A Complete Invariant Analysis of the Kerr Spacetime and its Photon Region

TL;DR

This paper develops an invariant characterization of Kerr spacetime, providing a new method to identify photon surfaces and spherical photon orbits, and offering computational tools for analyzing geodesics in the photon region.

Contribution

It introduces a novel invariant function that characterizes photon surfaces and spherical photon orbits in Kerr spacetime, enhancing geometric understanding and computational analysis.

Findings

Invariant function identifies all spherical photon orbits

Invariant determines constants of motion for spherical orbits

Invariant characterizes photon surfaces and other key spacetime surfaces

Abstract

We present an invariant characterization of the Kerr spacetime, and utilize the invariant structure of the spacetime to derive a function whose zeros identify a special family of null geodesics. Each member of this family is tangent to every photon surface in the Kerr photon region, offering a method of invariantly characterizing photon surfaces in axially symmetric spacetimes and thereby a providing a computational tool for efficiently computing the geodesic equations for any part of the photon region. The invariant that identifies all of the spherical photon orbits is parameterized by a Lorentz parameter, where the parameter is effectively an inclination angle of the spherical photon orbits through the equatorial plane. We also show how the invariant determines the constants of motion for all spherical orbits in the photon region. Finally, we briefly derive invariants which identify the other geometrically important surfaces such as the ergosurfaces and local horizons.