Abstract
We construct the Komar integral for axion-dilaton gravity using Wald’s formalism and momentum maps and we use it to derive a Smarr relation for stationary axion-dilaton black holes. While the Wald-Noether 2-form charge is not invariant under SL(2, ℝ) electric-magnetic duality transformations because Wald’s formalism does not account for magnetic charges and potentials, the Komar integral constructed with it turns out to be invariant and, in more general theories, it will be fully symplectic invariant. We check the Smarr formula obtained with the most general family of static axion-dilaton black holes.
Highlights
Terms associated to the variations of the asymptotic values of the scalars such as those found in ref. [14], will not appear, either
In this paper we want to study if and how this electric-magnetic duality invariance of the Smarr formula arises from a formalism (Wald’s) which is not electric-magnetic symmetric because only the gauge transformations which imply the conservation of the electric charges are taken into account
In this paper we have shown how the momentum maps introduced in refs. [5,6,7] in the context of black-hole thermodynamics can be used to express the Komar integral obtained in the context of Wald’s formalism [22] as a surface integral in a manifestly covariant way, generalizing the results of [17, 18, 23]
Summary
The number of 1-forms does not play a relevant rôle if it is larger than one, and can be left undetermined it has to be set to six if one wants to embed the solutions of the theory into the Heterotic Superstring (HST) effective action compactified on a T6. The model with just two 1-forms can be viewed as a model of N = 2, d = 4 supergravity coupled to a single vector multiplet, and one can use the powerful solution-generating techniques developed in that class of models to construct extremal [26, 27] and non-extremal [27, 28] blackhole solutions. Since the Maxwell equations tell us that the Fms are closed on-shell, we can introduce a dual 1-form field Am defined by
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