Non-minimal coupling of scalar and gauge fields with gravity: an entropy current and linearized second law

Abstract

This work extends the proof of a local version of the linearized second law involving an entropy current with non-negative divergence by including the arbitrary non-minimal coupling of scalar and U(1) gauge fields with gravity. In recent works, the construction of entropy current to prove the linearized second law rested on an important assumption about the possible matter couplings to gravity: the corresponding matter stress tensor was assumed to satisfy the null energy conditions. However, the null energy condition can be violated, even classically, when the non-minimal coupling of matter fields to gravity is considered. Considering small dynamical perturbations around stationary black holes in diffeomorphism invariant theories of gravity with non-minimal coupling to scalar or gauge fields, we prove that an entropy current with non-negative divergence can still be constructed. The additional non-minimal couplings that we have incorporated contribute to the entropy current, which may even survive in the equilibrium limit. We also obtain a spatial current on the horizon apart from the entropy density in out-of-equilibrium situations. We achieve this by using a boost symmetry of the near horizon geometry, which constraints the off-shell structure of a specific component of the equations of motion with newer terms due to the non-minimal couplings. The final expression for the entropy current is U(1) gauge-invariant for gauge fields coupled to gravity. We explicitly check that the entropy current obtained from our abstract arguments is consistent with the expressions already available in the literature for specific model theories involving non-minimal coupling of matter with higher derivative theories of gravity. Finally, we also argue that the physical process version of the first law holds for these theories with arbitrary non-minimal matter couplings.

Noether Gauge Symmetry Approach Applying for the Non‐Minimally Coupled Gravity to the Maxwell Field

“…It has been shown that one way to investigate the thermodynamic properties of dynamic black holes is working in the limit of weak gravity and considering the non-minimal term as a perturbation around the Einstein gravity, known as the linearized amplitude expansion. [66] In the context of the field equation, we can recast the field equation (35) in a way that the higher order corrections are written as an energy-momentum tensor of geometrical origin describing an effective source term on the right hand side of the standard Einstein field equations, namely…”

“…It has been shown that one way to investigate the thermodynamic properties of dynamic black holes is working in the limit of weak gravity and considering the non‐minimal term as a perturbation around the Einstein gravity, known as the linearized amplitude expansion. [ 66 ] In the context of the field equation, we can recast the field equation () in a way that the higher order corrections are written as an energy‐momentum tensor of geometrical origin describing an effective source term on the right hand side of the standard Einstein field equations, namely $$$$\begin{equation} G_{\mu \nu }=T_{\mu \nu }^{m}+T_{\mu \nu }^{g}, \end{equation}$$$$where $$$$\begin{equation} T_{\mu \nu }^{m}=\frac{-8\pi G Y_{\mathcal {R}}(\mathcal {R})}{1 - Y_{\mathcal {R}}(\mathcal {R}) {F^2} }T_{\mu \nu }, \end{equation}$$$$and $$$$\begin{align} &T_{\mu \nu }^{g}=\frac{1}{1 - Y_{\mathcal {R}}(\mathcal {R}) {F^2} }\nonumber \\ &\;\times{\left[{1 \over 2} {\left[\mathcal {R} - {\left(1 -...$$…”

Generalized second law for non-minimally coupled matter theories

“…The thermodynamic properties of black holes in higher curvature theories of gravity have been a subject of active research [1][2][3][4][5][6][7][8][9][10][11][12][13]. The area law [14][15][16][17][18] for the entropy of black holes in the presence of higher curvature terms is modified to the Iyer-Wald entropy for stationary black holes [4,5].…”

“…In previous proofs of the second law [8][9][10], the approach was to fix the JKM ambiguities such that the new entropy functional satisfied a second law. In [11][12][13], building on [8][9][10], a local version of the second law of black hole thermodynamics was established for arbitrary diffeomorphic invariant theories of gravity minimally coupled to matter fields. By local version, we mean one has a local 'entropy current' with non-negative divergence on the black hole's dynamic horizon.…”

“…For non-equilibrium configurations, these terms denote the dynamical corrections to the Iyer-Wald entropy needed for the entropy functional to satisfy a second law. By fixing, we mean that the numerical coefficient multiplying these terms gets fixed [10][11][12][13]. The spatial components of the current facilitated the redistribution of entropy across different spatial sections of the horizon.…”