Abstract
We discuss black hole thermodynamics in the manifestly duality invariant formalism of double field theory (DFT). We reformulate and prove the first law of black hole thermodynamics in DFT, using the covariant phase space approach. After splitting the full O(D, D) invariant DFT into a Kaluza–Klein-inspired form where only n coordinates are doubled, our results provide explicit duality invariant mass and entropy formulas. We illustrate how this works by discussing the black string solution and its T-duals.
Highlights
The massless spectrum of any of the closed string theories has a common sector consisting of the NSNS fields: the spacetime metric gμν, 2-form Bμν and dilaton φ
[25] Iyer and Wald provide a calculation of black hole entropy which goes through for any action where the gravitational degrees of freedom are encoded in the spacetime metric gμν; this has become known as the Wald entropy formula
In the Lee–Iyer–Wald covariant phase space formalism the first law of black hole thermodynamics is derived from a variational identity which sets the infinitesimal Noether charges of the previous section equal to an integral over the horizon; the last integral is proportional to the variation of the entropy, plus any charge contrib utions if the solution is supported by non-vanishing gauge fields
Summary
The massless spectrum of any of the closed string theories has a common sector consisting of the NSNS fields: the spacetime metric gμν, 2-form Bμν and dilaton φ. The issue of duality invariance of entropy and other thermodynamic quantities has been looked into from a semi-classical gravity (or, macroscopic) perspective in a few works, of which [22] by Horowitz and Welch appears to be the earliest They verify the invariance of the surface gravity and horizon area of a black hole with bifurcate Killing horizon under a Buscher transformation (1.2) by an explicit component calculation in spacetime. We make use of the ‘covariant phase space’ approach due to Lee et al [23,24,25] In this approach, the first law of black hole thermodynamics is re-expressed in a ‘differential’ form, as the vanishing of the exterior derivative of a certain (D − 2)-form constructed out of the fields and their variations; Stokes’ theorem sets the integral of this form on a horizon cross-section (which is related to the variation of the entropy) equal to the integral on a sphere at infinity (which yields variations of energy, angular momentum, etc), recovering the usual, integrated form of the first law (4.31). We provide appendices containing additional results, including a discussion of Stokes’ theorem in DFT
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