Thermodynamic geometry of charged AdS black holes with a string cloud in Lorentz-violating Einstein-Kalb-Ramond gravity

Abstract

We investigate the thermodynamic microstructure of electrically charged AdS black holes in Einstein-Kalb-Ramond bumblebee gravity in the presence of a spherically symmetric cloud of strings. Employing Weinhold and Ruppeiner thermodynamic geometries in complementary thermodynamic representations, we show that curvature singularities consistently track the spinodal boundaries and the second-order critical point associated with a Van der Waals-like small/large black-hole phase transition. Moreover, the sign structure of the Ruppeiner curvature provides a transparent characterization of the competing microscopic interactions across the coexisting phases. We find that the Lorentz-violating parameter $\ell$ and the string-cloud parameter $α$ shift the critical scales and rescale correlation measures while preserving the universality class of the critical behavior. We further comment on dynamical and holographic implications and contrast the thermodynamic sensitivity to $(\ell,α)$ with thin-disk optical signatures.

Figures (15)

• Figure 1: (Color online) Critical thermodynamic quantities ($v_c$, $T_c$, and $P_c$) of the charged AdS black hole with a cloud of strings in Kalb-Ramond gravity, obtained from the criticality condition in Eq. (\ref{['eq:review_critical']}). We fix $Q = 0.6$ and show the dependence of the critical specific volume $v_c$ (a), the critical temperature $T_c$ (b), and the critical pressure $P_c$ (c) on the deformation parameters. Solid blue curves correspond to varying $\ell$ at fixed $\alpha = 0.05$, whereas dashed red curves correspond to varying $\alpha$ at fixed $\ell = 0.10$.
• Figure 2: (Color online) Corrected entropy $S_c$ as a function of the horizon radius $r_+$, obtained from Eq. (\ref{['eq:entropy_corrected_explicit']}).Panel (a) shows $S_c$ for fixed $\alpha = 0.05$ and different values of the cloud-of-strings parameter $\ell$, while panel (b) displays the complementary behaviour for fixed $\ell = 0.10$ and varying Lorentz-violating parameter $\alpha$.. In both panels we set $Q = 0.6$, $P = 0.01$, $\lambda_1 = 1$ and $\lambda_2 = 0$, and we use the Bekenstein--Hawking entropy $S_0 = \pi r_+^2$ as a reference curve.
• Figure 3: (Color online) Three--dimensional representation of the corrected entropy obtained from Eq. (\ref{['eq:entropy_corrected_explicit']}). Figure (a) displays the surface $S_c(r_+, \ell)$ for fixed $\alpha = 0.05$, $Q = 0.6$, $P = 0.01$, $\lambda_1 = 1$ and $\lambda_2 = 0$, while Fig. (b) shows $S_c(r_+, \alpha)$ for fixed $\ell = 0.10$ and the same remaining parameters. In both cases the entropy grows monotonically with $r_+$ but is suppressed as the deformation parameters $\ell$ and $\alpha$ increase, illustrating the combined geometric effect of the string cloud and the Lorentz--violating background.
• Figure 4: (Color online) Corrected enthalpy $H_c$ as a function of the horizon radius $r_+$, obtained from Eq. (\ref{['mass-2']}). Figure (a) shows the effect of the pressure for fixed $\ell = 0.10$, $\alpha = 0.05$, $Q = 0.6$, $\lambda_1 = 1$ and $\lambda_2 = 0$. Figure (b) displays the dependence on the cloud--of--strings parameter $\ell$ for fixed $P = 0.01$ and the same remaining parameters.
• Figure 5: (Color online) Corrected thermodynamic volume $V_c$ as a function of the horizon radius $r_+$, as given by Eq. (\ref{['volume-2']}). In Fig. (a) we fix $\lambda_2 = 0$ and vary the leading fluctuation parameter $\lambda_1$, whereas in Fig. (b) we fix $\lambda_1 = 1$ and vary the subleading parameter $\lambda_2$. The dashed curve in both panels corresponds to the uncorrected volume $V_0 = \tfrac{4\pi}{3} r_+^3$.