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Register a new userThe First Law of Black Hole Mechanics
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Lorenzo Rossi
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The first law of black hole mechanics has been the main motivation for investigating thermodynamic properties of black holes. The first version of this law was proved in cite{Bardeen:1973gs} by considering perturbations of an asymptotically flat, stationary black hole spacetime to other stationary black hole spacetimes. This result was then extended to fully general perturbations, first in the context of Einstein-Maxwell theory in cite{Sudarsky:1992ty},cite{Wald:1993ki}, and then in the context of a general diffeomorphism invariant theory of gravity with an arbitrary number of matter fields in cite{Wald:1993nt},cite{Iyer:1994ys}. Here a review of these two generalizations of the first law is presented, with particular attention to outlining the necessary formalisms and calculations in an explicit and thorough way, understandable at a graduate level. The open problem of defining the entropy for a dynamical black hole that satisfies a form of the second law of black hole mechanics is briefly discussed.
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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 $math{L}(g_{ab}, R_{abcd}, F_{ab})$. 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 axisymmetrical 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.
It is well known that in general theories of gravity with the diffeomorphism symmetry, the black hole entropy is a Noether charge. But what will happen if the symmetry is explicitly broken? By investigating the covariant first law of black hole mechanics with background fields, we show that the Noether entropy is still applicable due to the local nature of the black hole entropy. Moreover, motivated by the proposal that the cosmological constant behaves as a thermodynamic variable, we allow the non-dynamical background fields to be varied. To illustrate this general formalism, we study a generic static black brane in the massive gravity. Using the first law and the scaling argument, we obtain two Smarr formulas. We show that both of them can be retrieved without relying on the first law, hence providing a self-consistent check of the theory.
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The black hole area theorem implies that when two black holes merge, the area of the final black hole should be greater than the sum of the areas of the two original black holes. We examine how this prediction can be tested with gravitational-wave observations of binary black holes. By separately fitting the early inspiral and final ringdown stages, we calculate the posterior distributions for the masses and spins of the two initial and the final black holes. This yields posterior distributions for the change in the area and thus a statistical test of the validity of the area increase law. We illustrate this method with a GW150914-like binary black hole waveform calculated using numerical relativity, and detector sensitivities representative of both the first observing run and the design configuration of Advanced LIGO. We obtain a $sim74.6%$ probability that the simulated signal is consistent with the area theorem with current sensitivity, improving to $sim99.9%$ when Advanced LIGO reaches design sensitivity. An important ingredient in our test is a method of estimating when the post-merger signal is well-fit by a damped sinusoid ringdown waveform.
We present observational confirmation of Hawkings black-hole area theorem based on data from GW150914, finding agreement with the prediction with 97% (95%) probability when we model the ringdown including (excluding) overtones of the quadrupolar mode. We obtain this result from a new time-domain analysis of the pre- and postmerger data. We also confirm that the inspiral and ringdown portions of the signal are consistent with the same remnant mass and spin, in agreement with general relativity.
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