On black hole entropy

Abstract

Two techniques for computing black hole entropy in generally covariant gravity theories including arbitrary higher derivative interactions are studied. The techniques are Wald’s Noether charge approach introduced recently, and a field redefinition method developed in this paper. Wald’s results are extended by establishing that his local geometric expression for the black hole entropy gives the same result when evaluated on an arbitrary cross section of a Killing horizon (rather than just the bifurcation surface). Further, we show that his expression for the entropy is not affected by ambiguities which arise in the Noether construction. Using the Noether charge expression, the entropy is evaluated explicitly for black holes in a wide class of generally covariant theories. For a Lagrangian of the functional form L̃=L̃(${\mathrm{\psi }}_{\mathit{m}}$, ${\mathrm{\nabla }}_{\mathit{a}}$ ${\mathrm{\psi }}_{\mathit{m}}$,${\mathit{g}}_{\mathit{a}\mathit{b}}$,${\mathit{R}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$, ${\mathrm{\nabla }}_{\mathit{e}}$ ${\mathit{R}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$), it is found that the entropy is given by S=-2π∮(${\mathit{Y}}^{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$-${\mathrm{\nabla }}_{\mathit{e}}$ ${\mathit{Z}}^{\mathit{e}:\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$) ε${\mathrm{^}}_{\mathit{a}\mathit{b}}$ε${\mathrm{^}}_{\mathit{c}\mathit{d}}$ε¯, where the integral is over an arbitrary cross section of the Killing horizon, ε${\mathrm{^}}_{\mathit{a}\mathit{b}}$ is the binormal to the cross section, ${\mathit{Y}}^{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$=∂L̃/∂${\mathit{R}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$, and ${\mathit{Z}}^{\mathit{e}:\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$=∂L̃/∂${\mathrm{\nabla }}_{\mathit{e}}$ ${\mathit{R}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$.