Topological Mod(A)Max AdS black holes

In this work, we construct new classes of topological black hole solutions in anti-de Sitter (AdS) spacetime using a novel model of nonlinear electrodynamics called Modification Maxwell (ModMax) and Modification phantom or Modification anti-Maxwell (ModAMax). We then evaluate the thermodynamic quantities and verify the first law of thermodynamics. Our study examines how the parameters of the ModMax and ModAMax fields, as well as the topological constant, affect the black hole solutions, thermodynamic quantities, and local and global thermal stabilities. Furthermore, within the framework of extended phase space thermodynamics, we analyze the Joule-Thomson expansion process and determine the inversion curves. This analysis reveals that the ModMax and ModAMax parameters significantly alter the cooling and heating behavior of these AdS black holes, depending on their topology. Finally, by treating these topological Mod(A)Max AdS black holes as heat engines, we assess their efficiencies, demonstrating that the parameters of nonlinear electrodynamics and horizon topology play crucial roles in enhancing or suppressing the system’s thermodynamic performance.

💡 Research Summary

In this paper the authors construct new families of topological black‑hole solutions in four‑dimensional anti‑de Sitter (AdS) spacetime by coupling Einstein gravity with a recently proposed nonlinear electrodynamics (NED) model known as Modification of Maxwell (ModMax) and its phantom counterpart, Modification anti‑Maxwell (ModAMax). The action contains the usual Einstein–Hilbert term with a cosmological constant Λ, and a matter sector described by the Lagrangian

L = S cosh γ − p S² + P² sinh γ,

where γ is a dimensionless ModMax parameter, S = FμνFμν/4 and P = εμναβFμναFβ/4. By setting the pseudoscalar invariant P to zero the authors focus on purely electric configurations. The sign η = +1 corresponds to the standard ModMax field, while η = –1 yields the phantom (anti‑Maxwell) version, collectively denoted Mod(A)Max.

Assuming a static, spherically‑symmetric metric with a constant curvature horizon

ds² = –f(r)dt² + dr²/f(r) + r² dΩ²_k,

where k = 1, 0, –1 encodes spherical, planar, or hyperbolic topology, they solve the Maxwell‑type equation for the gauge potential A_t = h(r) and obtain h(r) = –q/r, with q the electric charge. Substituting this into the Einstein equations leads to a single ordinary differential equation for f(r) whose solution is

f(r) = k – m r – (Λ/3) r² + η q² e^{–γ}/r².

Here m is an integration constant related to the black‑hole mass. When γ → 0 the solution reduces to the usual Reissner–Nordström‑AdS black hole (η = +1) or its phantom counterpart (η = –1). The Kretschmann scalar shows a curvature singularity at r = 0 and asymptotically approaches a constant determined by Λ, confirming the AdS (Λ < 0) or dS (Λ > 0) nature of the spacetime.

Thermodynamic quantities are derived in the extended phase‑space formalism where the cosmological constant is identified with pressure P = –Λ/(8π) and the black‑hole mass M is interpreted as enthalpy H. The temperature, entropy, electric potential, thermodynamic volume and specific heat are obtained as

T = f′(r_h)/(4π), S = π r_h², Φ = q e^{–γ}/r_h, V = (4π/3) r_h³,

with r_h the outer horizon radius defined by f(r_h)=0. The first law dM = T dS + Φ dq + V dP is explicitly verified.

Stability analysis proceeds in two steps. Local stability is examined via the heat capacity at constant pressure C_P; global stability is probed through the Gibbs free energy G = M – TS. For η = +1 (ModMax) the metric can admit two horizons (inner and outer) depending on the values of k, γ and q. Increasing γ generally pushes the outer horizon outward for spherical (k = 1) and planar (k = 0) topologies, while for hyperbolic (k = –1) the effect is more subtle. In contrast, the phantom case (η = –1) always yields a single horizon, and the horizon radius shrinks as γ grows.

The authors then explore the Joule‑Thomson (JT) expansion in the extended phase space, treating the black hole as a thermodynamic system undergoing an iso‑enthalpic process. The JT coefficient μ_JT = (∂T/∂P)_H is computed, and inversion curves T_i(P_i) are plotted for various γ, η and k. For ModMax black holes the inversion curve shifts upward with larger γ, indicating an enlarged cooling region; for ModAMax the opposite trend appears, with the cooling region diminishing as γ increases. The topology influences the shape of the inversion curve: hyperbolic horizons (k = –1) tend to produce broader cooling zones compared with spherical ones.

Finally, the paper treats the black holes as holographic heat engines. A rectangular cycle in the P‑V plane is defined, and the work output W and heat input Q_H are evaluated analytically. The efficiency η = W/Q_H is found to depend sensitively on γ, η and the horizon topology. In the ModMax case, larger γ enhances efficiency, especially for spherical horizons (k = 1), whereas in the ModAMax case efficiency decreases with γ, with hyperbolic horizons (k = –1) offering relatively higher performance. These results demonstrate that the nonlinear electrodynamics parameters and the horizon geometry can be tuned to control thermodynamic performance.

Overall, the study provides a comprehensive analysis of how ModMax and ModAMax fields modify the structure, thermodynamics, phase transitions, JT expansion behavior, and heat‑engine efficiency of topological AdS black holes. It highlights the rich interplay between nonlinear electromagnetic dynamics, horizon topology, and extended thermodynamic variables, offering new avenues for exploring holographic dualities and potential experimental analogues in laboratory NED systems.