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Photon surface

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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 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∈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

  1. Virbhadra, K. S.; Ellis, George F. R. (2000-09-08). "Schwarzschild black hole lensing". Physical Review D. 62 (8). American Physical Society (APS): 084003. arXiv:astro-ph/9904193v2. Bibcode:2000PhRvD..62h4003V. doi:10.1103/physrevd.62.084003. ISSN 0556-2821. S2CID 15956589.
  2. Virbhadra, K. S.; Ellis, G. F. R. (2002-05-10). "Gravitational lensing by naked singularities". Physical Review D. 65 (10). American Physical Society (APS): 103004. Bibcode:2002PhRvD..65j3004V. doi:10.1103/physrevd.65.103004. ISSN 0556-2821.
  3. ^ Claudel, Clarissa-Marie; Virbhadra, K. S.; Ellis, G. F. R. (2001). "The geometry of photon surfaces". Journal of Mathematical Physics. 42 (2): 818–838. arXiv:gr-qc/0005050. Bibcode:2001JMP....42..818C. doi:10.1063/1.1308507. ISSN 0022-2488. S2CID 119457077.


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