Wald Entropy in Extended Modified Myrzakulov Gravity Theories: \(f(R, T, Q, R_{μν}T^{μν}, R_{μν}Q^{μν}, \dots)\)

TL;DR

The paper develops a covariant framework to compute black hole entropy in extended gravity theories that include curvature, torsion, and non-metricity within the Myrzakulov class. Using the Wald entropy formalism in the vielbein setting, it derives explicit entropy corrections for Lagrangians of the form $f(R,T,Q,R_{\mu\nu}T^{\mu\nu},R_{\mu\nu}Q^{\mu\nu},\dots)$, showing how derivatives of the Lagrangian with respect to these invariants modify the Bekenstein–Hawking area law. The work systematically catalogues entropy expressions for various subcases (e.g., $f(R)$, $f(R,T)$, $f(R,T,Q)$, and mixed contractions), and discusses the implications for thermodynamics, nonequilibrium effects, and potential microscopic interpretations. By connecting diffeomorphism invariance, Noether charges, and extended geometric degrees of freedom, the results lay a groundwork for gravitational thermodynamics in non-Riemannian spacetimes and offer avenues for phenomenology in cosmology, gravitational waves, and quantum gravity. The conclusions emphasize that extended geometric structures yield consistent entropy corrections and suggest future explorations in cosmology, compact objects, and unification schemes.

Abstract

We investigate black hole entropy in a broad class of modified Myrzakulov gravity theories defined by generalized Lagrangians of the form \( \mathcal{L} = αR + F(T, Q, R_{μν}T^{μν}, R_{μν}Q^{μν}, \dots) \), where \( R \), \( T \), and \( Q \) represent curvature, torsion, and non-metricity scalars. Using the vielbein formalism, we derive the Wald entropy for various subclasses of these models, extending the classical entropy formula to accommodate non-Riemannian geometry. Our focus is on how the additional geometric degrees of freedom modify the entropy expression. The analysis shows that such corrections arise systematically from the extended structure of the action and preserve diffeomorphism invariance. These results refine the theoretical framework for gravitational thermodynamics in extended geometry settings.