Lesson plan
Photon surface
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 generalizes 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 p S, 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.
Article summary
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 generalizes 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 p S, 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.
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Discussion questions
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