Entropy formula in Einstein-Maxwell-dilaton theory and its validity for black strings

Abstract

We consider near horizon fall-off conditions of stationary black holes in Einstein-Maxwell-Dilaton theory and find conserved charge conjugate to symmetry generator that preserves near horizon fall-off conditions. Subsequently, we find supertranslation, superrotation and multiple-charge modes. We apply the obtained results on a typical static dilaton black hole and on a charged rotating black string, as examples. In this case, supertranslation doublezero-mode charge T (0,0) is not equal to black hole entropy times Hawking temperature. This may be seen as a problem but it is not, because, in Einstein-Maxwell-Dilaton theory, we have a U(1) gauge freedom and we use an appropriate gauge fixing to fix that problem. We show that new entropy formula 4πĴ + 0Ĵ − 0 , proposed in [5], is valid for black strings as well as black holes. * Electronic address: rezakord@ipm.ir † Electronic address: hamed.adami@yahoo.com

Lee-Wald Charge and Asymptotic Behaviors of the Weyl-invariant Topologically Massive Gravity

“…7 In our calculation, since the Lagrangian of the Weyl-TMG contains extra dofs together with the graviton field as in [45] where the LW charges in the Einstein-Maxwell-Dilation theory are obtained, we will follow it, too.…”

“…Now we wish to study the near horizon behavior of a stationary black hole in the Weyl-invariant TMG in order to particularly reveal the behavior of the Weyl's gauge symmetry and check if there comes any residual symmetry in this region of the spacetime. In doing so, we assume that the geometry of the near horizon is described by the following generic metric in the Gaussian null coordinates [24,25,45,[47][48][49]…”

“…As is apparent in the Eq. ( 35), we start with the general situation where all functions κ, θ, Ω, λ are assumed to be x-dependent [24,25,45,[47][48][49]. By picking the gauge-fixing condition to be A ρ = 0, one is allowed to set up the ensuing decays of the Weyl's real scalar and gauge fields up to O(ρ 2 ) in the vicinity of horizon, respectively…”

“…To get the asymptotic generators for the enhanced symmetry, we first presume that the boundary conditions are state-independent which is achieved by taking the leading terms to be independent of the propagating modes as in [45,48]. It is actually straightforward to show that by solving…”

“…Later, with the help of Eq. ( 40) and δ ξ g µν = 2∇ (µ ξ ν) , one eventually gets the variation of the propagating fields with respect to the enlarged symmetry generator as follows [45,48] (1) x = £ Y A (1) x + A (1) ν ∂ x T. (41) Observe that the gauge-parameters are field-dependent. Therefore, before computing the algebra among the asymptotic charges, let us first search for the algebra among the generators: as is wellknown, the algebra among the generators in general will not be closed in the field-dependent case.…”