Large Gauge Transformations and Black Hole Entropy
Diffeomorphisms in general relativity can act non-trivially on the boundary of spacetimes. Of particular importance are the $\textit{BMS group}$ of transformations, originally found by Bondi, Metzner, van der Burg and Sachs. They describe the symmetries of asymptotically flat spacetimes. We first consider symmetries such as these that act non-trivially at null infinity. We look in detail at the BMS group and then describe the conformal symmetries of asymptotically flat spacetime. These represent an extension of the BMS group that we call the $\textit{conformal BMS group}$. Second, we explore the action of these large diffeomorphisms in the context of black holes. The emergence of Virasoro algebras as the asymptotic symmetry algebras of black hole spacetimes suggests a fundamental link to conformal field theory. For the case of the generic Kerr black hole, we hypothesize that the black hole is itself a thermal conformal field theory which transforms under a Virasoro action. A set of infinitesimal ${\rm Virasoro_{\,L}}\otimes{\rm Virasoro_{\,R}}$ diffeomorphisms are presented which act non-trivially on the horizon. Using the covariant phase space formalism, we can construct the corresponding surface charges on the black hole horizon and find the central terms in their algebras. Ambiguities in the construction of the charges allow for the addition of extra terms. Wald and Zoupas have provided a general framework for these counterterms, although the precise form is left undetermined. In computing the horizon charges, certain obstructions to the integrability and associativity of the charge algebra arise, calling for some counterterm to be used. A consistent counterterm is found that removes these obstructions and gives rise to central charges $c_L=c_R=12J$. On the assumption that there exists a quantum Hilbert space on which these charges generate the symmetries, one can use the Cardy formula to compute the entropy of the conformal field theory. This Cardy entropy turns out to be exactly equal to the Bekenstein-Hawking entropy, providing a potential microscopic interpretation for this macroscopic area-entropy law. The results are generalised to the Kerr-Newman black hole with the addition of charge.
Black holes, entropy, symmetries, Information Paradox
Networked Digital Library of Theses and Dissertations