Photon surface

{{Short description|Surface of a black hole}}

{{Technical|date=July 2011}}

Photon sphere (definition{{cite journal | last1=Virbhadra | first1=K. S. | last2=Ellis | first2=George F. R. | title=Schwarzschild black hole lensing | journal=Physical Review D | publisher=American Physical Society (APS) | volume=62 | issue=8 | date=2000-09-08 | issn=0556-2821 | doi=10.1103/physrevd.62.084003 | page=084003|arxiv=astro-ph/9904193v2| bibcode=2000PhRvD..62h4003V | s2cid=15956589 }}{{cite journal | last1=Virbhadra | first1=K. S. | last2=Ellis | first2=G. F. R. | title=Gravitational lensing by naked singularities | journal=Physical Review D | publisher=American Physical Society (APS) | volume=65 | issue=10 | date=2002-05-10 | issn=0556-2821 | doi=10.1103/physrevd.65.103004 | page=103004| bibcode=2002PhRvD..65j3004V }}):

A photon sphere of a static spherically symmetric metric is a timelike hypersurface $\backslash \{r=r\_\{ps\}\backslash \}$ if the deflection angle of a light ray with the closest distance of approach $r\_o$ diverges as $r\_o\; \backslash rightarrow\; r\_\{ps\}.$

For a general static spherically symmetric metric

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

the photon sphere equation is:

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

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

Photon surface (definition{{cite journal | last1=Claudel | first1=Clarissa-Marie | last2=Virbhadra | first2=K. S. | last3=Ellis | first3=G. F. R. | title=The geometry of photon surfaces | journal=Journal of Mathematical Physics | volume=42 | issue=2 | year=2001 | issn=0022-2488 | doi=10.1063/1.1308507 | pages=818–838| arxiv=gr-qc/0005050 | bibcode=2001JMP....42..818C | s2cid=119457077 }}) :

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 $\{\backslash gamma\}$:(-ε,ε)→M of (M,g) such that $\{\backslash dot\{\backslash gamma\}\}$(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.