Photon regions and umbilic conditions in stationary axisymmetric spacetimes

Photon regions and umbilic conditions in stationary axisymmetric spacetimes

Blog Article

Abstract Photon region (PR) in the strong gravitational field is defined as a compact region where photons can travel endlessly without going to infinity or disappearing at the event horizon.In Schwarzschild metric PR degenerates to the two-dimensional photon sphere $$r=3r_g/2$$ r=3rg/2 where closed circular photon orbits are located.The photon sphere as a three-dimensional hypersurface in spacetime is umbilic (its second quadratic form is pure trace).

In Stove Simmerstat Kerr metric the equatorial circular orbits have different radii for prograde, $$r_p$$ rp , and retrograde, $$r_r$$ rr , motion (where r is Boyer–Lindquist radial variable), while for $$r_pCamping Beds the existence of PR as well, but the underlying geometric structure was not fully identified so far.

Here we suggest geometric description of PR in generic stationary axisymmetric spacetimes, showing that PR can be foliated by partially umbilic hypersurfaces, such that the umbilic condition holds for classes of orbits defined by the foliation parameter.New formalism opens a way of analytic description of PR in stationary axisymmetric spacetimes with non-separable geodesic equations.