Static Spherically Symmetric Kerr-Schild Metrics and Implications for the Classical Double Copy

TL;DR

The paper analyzes static, spherically symmetric Kerr-Schild metrics lacking horizons or singularities through the classical double copy lens, linking gravity sources to EM sources. It shows the EM charge density maps to the Komar energy density and finds the resulting stress-energy tensors are typically anisotropic and include shear, not being perfect fluids. A decomposition emerges: part fixed by EM sources and a conservation-determined stabilization piece, with energy conditions (weak vs strong) unable to be satisfied simultaneously for these solutions. These results clarify limitations of the classical double copy in static spherical settings and suggest exploring alternative coordinate realizations for potential different outcomes.

Abstract

We discuss the physical interpretation of stress-energy tensors that source static spherically symmetric Kerr-Schild metrics. We find that the sources of such metrics with no curvature singularities or horizons do not simultaneously satisfy the weak and strong energy conditions. Sensible stress-energy tensors usually satisfy both of them. Under most circumstances these sources are not perfect fluids and contain shear stresses. We show that for these systems the classical double copy associates the electric charge density to the Komar energy density. In addition, we demonstrate that the stress-energy tensors are determined by the electric charge density and their conservation equations.