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Photon surface

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Photon sphere (definition[1][2]):
A photon sphere of a static spherically symmetric metric is a timelike hypersurface if the deflection angle of a light ray with the closest distance of approach diverges as

For a general static spherically symmetric metric

the photon sphere equation is:

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 kTpS, there exists a null geodesic :(-ε,ε)→M of (M,g) such that (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.

References

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  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. ^ a b c 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.