Black hole mass and Hamilton-Jacobi counterterms

We apply the method of holographic renormalization to computing black hole masses in asymptotically anti-de Sitter spaces. In particular, we demonstrate that the Hamilton-Jacobi approach to obtaining the boundary action yields a set of counterterms sufficient to render the masses finite for four, five, six and seven-dimensional R-charged black holes in gauged supergravities. In addition, we prove that the familiar black hole thermodynamical expressions and in particular the first law continues to holds in general in the presence of arbitrary matter couplings to gravity.

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