Photon sphere (definition): A photon sphere of a static spherically symmetric metric is a timelike hypersurface { r = r p s } {\displaystyle \{r=r_{ps}\}} if the deflection angle of a light ray with the closest distance of approach r o {\displaystyle r_{o}} diverges as r o → r p s . {\displaystyle r_{o}\rightarrow r_{ps}.}
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 ) , {\displaystyle g=-\beta \left(r\right)dt^{2}-\alpha (r)dr^{2}-\sigma (r)r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{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. {\displaystyle 2\sigma (r)\beta +r{\frac {d\sigma (r)}{dr}}\beta (r)-r{\frac {d\beta (r)}{dr}}\sigma (r)=0.}
The concept of a photon sphere in a static spherically metric generalizes 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∈TpS, there exists a null geodesic γ {\displaystyle {\gamma }} :(-ε,ε)→M of (M,g) such that γ ˙ {\displaystyle {\dot {\gamma }}} (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.
References