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| | '''Photon sphere''' (''definition''<ref>K.S. Virbhadra and G. F.R. Ellis, Schwarzschild black hole lensing, Phys. Rev. D62, 084003 (2000); K.S. Virbhadra and G. F.R. Ellis, Gravitational lensing by naked singularities, Phys. Rev. D65, 103004 (2002).</ref>):<br /> |
| | A [[photon sphere]] of a static spherically symmetric [[Metric tensor|metric]] is a [[timelike]] hypersurface <math>\{r=r_{ps}\}</math> if the deflection angle of a light ray with the closest distance of approach <math>r_o</math> diverges as <math>r_o \rightarrow r_{ps}.</math> |
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| | For a general static spherically symmetric metric |
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| | <math>g = - \beta\left(r\right) dt^2 - \alpha(r) dr^2 - \sigma(r) r^2 (d\theta^2 + \sin^2\theta d\phi^2),</math> |
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| | the photon sphere equation is: |
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| | <math>2\sigma(r) \beta + r \frac{d\sigma(r)}{dr} \beta(r) - r \frac{d\beta(r)}{dr} \sigma(r) = 0.</math> |
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| | The concept of a photon sphere in a static spherically metric was generalized to a photon surface of any metric. |
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| | '''Photon surface''' (''definition''<ref>Clarissa-Marie Claudel, K.S. Virbhadra, and G.F.R. Ellis, The geometry of photon surfaces, J. Math. Phys. 42, 818-838 (2001).</ref>) : <br /> |
| | 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''<sub>p</sub>S, there exists a null [[geodesic]] <math>{\gamma}</math>:(-ε,ε)→M of (M,'''g''') such that <math>{\dot{\gamma}}</math>(0)='''k''', |γ|⊂S. |
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| | Both definitions give the same result for a general static spherically symmetric metric.<ref>See in [2].</ref> |
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| | '''Theorem''':<ref>See in [2].</ref><br /> |
| | 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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| | ==References== |
| | {{Reflist}} |
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| | [[Category:Black holes]] |
| | [[Category:General relativity]] |
Revision as of 05:05, 29 December 2013
Template:Multiple issues
Photon sphere (definition[1]):
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[2]) :
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 :(-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:[4]
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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- ↑ K.S. Virbhadra and G. F.R. Ellis, Schwarzschild black hole lensing, Phys. Rev. D62, 084003 (2000); K.S. Virbhadra and G. F.R. Ellis, Gravitational lensing by naked singularities, Phys. Rev. D65, 103004 (2002).
- ↑ Clarissa-Marie Claudel, K.S. Virbhadra, and G.F.R. Ellis, The geometry of photon surfaces, J. Math. Phys. 42, 818-838 (2001).
- ↑ See in [2].
- ↑ See in [2].