Black hole spacetimes with dark matter spikes: Energy-momentum tensor and backreaction effects

TL;DR

This paper analyzes the energy-momentum tensor of dark matter spikes around black holes, revealing significant backreaction effects on spacetime geometry that are crucial for accurate modeling.

Contribution

It derives the complete energy-momentum tensor of DM spikes, including kinetic and anisotropic pressure effects, and demonstrates their impact on black hole spacetime geometry.

Findings

Kinetic term increases energy density by ~50% near the spike

Radial pressure causes mild anisotropy in stress tensor

Backreaction leads to metric deviations more than twice those from mass density alone

Abstract

We study the energy-momentum tensor of a dark matter (DM) spike formed during the adiabatic growth of a black hole embedded in a DM halo, and investigate its backreaction on the spacetime geometry. Within the Einstein cluster framework, we derive the complete tensor, explicitly incorporating the kinetic contribution to the energy density and the anisotropic pressure arising from noncircular particle orbits. Adopting the Hernquist profile as an illustrative model of DM halo and employing parameters appropriate to the Milky Way, we find that near the spike, the kinetic term enhances the total energy density by approximately $50\%$ relative to the rest-mass component, while the nonzero radial pressure induces a mild anisotropy in the stress tensor. The derived tensor satisfies all standard energy conditions. By treating it as a fixed source in Einstein' s equations, we numerically obtain a static, spherically symmetric metric that deviates from the Schwarzschild solution by an amount more than twice that found when only the mass density is considered. These results demonstrate that including the full dynamical structure of the DM spike is essential for accurately modeling the backreaction of DM on black hole spacetimes.