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A perturbative study on the analytic continuation for generalized gravitational entropy

We study the analytic continuation used by Lewkowycz and Maldacena to prove the Ryu-Takayanagi formula for entanglement entropy, which is the holographic dual of the trace of the $β$-power of the time evolution operator when $β\in \mathbb{R}$. This will be done perturbatively by using a weakly time dependent Hamiltonian, corresponding to a small shift of the dual static background. Depending on the periodicity we impose on the gravitational solution, we consider two different possibilities and compare the associated entropies with the results obtained through a minimal area computation. To our surprise we discover that, at first order, both choices correctly reproduce the associated entanglement entropy. Furthermore we find unexpected divergent contributions that we have to discard in order to fit the minimal area computation, and an additional requirement that needs to be imposed on the $β$ dependence on the metric.