A perturbative study on the analytic continuation for generalized gravitational entropy

Understand

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 $\beta$-power of the time evolution operator when $\beta\in \mathbb{R}$.

• 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.