Deformed black hole immersed in dark matter spike

If a lot of dark matter particles accumulate near the black hole, then the chances of detecting dark matter signals near a black hole are greatly increased. These effects may be observed by the Event Horizon Telescope (EHT), Tianqin project, Taiji project, Laser Interferometer Space Antenna (LISA) and Laser Interferometer Gravitational-Wave Observatory (LIGO). In this work, we explore the effects of dark matter spikes on black hole space-time. For the Schwarzschild-like black hole case, we consider Newton$'$s approximation and perturbation approximation. This makes it possible to use Xu$'$s method to solve the Einstein field equation, and extend Schwarzschild-like black hole to Kerr-like black hole (BH) via Newman-Janis (NJ) algorithm. By analyzing the dark matter spike on the black hole event horizon (EH), stationary limit surfaces (SLS), ergosphere and energy-momentum tensors (EMT), we found that compared with the dark matter halo, the dark matter spike would have a higher effect on the black hole by several orders of magnitude. Therefore, if there is a dark matter spike near the black hole, it is very possible to test the dark matter model through gravitational wave (GW) observation and EHT observation.