3.8 General formula for entropy in the replica method, relation to the Wald entropy

As a consequence of the expressions (55View Equation) for the curvature of space with a conical singularity that were presented in Section 3.7.1 one obtains a general expression for the entropy. Consider a Euclidean general covariant action
∫ d √ -- αβ αβ W [gμν,φA ] = − d x g ℒ(gμν,R μν,∇ σR μν,...,φA ), (71 )
which describes the gravitational field coupled to some matter fields φA. In the replica trick we first introduce a conical singularity at the horizon surface Σ with a small angle deficit δ = 2π (1 − α ) so that the Riemann curvature obtains a delta-like surface contribution (55View Equation) and the gravitational action (71View Equation) becomes a function of α. Then applying the replica formula
S = (α ∂ − 1)W (α )| α α=1
we get
∫ ( μ α ν β μ β ν α ) S = 2 π Σ Q αβμν (n n )(n n ) − (n n )(n n ) (72 )
for the entropy associated to Σ, where tensor Q αβμν is defined as a variation of action (71View Equation) with respect to the Riemann tensor,
Qμν = -1--δW-[gμν,φA-]. (73 ) αβ √ g δR αβμν
If action (71View Equation) is local and it does not contain covariant derivatives of the Riemann tensor, then the tensor μν Q αβ is a partial derivative of the Lagrangian,
Q μν = --∂-ℒ-- . (74 ) αβ ∂R αβμν
Now, as was observed by Myers and Sinha [181Jump To The Next Citation Point] (see also [4Jump To The Next Citation Point]), one can re-express
∑2 μ β μ β (ni nαi )(nνjnj ) − (ni n i )(nνjnαj) = 𝜖μν𝜖αβ , (75 ) i,j=1
where 𝜖αβ = nαn β− n αnβ 1 2 2 1 is the two-dimensional volume form in the space transverse to the horizon surface Σ. Then, for a local action (71View Equation) polynomial in the Riemann curvature, the entropy (72View Equation) takes the form
∫ ∂ℒ S = 2 π ---αβ--𝜖μν𝜖αβ , (76 ) Σ ∂R μν
which is exactly the Wald entropy [221Jump To The Next Citation Point, 144Jump To The Next Citation Point]. It should be noted that Wald’s Noether charge method is an on-shell method so that the metric in the expression for the Wald entropy is supposed to satisfy the field equations. On the other hand, the conical singularity method is an off-shell method valid for any metric that describes a black-hole horizon. The relation between the on-shell and the off-shell descriptions will be discussed in Section 4.1.
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