Astronomy:Photon surface

Photon sphere (definition):
A photon sphere of a static spherically symmetric metric is a timelike hypersurface [math]\displaystyle{ \{r=r_{ps}\} }[/math] if the deflection angle of a light ray with the closest distance of approach [math]\displaystyle{ r_o }[/math] diverges as [math]\displaystyle{ r_o \rightarrow r_{ps}. }[/math]

For a general static spherically symmetric metric

[math]\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), }[/math]

the photon sphere equation is:

[math]\displaystyle{ 2\sigma(r) \beta + r \frac{d\sigma(r)}{dr} \beta(r) - r \frac{d\beta(r)}{dr} \sigma(r) = 0. }[/math]

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 [math]\displaystyle{ {\gamma} }[/math]:(-ε,ε)→M of (M,g) such that [math]\displaystyle{ {\dot{\gamma}} }[/math](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.