On the black hole mass decomposition in nonlinear electrodynamics

In the weak field limit of nonlinear Lagrangians for electrodynamics, i.e. theories in which the electric fields are much smaller than the scale (threshold) fields introduced by the nonlinearities, a generalization of the Christodoulou–Ruffini mass formula for charged black holes is presented. It proves that the black hole outer horizon never decreases. It is also demonstrated that reversible transformations are, indeed, fully equivalent to constant horizon solutions for nonlinear theories encompassing asymptotically flat black hole solutions. This result is used to decompose, in an analytical and alternative way, the total mass-energy of nonlinear charged black holes, circumventing the difficulties faced to obtain it via the standard differential approach. It is also proven that the known first law of black hole thermodynamics is the direct consequence of the mass decomposition for general black hole transformations. From all the above we finally show a most important corollary: for relevant astrophysical scenarios nonlinear electrodynamics decreases the extractable energy from a black hole with respect to the Einstein–Maxwell theory. Physical interpretations for these results are also discussed.