Photon surface

Photon sphere (definition):
A photon sphere of a static spherically symmetric metric is a timelike hypersurface { r = r p s } if the deflection angle of a light ray with the closest distance of approach r o diverges as r o → r p s .

For a general static spherically symmetric metric

g = − β ( r ) d t 2 − α ( r ) d r 2 − σ ( r ) r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) ,

the photon sphere equation is:

2 σ ( r ) β + r d σ ( r ) d r β ( r ) − r d β ( r ) d r σ ( r ) = 0.

The concept of a photon sphere in a static spherically metric was generalized to a photon surface of any metric.

Photon surface (definition) :
A photon surface of (M,g) is an immersed, nowhere spacelike hypersurface S of (M, g) such that, for every point p∈S and every null vector k∈T_pS, there exists a null geodesic γ :(-ε,ε)→M of (M,g) such that γ ˙ (0)=k, |γ|⊂S.

Both definitions give the same result for a general static spherically symmetric metric.

Theorem:
Subject to an energy condition, a black hole in any spherically symmetric spacetime must be surrounded by a photon sphere. Conversely, subject to an energy condition, any photon sphere must cover more than a certain amount of matter, a black hole, or a naked singularity.