Abstract
After considering the quantum corrections of Einstein-Maxwell theory, the effective theory will contain some higher-curvature terms and nonminimally coupled electromagnetic fields. In this paper, we study the first law of black holes in the gravitational electromagnetic system with the Lagrangian ℒ(gab, Rabcd, Fab). Firstly, we calculate the Noether charge and the variational identity in this theory, and then generically derive the first law of thermodynamics for an asymptotically flat stationary-axisymmetric symmetric black hole without the requirement that the electromagnetic field is smooth on the bifurcation surface. Our results indicate that the first law of black hole thermodynamics might be valid for the Einstein-Maxwell theory with some quantum corrections in the effective region.
Highlights
Noether charge in the gravitational electromagnetic systemWe first review the Noether current and Noether charge in the diffeomorphism covariant gravitational electromagnetic theory
Is the Wald entropy which can be expressed as a Noether charge of theory [6, 7], where ab is the binormal of the cross-section B of the event horizon
The first law of black hole in a diffeomorphism covariant theory is generally derived by Iyer and Wald [6]
Summary
We first review the Noether current and Noether charge in the diffeomorphism covariant gravitational electromagnetic theory. The Noether current (n − 1)-form related to the vector field ζa is defined as (2.21). If the dynamical field φ satisfy the on-shell condition Eφ = 0, the Noether current is a closed form, i.e., dJζ = 0, which implies there is a Noether charge (n − 2)-form Qζ such that J = dQ. When the dynamical field φ satisfies the on-shell condition, we have Cζ = 0. Together with the equation of motion (2.19), we can get vζc − ζcL =ζeTec + ζeAejc + 2∇d(ERabcd∇bζa + 2ζb∇aERabcd + EFcdζaAa). We consider a one-parameter family φ(λ) in which any φ(λ) satisfy the on-shell condition, i.e., we have C(λ) = Eφ(λ) = 0 and δC = Eφ = 0. Using the equation of motion (2.19), the electric charge of the spacetime is defined by
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