Extremal Black Holes in Diffeomorphism Covariant Gravity

Abstract

This thesis explores some classical aspects of extremal black holes in a diffeomorphism covariant formulation of gravity (of which general relativity is one). In particular, we first study the existence of Noether currents in the presence of Killing or gauge symmetries, and then generalize it to a theory exhibiting diffeomorphism-invariance in the context of the Noether-Wald formalism. We also explore the implication of the latter on black hole entropy for an arbitrary theory of gravity which may or may not include higher derivatives. We then study the dynamics of charged static extremal black holes in Eddington-Finkelstein coordinates and encounter the problem of stability of the horizon. From a thermodynamics standpoint, we also realize that a non-extremal black hole cannot be made extremal in a manner consistent with the third law of black hole thermodynamics. To that end, we explore the procedure of studying the near-horizon geometry in the extremal limit of a non-extremal black hole and arrive at a well-known universal result (with a caveat on some exceptions) that all extremal black holes have an AdS_2 factor in their near-horizon geometry. Motivated by the fact that the near-horizon geometry of an extremal black hole is consistent with the definitions of Wald, we study the entropy function formalism for a theory of gravity coupled to abelian gauge fields, and neutral scalars. A derivation of Wald entropy for the extremal Reissner-Nordstrom black hole in D=4 dimensions is also given. We also discuss how the entropy function formalism leads to a generalization of the attractor phenomena of black holes without explicit reference to supersymmetry or string theory. Towards the end, we provide some additional evidence for the existence of the horizon instability of an extremal black hole by looking at the redshift effect, pair production at the horizon, and mass inflation. This thesis has been written keeping in mind an advanced beginner and also includes a useful appendix that could serve as a short "primer" to delve into the field of black hole physics. There, we explore some important geometrical and (classical) thermodynamic features of rotating black holes via brute-force computation.