summary

Stationary Black Holes and Light Rings

The ringdown and shadow of the astrophysically significant Kerr black hole (BH) are both intimately connected to a special set of bound null orbits known as light rings (LRs). Under reasonable assumptions, we shall establish the following theorem: A stationary, axisymmetric, asymptotically flat, 1 þ 3 dimensional BH spacetime, ðM; gÞBH, with a nonextremal, topologically spherical Killing horizon, H, admits at least one standard LR outside the horizon for each rotation sense. We assume that the metric is at least C2-smooth on and outside H, and circular The latter, together with asymptotic flatness, implies the spacetime admits a 2-space orthogonal to f∂t; ∂φg—see, e.g., Theorem 7.11 in Ref. Our task is to show that the total LR topological charge in the region outside a BH (under the assumptions stated above) is w 1⁄4 −1, regardless of choosing Hþ or H− This implies that at least one standard LR must exist within that region, for each rotation sense of the BH, and establishes the theorem.

This leads to

ΩHπ Þ