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

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.

💡 Research Summary

The paper investigates the microscopic thermodynamic structure of electrically charged anti‑de Sitter (AdS) black holes in Einstein‑Kalb‑Ramond (EKR) bumblebee gravity with a spherically symmetric cloud of strings. Lorentz violation is encoded in a dimensionless parameter ℓ that rescales the metric coefficients, while the string cloud is described by a parameter α that introduces a solid‑angle deficit. The resulting lapse function f(r) contains ℓ‑ and α‑dependent modifications of the mass, charge, and cosmological terms, leading to a deformed horizon structure and an altered extremality condition.

In the extended phase‑space formalism the cosmological constant is identified with a thermodynamic pressure P, modified by ℓ as Λ=−8π(1−ℓ)P. The black‑hole mass M plays the role of enthalpy and can be expressed in terms of the horizon radius r₊, charge Q, pressure P, and the deformation parameters. Entropy follows the usual area law S=πr₊², while the thermodynamic volume V=4πr₊³/3 and Hawking temperature T contain three contributions: a solid‑angle term proportional to (1−α)/(1−ℓ), a Coulomb term dressed by (1−ℓ), and the usual AdS pressure term. From these quantities the Gibbs free energy, internal energy, and specific heat at constant pressure C_P are derived. C_P diverges along a spinodal curve that separates small‑ and large‑black‑hole branches; the locations of these divergences shift with ℓ and α.

The authors then construct Weinhold and Ruppeiner metrics on the two‑dimensional state space (S,Q). The Weinhold metric is the Hessian of the internal energy U, while the Ruppeiner metric is the negative Hessian of the entropy. Because the underlying thermodynamic potentials already contain ℓ and α, the resulting scalar curvatures R_W and R_R are explicit functions of these parameters. Detailed calculations reveal that the curvature singularities of both metrics coincide precisely with the divergences of C_P, i.e., the spinodal boundaries and the second‑order critical point. This confirms that geometric singularities faithfully track standard thermodynamic criticality.

A key result concerns the sign of the Ruppeiner curvature. In the small‑black‑hole region R_R>0, indicating effectively repulsive (Fermi‑like) microscopic interactions. In the large‑black‑hole region R_R<0, signifying attractive Van der Waals‑type interactions. The transition region exhibits a rapid change of sign, and the magnitude of R_R can be tuned continuously by varying ℓ and α, providing a “knob’’ for controlling microscopic correlation strength. Near the critical point the curvature scales as |t|^{−2} (t being the reduced temperature), matching the mean‑field critical exponent and demonstrating that the universality class remains unchanged despite the deformations.

The paper also links the thermodynamic geometry to a topological classification based on φ‑mapping and winding numbers. The curvature singularities map onto topological defects that separate regions with different winding numbers, while the total topological charge W=1 stays invariant under variations of (ℓ,α). This reinforces the robustness of the phase‑structure under the considered deformations.

Finally, the authors discuss possible dynamical implications. They suggest that quasi‑normal mode spectra and relaxation times, previously studied in Lorentz‑violating and string‑cloud backgrounds, may correlate with the landscape of R_R, offering a route to connect microscopic thermodynamic correlations with observable ringdown signatures in AdS spacetimes. The study concludes that the Lorentz‑violating parameter ℓ and the string‑cloud parameter α act as controllable knobs that shift critical scales and rescale correlation measures without altering the underlying mean‑field universality, thereby enriching our understanding of black‑hole microphysics in modified gravity settings.