Abstract
In this note we identify photon surfaces and anti-photon surfaces in some physically interesting spacetimes, which are not spherically symmetric. All of our examples solve physically reasonable field equations, including for some cases the vacuum Einstein equations, albeit they are not asymptotically flat. Our examples include the vacuum C-metric, the Melvin solution of Einstein-Maxwell theory and generalisations including dilaton fields. The (anti-)photon surfaces are not round spheres, and the lapse function is not always constant.
Highlights
It is well known that the Schwarzschild solution contains circular photon orbits at r = 3M, where M > 0 is the ADM mass
These circular photon orbits are the projection onto the spatial manifold t = constant of null geodesics in the spacetime
If the projection of the tangent vector of any null geodesic is tangent to the sphere at one time it remains tangent to the sphere at all future times
Summary
It is well known that the Schwarzschild solution contains circular photon orbits at r = 3M , where M > 0 is the ADM mass These circular photon orbits are the projection onto the spatial manifold t = constant of null geodesics in the spacetime. Photon surfaces have attracted attention recently, in particular in the last two years there have been several results establishing the uniqueness of spacetimes admitting a photon surface under certain conditions [1,2,3,4,5,6,7,8] These works typically assume that the spacetime is complete, asymptotically flat and with the exception of [8] assume that the lapse, N , is constant on the surface. In a class of cylindrically symmetric spacetimes of Melvin type [11] we present anti-photon cylinders
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