Structural acoustics

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

For a general static spherically symmetric metric

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

the photon sphere equation is:

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

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

Photon surface (definition) :
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 ${\gamma }$:(-ε,ε)→M of (M,g) such that ${\dot {\gamma }}$(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

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