Black hole surrounded by perfect fluid dark matter with a background Kalb-Ramond field

TL;DR

This work addresses how spontaneous Lorentz symmetry breaking from a Kalb-Ramond field interacts with perfect fluid dark matter to modify black hole spacetimes. It derives an exact static, spherically symmetric KRPFDM black hole metric with $A(r)=\frac{1}{1-α}-\frac{2M}{r}+\frac{β}{(1-α)r}\log\frac{r}{|β|}$, analyzes curvature invariants to characterize singularities, and shows how the horizon position $r_h$ depends on $α$ and $β$ via $r_h=β \mathrm{ProductLog}\left(e^{-\frac{2(α-1)M}{β}}\right)$. Employing Bozza's strong-field lensing formalism, it computes photon-sphere radius $x_m$, the logarithmic deflection coefficient $\bar{a}$ and residue $\bar{b}$, and the lensing observables $\theta_∞$, $s$, and $r_{mag}$ for M87$^*$ and Sgr A$^*$, demonstrating that both $α$ and $β$ suppress lensing signals relative to Schwarzschild and can cause nontrivial shifts in image positions. Shadow-based constraints are then used to bound $α$ and $β$ against EHT/Keck/VLTI data, illustrating that a finite parameter space remains compatible with observations, though current data cannot decisively distinguish KRPFDM black holes from Schwarzschild. The study provides a concrete framework to test Lorentz-violating and PFDM effects in strong gravity with current and upcoming observations such as ngEHT.

Abstract

With an intent to explore the interplay between the Lorentz symmetry breaking (LSB) and the presence of dark matter (DM), we obtain a static and spherically symmetric black hole (BH) solution in the background of nonminimally coupled Kalb-Ramond (KR) field surrounded by perfect fluid dark matter (PFDM). The KR field is frozen to a non-zero vacuum expectation value (VEV) that breaks the particle Lorentz symmetry spontaneously. We explore scalar invariants, Ricci Scalar, Ricci squared, and Kretschmann Scalar, to probe the nature of singularities in the obtained solution. We then study strong gravitational lensing in the background of our BH, i.e., KRPFDM BH, revealing the adverse impact of LSB parameter $α$ and PFDM parameter $β$ on the lensing coefficients. The significant effect of our model parameters is evident in strong lensing observables. Bounds on the deviation from Schwarzschild, $δ$, for supermassive BHs (SMBHs) $M87^*$ and $SgrA^*$ from the EHT, Keck, and VLTI observatories are then utilized to put our BH model to the test and extract possible values of model parameters $α$ and $β$ that generate theoretical predictions in line with experimental observations within $1σ$ confidence level. Our study sheds light on the combined effect of LSB and PFDM and may be helpful in finding their signature.

Figures (15)

• Figure 1: Variation of event horizon with $\alpha$ for different values of $\tilde{\beta}$ (left panel) and with $\tilde{\beta}$ for different values of $\alpha$ (right panel).
• Figure 2: Variation of photon radius with $\alpha$ for different values of $\tilde{\beta}$ (left panel) and with $\tilde{\beta}$ for different values of $\alpha$ (right panel).
• Figure 3: Variation of critical impact parameter $b_m$ with $\alpha$ for different values of $\tilde{\beta}$ (left panel) and with $\tilde{\beta}$ for different values of $\alpha$ (right panel).
• Figure 4: Variation of lensing coefficient $\overline{a}$ with $\alpha$ for different values of $\tilde{\beta}$ (left panel) and with $\tilde{\beta}$ for different values of $\alpha$ (right panel).
• Figure 5: Variation of lensing coefficient $\overline{b}$ with $\alpha$ for different values of $\tilde{\beta}$ (left panel) and with $\tilde{\beta}$ for different values of $\alpha$ (right panel).