# Distinguishing a Kerr-like black hole and a naked singularity in perfect fluid dark matter via precession frequencies

## TL;DR

This paper investigates how precession frequencies can distinguish Kerr-like black holes from naked singularities in perfect fluid dark matter, highlighting the influence of dark matter parameters on observable frequencies.

## Contribution

It introduces a method using precession frequencies to differentiate black holes from naked singularities in a dark matter context, considering the effects of the PFDM parameter.

## Key findings

- Precession frequency diverges near black hole centers but remains finite for naked singularities.
- Dark matter parameter $eta$ significantly alters Kepler, epicyclic, and nodal precession frequencies.
- A critical spin parameter $a_c$ determines the transition between black hole and naked singularity.

## Abstract

We study a Kerr-like black hole and naked singularity in perfect fluid dark matter (PFDM). The critical value of spin parameter $a_c$ is presented to differentiate the black hole from naked singularity. It is seen that for any fixed value of dark matter parameter $\alpha$ the rotating object is black hole if $a\leq a_c$ and naked singularity if $a>a_c$. Also for $-2\leq\alpha<2/3$ the size of the black hole horizons decrease whereas for $2/3<\alpha$ it increases. We also study spin precession frequency of a test gyroscope attached to stationary observer to differentiate a black hole from naked singularity in perfect fluid dark matter. For the black hole, spin precession frequency blows up as the observer reaches the central object while for naked singularity it remains finite except at the ring singularity. Moreover, we study Lense-Thirring precession for a Kerr-like black hole and geodetic precession for Schwarzschild black hole in perfect fluid dark matter. To this end, we have calculated the Kepler frequency (KF), the vertical epicyclic frequency (VEF), and the nodal plane precession frequency (NPPF). Our results show that, the PFDM parameter $\alpha$ significantly affects those frequencies. This difference can be used by astrophysical observations in the near future to shed some light on the nature of dark matter.