Photon surface

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Photon sphere (definition^[1][2]):
A photon sphere of a static spherically symmetric metric is a timelike hypersurface ${\displaystyle \{r=r_{ps}\}}$ if the deflection angle of a light ray with the closest distance of approach ${\displaystyle r_{o}}$ diverges as ${\displaystyle r_{o}\rightarrow r_{ps}.}$

For a general static spherically symmetric metric

${\displaystyle g=-\beta \left(r\right)dt^{2}-\alpha (r)dr^{2}-\sigma (r)r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}),}$

the photon sphere equation is:

${\displaystyle 2\sigma (r)\beta +r{\frac {d\sigma (r)}{dr}}\beta (r)-r{\frac {d\beta (r)}{dr}}\sigma (r)=0.}$

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

Photon surface (definition^[3]) :
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 ${\displaystyle {\gamma }}$:(-ε,ε)→M of (M,g) such that ${\displaystyle {\dot {\gamma }}}$(0)=k, |γ|⊂S.

Both definitions give the same result for a general static spherically symmetric metric.^[3]

Theorem:^[3]
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.

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