On black hole entropy

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 L\ifmmode \tilde{}\else \~{}\fi{}=L\ifmmode \tilde{}\else \~{}\fi{}(${\mathrm{\ensuremath{\psi}}}_{\mathit{m}}$, ${\mathrm{\ensuremath{\nabla}}}_{\mathit{a}}$${\mathrm{\ensuremath{\psi}}}_{\mathit{m}}$,${\mathit{g}}_{\mathit{a}\mathit{b}}$,${\mathit{R}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$, ${\mathrm{\ensuremath{\nabla}}}_{\mathit{e}}$${\mathit{R}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$), it is found that the entropy is given by S=-2\ensuremath{\pi}\ensuremath{\oint}(${\mathit{Y}}^{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$-${\mathrm{\ensuremath{\nabla}}}_{\mathit{e}}$${\mathit{Z}}^{\mathit{e}:\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$) \ensuremath{\epsilon}${\mathrm{^}}_{\mathit{a}\mathit{b}}$\ensuremath{\epsilon}${\mathrm{^}}_{\mathit{c}\mathit{d}}$\ensuremath{\epsilon}\ifmmode\bar\else\textasciimacron\fi{}, where the integral is over an arbitrary cross section of the Killing horizon, \ensuremath{\epsilon}${\mathrm{^}}_{\mathit{a}\mathit{b}}$ is the binormal to the cross section, ${\mathit{Y}}^{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$=\ensuremath{\partial}L\ifmmode \tilde{}\else \~{}\fi{}/\ensuremath{\partial}${\mathit{R}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$, and ${\mathit{Z}}^{\mathit{e}:\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$=\ensuremath{\partial}L\ifmmode \tilde{}\else \~{}\fi{}/\ensuremath{\partial}${\mathrm{\ensuremath{\nabla}}}_{\mathit{e}}$${\mathit{R}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}}$.Further, it is shown that the Killing horizon and surface gravity of a stationary black hole metric are invariant under field redefinitions of the metric of the form g${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{a}\mathit{b}}$\ensuremath{\equiv}${\mathit{g}}_{\mathit{a}\mathit{b}}$+${\mathrm{\ensuremath{\Delta}}}_{\mathit{a}\mathit{b}}$, where ${\mathrm{\ensuremath{\Delta}}}_{\mathit{a}\mathit{b}}$ is a stationary tensor field that vanishes at infinity and is regular on the horizon (including the bifurcation surface). Using this result, a technique is developed for evaluating the black hole entropy in a given theory in terms of that of another theory related by field redefinitions. Remarkably, it is established that certain perturbative, first order, results obtained with this method are in fact exact. A particular result established in this fashion is that a scalar matter term of the form ${\mathrm{\ensuremath{\nabla}}}^{2\mathit{p}}$\ensuremath{\varphi}${\mathrm{\ensuremath{\nabla}}}^{2\mathit{q}}$\ensuremath{\varphi} in the Lagrangian makes no contribution to the black hole entropy. The possible significance of these results for the problem of finding the statistical origin of black hole entropy is discussed.