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

TL;DR

This work addresses how a dark matter spike formed by adiabatic growth of a black hole backreacts on the surrounding spacetime. It builds a self-consistent description by computing the full energy-momentum tensor of the spike within the Einstein cluster framework, using a Hernquist halo and Milky Way parameters, and then solving the Einstein equations with this fixed source to obtain a slightly perturbed Schwarzschild metric. The key findings are that the kinetic energy of bound DM particles increases the spike's energy density by about $50\%$ relative to the rest-mass density, while a nonzero radial pressure introduces mild anisotropy, and the resulting metric deviates from Schwarzschild by a larger amount than in density-only treatments, while all energy conditions remain satisfied. This demonstrates that incorporating the complete dynamical structure of the DM spike is essential for accurate modeling of DM backreaction on BH spacetimes and provides a pathway to generalize to other DM halos and rotating black holes in future work.

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.

Figures (5)

• Figure 1: The relative potential for a BH and an isolated point mass, respectively. This plot shows that these two potentials are similar before $r=10^{5}$, where we set $M_{BH}=1$.
• Figure 2: This plot shows the integration region $(E,L)$ of integrals (\ref{['eq:4 flux on considered quantities']}) and (\ref{['eq:energy momentum tensor on considered quantities']}) for each fixed $r$. The colored lines represent the maximal $L^2$ for a given $r$ and $E$.
• Figure 3: The mass density (upper) and energy-momentum tensor (bottom) for the DM profile around a Schwarzschild BH. The black line in the upper panel shows the initial mass density for the Hernquist profile. After the adiabatic growth of a BH, $\rho_{M}$ forms a "spike" near the BH. Components of $T_{\mu}^{\nu}$ exhibit a similar spike, although their peak values differ by orders of magnitude.
• Figure 4: The comparison between the mass density $\rho_{M}$ (blue) and the energy density $\rho_{E}$ (red) of the DM spike. At the peak, the maximum of $\rho_{E}$ is approximately $1.50$ times the maximum of $\rho_{M}$.
• Figure 5: The deviation of the numerical solutions $g(r)$ (upper left) and $m(r)$ (bottom left) from their Schwarzschild counterparts $g_{\textrm{Sch}}$ and $m_{\textrm{Sch}}$, and the ratio of deviations of our resulting metric to those of $g_{\textrm{EC}}$ and $m_{\textrm{EC}}$ obtained by an alternative approach from Chakraborty:2024gcr in the right panels. The subscript "EC" represents that the alternative approach with the Einstein cluster model. The gray thin lines in right panels show the radius where the energy density $\rho_{E}$ becomes a maximum value.