Generalizations of the Smarr formula for black holes with nonlinear electromagnetic fields

Abstract

We present a direct, geometric derivation of the generalized Smarr formula for the stationary axially symmetric black holes with nonlinear electromagnetic fields. The additional term is proven to be proportional to the integral of the trace of the electromagnetic energy-momentum tensor and can be written as a product of two conjugate variables. From the novel relation we can deduce all previously proposed forms of the generalized Smarr formula, which were derived only for the spherically symmetric black holes, and provide the lowest order quantum correction to the classical relation from the Euler-Heisenberg Lagrangian. We use adjective "geometric" just to emphasize that in these derivations auxiliary physical assumptions are avoided by the approach based on the quantities of geometric origin.

Schwarzschild spacetime immersed in test nonlinear electromagnetic fields

“…Formally, this has exactly the same functional form as (44). As Schwarzschild spacetime metric is asymptotically flat, this immediately proves that field (45) asymptotically represents homogeneous magnetic field.…”

“…Just as in classical electrostatics and magnetostatics, a useful strategy for problem solving is introduction of scalar potentials, whenever this is possible [42][43][44]. Magnetic field 1-form defined with respect to a vector field X a is given by B[X] a ≡ X b * F ba .…”

Quasilocal Smarr relations for static black holes

“…However, regarding the quasilocal Smarr relation, the main issue to extend the results shown in this work to the rotating black hole case is related with a more fundamental property of the Kerr-Newman metric, namely, the impossibility to simultaneously set constant values on a specific surface B for β and the chemical potentials for angular momentum and other conserved charges [50]. This issue implies that the quasilocal first law (A13) can not be easily integrated on B to obtain a bilinear form in conjugated extensive and intensive variables through homogeneity arguments; therefore, there is no quasilocal Smarr relation in a simple form as in the static cases considered above, but it has a convoluted form that resembles, for example, the corresponding expression in black holes coupled with nonlinear electrodynamics [14]. Usual derivations of the Smarr relation for rotating black holes [27] are not affected by this problem since the surfaces of constant β and chemical potentials coincide at infinity.…”

“…We must compute τ ij for the metric (14). In this case, the hypersurfaces Σ are surfaces defined by t = constant, with normal vector u µ = −N δ 0 µ .…”

“…where the factors of 2 signal that the scaling properties of the Kerr spacetime are different from the usual thermodynamical systems, which in turn is interpreted as a consequence of the range and the absence of screening that characterize gravity. In the literature, Smarr relations are usually obtained through two ways: by using the Euler theorem for homogeneous functions in the first law of thermodynamics [27], and, on the other hand, by considering Komar-like integrals for the conserved charges (see for example [14][15] [28]). In the context of quasilocal variables, the analysis of the corresponding Smarr relations have been few in number.…”