Timelike Entanglement Entropy from Rindler Method

For a Lorentzian invariant theory, the entanglement entropy should be a function of the dependence of domain of the subregion under consideration. More precisely, it should be a function of the dependence of domain and the appropriate cut-off. In this paper, we define the concept of cut-off to make it applicable to timelike regions and assume that the usual entanglement entropy formula also applies to timelike intervals. Using the Rindler method, the timelike entanglement entropy can be regarded as the thermal entropy of the CFT after the Rindler transformation plus a constant $ic\pi/6$ with $c$ the central charge of the CFT. The gravitational dual of the timelike entanglement entropy is finally presented following this method.Comment: 14 pages, 3 figure