Remarks on Nöther Charges and Black Holes Entropy

Abstract

Abstract.We criticize and generalize some properties of Nöther charges presented in a paper by V. Iyer and R. M. Wald and their application to entropy of black holes. The first law of black holes thermodynamics is proven for any gauge-natural field theory. As an application charged Kerr-Newman solutions are considered. As a further example we consider a (1 + 2) black hole solution.

Nöther charges, Brown–York quasilocal energy, and related topics

“…In fact, the quasilocal energy as well as the action functional (1.3) appeared also as the starting point of a statistically-oriented approach to black hole entropy (see [21], [22], [24], [26], [36], [37] and references quoted therein). A different approach to black hole entropy based on Nöther approach (see [16], [27], [30], [38] and references quoted therein) may be found in literature. In view of the present comparison between the covariant first order approach (which the Nöther approach is based on) and the York's action functional (which quasilocal energy is based on) as well as between the Nöther charges and the quasilocal energy themselves, we believe that the two different approaches to entropy can be now successfully compared (see also [39], [40]).…”

“…The superpotential is defined modulo forms which are closed on-shell, while the reduced current is defined modulo forms vanishing along solutions. The decomposition (5.3) of Nöther's current is again well-defined only at bundle level, where a decomposition algorithm can be constructed (see [28], [27]). The conserved quantity in a region Ω is defined as:…”

“…the vector field which corresponds to asymptotic rotation; see e.g. [11], [15], [27]). Furthermore, the same results are achieved for non-asymptotically flat solutions by choosing suitable reference backgrounds.…”

“…when the field theory owns both covariance and gauge invariance (e.g. BCEA theory, Einstein-Maxwell theory; see [27], [31]). …”

“…Some details about this case are found in Section 5 . Further details can be found in [2], [28], [27]. The Nöther currents, as long as the superpotentials, are used in Section 5 to define covariantly conserved quantities in the particular case of General Relativity.…”