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

As a consequence of the expressions (55) 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
which describes the gravitational field coupled to some matter fields . In the replica trick we first
introduce a conical singularity at the horizon surface with a small angle deficit so that
the Riemann curvature obtains a delta-like surface contribution (55) and the gravitational action (71)
becomes a function of . Then applying the replica formula
we
get
for the entropy associated to , where tensor is defined as a variation of action (71) with respect
to the Riemann tensor,
If action (71) is local and it does not contain covariant derivatives of the Riemann tensor, then the tensor
is a partial derivative of the Lagrangian,
Now, as was observed by Myers and Sinha [181] (see also [4]), one can re-express
where is the two-dimensional volume form in the space transverse to the horizon
surface . Then, for a local action (71) polynomial in the Riemann curvature, the entropy (72) takes the
form
which is exactly the Wald entropy [221, 144]. 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.