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

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.