Thermodynamics of Magnetised Kerr-Newman Black Holes

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Abstract

The thermodynamics of a magnetised Kerr-Newman black hole is studied to all orders in the appended magnetic field \(B\). The asymptotic properties of the metric and other fields are dominated by the magnetic flux that extends to infinity along the axis, leading to subtleties in the calculation of conserved quantities such as the angular momentum and the mass. We present a detailed discussion of the implementation of a Wald-type procedure to calculate the angular momentum, showing how ambiguities that are absent in the usual asymptotically-flat case may be resolved by the requirement of gauge invariance. We also present a formalism from which we are able to obtain an expression for the mass of the magnetised black holes. The expressions for the mass and the angular momentum are shown to be compatible with the first law of thermodynamics and a Smarr type relation. Allowing the appended magnetic field \(B\) to vary results in an extra term in the first law of the form \(-\mu dB\) where \(\mu\) is interpreted as an induced magnetic moment. Minimising the total energy with respect to the total charge \(Q\) at fixed values of the angular momentum and energy of the seed metric allows an investigation of Wald's process. The Meissner effect is shown to hold for electrically neutral extreme black holes. We also present a derivation of the angular momentum for black holes in the four-dimensional STU model, which is \({\cal N}=2\) supergravity coupled to three vector multiplets.

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Black Hole Entropy is Noether Charge

Robert Wald (1993)

We consider a general, classical theory of gravity in \(n\) dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, \(\xi^a\), on spacetime one can associate a local symmetry and, hence, a Noether current \((n-1)\)-form, \({\bf j}\), and (for solutions to the field equations) a Noether charge \((n-2)\)-form, \({\bf Q}\). Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply \(2 \pi\) times the integral over \(\Sigma\) of the Noether charge \((n-2)\)-form associated with the horizon Killing field, normalized so as to have unit surface gravity. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the ``second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via ``Euclidean methods" also is explained.