Lorentz-violating ModMax black holes in phantom-enhanced Kalb–Ramond gravity: Thermodynamics and topological charges
Sekhmani Y. Maurya S.K. Rayimbaev J. Altanji M. Ibragimov I. Muminov S.
December 2025Elsevier B.V.
Physics of the Dark Universe
2025#50
We derive exact static, spherically symmetric black hole (BH) solutions within the framework of four-dimensional Einstein gravity, coupled to ModMax nonlinear electrodynamics and a Kalb–Ramond two-form. This coupling is governed by a discrete sign parameter, denoted as ζ=±1. In the ”ordinary” branch (ζ=+1), the metric features both Cauchy and event horizons, while the electromagnetic stress–energy adheres to all classical energy conditions. In contrast, the ”phantom” branch (ζ=−1) accommodates only a single horizon and simultaneously violates the weak, null, and strong conditions, which reflects genuine ghost-matter behaviour. The thermodynamic behaviour for ζ=+1 results in multiple divergences in heat capacity and presents a distinctive swallowtail pattern in the AdS Gibbs free energy, indicative of a first-order phase transition that encompasses small, intermediate, and large phases. In contrast, for ζ=−1, there is only a single divergence observed, accompanied by a smooth Gibbs curve with no van der Waals-type oscillatory behaviour, which precludes the possibility of true coexistence. Topological analysis using Duans φ-mapping in AdS further differentiates the branches by examining their winding numbers. For ζ=+1, the winding numbers are (+1,−1,+1), resulting in a total topological charge (net winding number) of W=+1. In contrast, for ζ=−1, the winding numbers are (−1,+1), leading to a total topological charge of W=0. This distinction assigns a clear topological charge, which differentiates between the first-order transitions associated with ζ=+1 and the smooth, latent-heat-free phase changes linked to ζ=−1. Overall, ζ serves as the primary determinant influencing causal structure, energy condition profiles, critical phenomena, and topological charge in ModMax-Kalb-Ramond BH.
Black holes , Exact solution , Kalb–Ramond gravity , Thermodynamics
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Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan
Centre for Research Impact & Outcome, Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, Rajpura, 140401, India
Department of Mathematical and Physical Sciences, College of Arts and Sciences, University of Nizwa, P.O. Box 33, Nizwa, 616, Oman
Institute of Nuclear Physics, Ibragimova, 1, Almaty, 050032, Kazakhstan
National University of Uzbekistan, Tashkent, 100174, Uzbekistan
Tashkent International University of Education, Imom Bukhoriy 6, Tashkent, 100207, Uzbekistan
University of Tashkent for Applied Sciences, Str. Gavhar 1, Tashkent, 100149, Uzbekistan
Urgench State University, Kh. Alimjan Str. 14, Urgench, 221100, Uzbekistan
Tashkent State Technical University, Tashkent, 100095, Uzbekistan
Department of Mathematics, College of Sciences, King Khalid University, Abha, 61413, Saudi Arabia
Kimyo International University in Tashkent, Shota Rustaveli street 156, Tashkent, 100121, Uzbekistan
Mamun University, Bolkhovuz Street 2, Khiva, 220900, Uzbekistan
Center for Theoretical Physics
Centre for Research Impact & Outcome
Department of Mathematical and Physical Sciences
Institute of Nuclear Physics
National University of Uzbekistan
Tashkent International University of Education
University of Tashkent for Applied Sciences
Urgench State University
Tashkent State Technical University
Department of Mathematics
Kimyo International University in Tashkent
Mamun University
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