Abstract
Using AdS/CFT an extended first law of entanglement has been previously derived for the vacuum reduced to a ball in Minkowski. The statement not only includes perturbations of the state but also of the conformal field theory (CFT), via variations of the generalized central charge. We clarify some subtleties previously overlooked and use simple arguments to generalize prior derivations to arbitrary gravity theories in the bulk as well as new regions in the boundary CFT. Our construction also applies to two-dimensional bulk theories and admits an interesting extension for a three-dimensional bulk, providing a curious result regarding the thermodynamic volume in extended black hole thermodynamics. We discuss future prospects regarding the extended first law of entanglement.
Highlights
The horizon entropy associated to this Killing horizon is identified as the entanglement entropy of the boundary conformal field theory (CFT), while the variation of the cosmological constant maps to changing the generalized central charge a∗d
Our computation is novel in its simplicity and the fact that it holds for arbitrary bulk gravity theories and Killing horizons in pure AdS, finding no need to resort to technical calculations as in refs. [8, 12,13,14]
From the bulk perspective we find some interesting results for extended black hole thermodynamics, where we obtain a curious formula for the thermodynamic volume (see eq (4.8)), the conjugate variable to the pressure p
Summary
Using (2.7) we can evaluate the variation in (2.5) and find δλi Sξ This expression relies on the fact that the pure AdS metric gμAνdS(L) is only a function of the length scale L = L(λi), which means the dimensionless horizon area Ahorizon = Ahorizon/Ld−1 is independent of λi. Our derivation generalizes to arbitrary covariant theories of gravity as well as any Killing horizon in pure AdS. Eξ [gμν , λi] = Eξ [gμν , λi] − Eξ gμAνdS(λi), λi While this normalization plays no role in (2.4) when considering metric perturbations, it gives the appropriate behavior under more general variations. This prescription is equivalent to subtracting the Casimir energy contribution in pure AdS, that is present for certain foliations of the space-time The procedure is common in extended black hole thermodynamics, where the Casimir energy is not included in the first law [5]
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