Black hole geometrothermodynamics and critical phenomena: a look from Tsallis entropy-based perspective

The paper addresses how non-additive Tsallis entropy modifies the thermodynamics and geometrothermodynamics of charged AdS black holes (BHs) with a global monopole, focusing on phase transitions and critical phenomena. Standard BH entropy S_BH scales with horizon area and is non-extensive; Tsallis entropy Sδ = (S_BH)^δ has been proposed to model systems with long-range interactions (like gravity) and can restore extensivity under suitable δ. AdS BHs in extended phase space (identifying cosmological constant with pressure) exhibit van der Waals–like phase transitions, and geometrothermodynamics (Weinhold/Ruppeiner) links thermodynamic curvature to microscopic interactions. Global monopoles arise from symmetry breaking in the early universe and modify BH geometry and thermodynamics. The study aims to understand qualitatively and quantitatively how Tsallis entropy impacts BH phase structure, criticality, thermodynamic curvature, and radiation sparsity, and to compare with van der Waals fluids.

The background spans: foundational BH thermodynamics (Bekenstein-Hawking), extended phase space thermodynamics where Λ ↔ pressure and mass ↔ enthalpy, and van der Waals fluid analogies for AdS BHs. Prior geometrothermodynamic analyses (Weinhold, Ruppeiner, Quevedo, HPEM) have connected curvature sign to microscopic interactions (repulsive for charged systems). Non-additive entropies (Tsallis, Rényi, Barrow) have been applied to cosmology and BHs, with constraints on δ and sometimes scale dependence. Global monopoles emerge from SO(3) → U(1) symmetry breaking and influence BH properties; charged monopole BHs have shown van der Waals-like transitions in standard entropy. Previous works also studied sparsity of Hawking radiation and its modification by generalized uncertainty principles and non-extensive statistics. However, a systematic Tsallis-based analysis for charged AdS BHs with global monopole, including critical phenomena and thermodynamic geometry, was lacking.

- Geometry and fields: Start from the Lagrangian for a global monopole (triplet scalar with symmetry breaking). Use the static spherically symmetric AdS charged BH solution with monopole parameter η (from η^2 = 8π η_0^2) and metric function f(r) = 1 − 2m/r + q^2/r^2 + r^2/l^2 after rescalings that factor the solid angle deficit 1 − η^2. - Thermodynamic setup: Horizon radius r_h solves f(r_h)=0. ADM mass and charge: M = (1 − η^2) m, Q = (1 − η^2) q. Surface gravity gives temperature; horizon area A_bh = 4π (1 − η^2) r_h^2. - Tsallis entropy: Replace Bekenstein-Hawking entropy by S = (S_BH)^δ = [π (1 − η^2) r_h^2]^δ. - Extended phase space: Identify P = 3/(8π l^2), V = (∂M/∂P)_{S,Q} = (4/3)π (1 − η^2) r_h^3. Verify first law dM = T dS + ϕ dQ + V dP and Smarr relation M = 2(TS − VP) + ϕQ. - Working ensemble: Canonical ensemble with fixed Q. Express M as a function of S, and obtain T(S) via T = (∂M/∂S)_{P,Q}. Analyze T(S) behavior and locate stationary points indicative of criticality. - Heat capacity and criticality: Compute C_p = T (∂S/∂T)_P. Discontinuities of C_p signal phase transitions. Determine the equation of state by combining T(S) with the P definition and expressing P in terms of r_h or specific volume v = 2 r_h. Find critical point from ∂P/∂v|_T = 0 and ∂^2P/∂v^2|_T = 0 and extract v_c, T_c, P_c, V_c. - Law of corresponding states: Rescale variables to reduced form p = P/P_c, ν = v/v_c, τ = T/T_c and derive a generalized corresponding-states relation dependent on δ. - Gibbs free energy: G = M − T S computed as a function of T and P using r_h(P,T) from the EoS. Plot G(T) at fixed P and in 3D to identify swallow-tail indicating first-order transitions and to extract the coexistence line where SBH and LBH phases have equal G. - Critical exponents: Expand the generalized corresponding-states EoS near the critical point using reduced variables. Use Maxwell’s equal area construction to obtain order parameter behavior and derive α, β, γ, Δ. - Sparsity: Define sparsity parameter η̃ = C g λ_t^2 / A_eff with λ_t = 2π/T, A_eff = 27A_bh/4, and insert T(S) to obtain η̃_δ(S). Analyze behavior versus S and δ. - Ruppeiner geometry: Compute Ruppeiner scalar curvature R_Rup on the S − P space for the AdS BH with monopole under Tsallis entropy, interpreting the sign as indicating repulsive (positive) or attractive (negative) interactions. - BTZ extension: Apply Tsallis prescription to charged (2+1)D BTZ black holes. Replace S_btz → (S_btz)^δ, derive modified M, T, V, ϕ and compute Ruppeiner curvature in S − P and S − V spaces. Check consistency across coordinate choices.

- Phase structure and equation of state: With Tsallis entropy, charged AdS BHs with a global monopole maintain van der Waals–like small–large BH phase transitions in the extended phase space. P–v isotherms show the standard oscillation below T_c, an inflection at T_c, and monotonic behavior above T_c. Gibbs free energy exhibits a swallow-tail below P_c, indicating a first-order SBH–LBH transition and a coexistence line analogous to fluids. - Critical point and parameters: The critical specific volume, temperature, pressure, and thermodynamic volume depend on δ and η. They are physically meaningful for 1/2 < δ < 3/2 (positivity and finiteness), recovering the standard δ = 1 results when δ → 1. Qualitative trends: for δ < 1, v_c increases and P_c decreases relative to δ = 1; for δ > 1 the opposite occurs; T_c is above the standard value at small and large δ and dips below it in an intermediate δ interval. As Q → 0, v_c → 0 while T_c, P_c → ∞, indicating the charged nature of the transition persists in the Tsallis framework (in Einstein gravity). - Critical coefficient and law of corresponding states: The ratio P_c v_c / T_c becomes δ- and η-dependent and differs from the universal 3/8 unless δ = 1. A generalized law of corresponding states is derived, reducing to the standard form when δ = 1. - Critical exponents: The exponents remain the same as in van der Waals fluids and standard AdS BHs: α = 0, β = 1/2, γ = 1, Δ = 3. Thus Tsallis non-additivity changes critical parameters but not universality class. - Heat capacity and stability: C_p shows one, two, or no divergences depending on parameters, with thermodynamically stable SBH and LBH branches (C_p > 0) separated by an unstable intermediate branch (C_p < 0). At a critical value of l the two divergences merge; for sufficiently small l, C_p remains positive and no transition occurs. - Sparsity of radiation: The Tsallis-based sparsity parameter η̃_δ(S) is larger than the standard Hawking value at small S (more sparse) and decreases monotonically with S, vanishing asymptotically (approaching blackbody-like emission). For δ < 1 sparsity decreases faster than for δ = 1; for δ > 1 it decreases more slowly. - Thermodynamic curvature (AdS BHs): Ruppeiner curvature R_Rup(S,P) is positive at small S (indicating repulsive microinteractions) and tends to zero for large S (effectively non-interacting). Tsallis parameter δ shifts the S-interval where repulsion prevails; δ < 1 tends to suppress the repulsive regime earlier relative to δ = 1, while δ > 1 extends it. The singularity of R_Rup coincides with the physical limitation point (T = 0) and is not a transition singularity. - BTZ black holes: For charged (2+1)D BTZ BHs under Tsallis entropy, the Ruppeiner curvature in S − P and S − V spaces is non-negative and asymptotically vanishing, indicating repulsive interactions persisting across the parameter space considered. Curvatures computed in different thermodynamic coordinate spaces agree.

The analysis shows that adopting a non-additive Tsallis entropy preserves the qualitative van der Waals correspondence for charged AdS BHs with global monopole while quantitatively shifting the critical parameters (v_c, T_c, P_c) and the corresponding reduced equation of state. The unchanged critical exponents indicate robustness of the universality class under Tsallis deformations, suggesting that non-additivity primarily rescales thermodynamic variables rather than altering the underlying mean-field critical behavior. The Ruppeiner curvature results support a microstructural picture where charge-induced repulsion dominates at smaller horizon radii/entropies and fades at larger sizes; Tsallis non-additivity modulates the range over which repulsion is significant. The sparsity results imply that non-additivity can make early-stage emission more sparse and late-stage emission less sparse than Hawking’s, potentially providing phenomenological handles to constrain δ. Extending to BTZ BHs shows the repulsive microinteraction behavior is robust across dimensions and persists under different coordinate choices in thermodynamic geometry. Overall, the findings reinforce the BH–van der Waals analogy and highlight where non-additive entropy leaves its imprint without changing universal scaling.

The paper develops a Tsallis entropy-based geometrothermodynamic framework for charged AdS black holes with a global monopole and extends it to charged BTZ black holes. Main contributions: (i) derivation of the Tsallis-modified equation of state, critical parameters, and coexistence structure exhibiting a first-order SBH–LBH transition; (ii) demonstration that the critical exponents remain α = 0, β = 1/2, γ = 1, Δ = 3, preserving the van der Waals universality class; (iii) computation of Ruppeiner curvature indicating predominantly repulsive microinteractions at small entropies, decaying to non-interacting behavior at large entropies, with δ shifting the onset and termination of these regimes; (iv) analysis of radiation sparsity under Tsallis statistics showing enhanced sparsity at small S and blackbody-like behavior at large S; (v) extension to BTZ BHs with consistent repulsive microstructure signals across thermodynamic coordinate spaces. Future work includes deriving modified field equations consistent with Tsallis statistics, extending to rotating and exotic BTZ BHs, exploring alternative non-extensive entropies (Rényi, Kaniadakis, Barrow), and relating Tsallis thermodynamics to generalized uncertainty principles to further understand the interplay between non-Gaussian statistics and quantum gravity effects.

- Entropy-only deformation: Tsallis prescription is assumed to modify only the entropy while leaving the gravitational action and field equations unchanged; a full theory with consistent modified dynamics is not developed here. - Parameter ranges: Physical results and criticality require 1/2 < δ < 3/2 and η^2 < 1; the study mainly considers small deviations |δ − 1| < 1/2. - Ensemble and theory: Analysis is performed in the canonical ensemble (fixed Q) within Einstein gravity; uncharged limits and higher-curvature corrections may change phase structure. - Model specifics: Numerical illustrations often use fixed representative values (e.g., Q, η, l), and explicit constraints on δ are inferred qualitatively rather than from observational data. - Geometric diagnostics: Ruppeiner geometry is used as a phenomenological indicator; alternative metrics (Quevedo, HPEM) can yield different singularity structures, not exhaustively explored here under Tsallis entropy.