Abstract
We derive the first law of black hole mechanics in the context of the Heterotic Superstring effective action to first order in α′ using Wald’s formalism. We carefully take into account all the symmetries of the theory and, as a result, we obtain a manifestly gauge- and Lorentz-invariant entropy formula in which all the terms can be computed explicitly. An entropy formula with these properties allows unambiguous calculations of macroscopic black-hole entropies to first order in α′ that can be reliably used in a comparison with the microscopic ones. Such a formula was still lacking in the literature.In the proof we use momentum maps to define covariant variations and Lie derivatives and restricted generalized zeroth laws which state the closedness of certain differential forms on the bifurcation sphere and imply the constancy of the associated potentials on it.We study the relation between our entropy formula and other formulae that have been used in the literature.
Highlights
In the proof we use momentum maps to define covariant variations and Lie derivatives and restricted generalized zeroth laws which state the closedness of certain differential forms on the bifurcation sphere and imply the constancy of the associated potentials on it
We derive the first law of black hole mechanics in the context of the Heterotic Superstring effective action to first order in α using Wald’s formalism
We study the relation between our entropy formula and other formulae that have been used in the literature
Summary
The Heterotic Superstring effective action can be described at first order in α as follows [3]:9 we start by defining the zeroth-order. If A is a p-form with components Aμ1···μp , ıaA is the (p − 1) form with components eaν Aνμ1···μp−1 Having made these definitions and adding the dilaton field φ, we can write the Heterotic Superstring effective action to first-order in α as. This action is defined in 10 dimensions, we have left the dimension arbitrary (d) because that allows us to use the results in other dimensions after trivial dimensional reduction on a torus. The variation of the action with respect to the torsionful spin connection takes exactly the same form as the YM equation, the only difference being the group indices and their contractions. This crucial property effectively reduces the degree of the differential equations to 2, avoiding the problems that arise with dynamical equations that involve derivatives of the fields of higher order
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