Geometric entropy and time-like entanglement entropy on a rotating BTZ black hole

In this paper, we analyze the double Wick rotation of a rotating BTZ black hole and the entanglement entropy. We derive the transition matrix dual to the double Wick-rotated BTZ black hole, which has the usual shape at an imaginary chemical potential. In the dual gravity side, the double Wick rotated BTZ black hole, which is obtained as a quotient, is equal to a rotating BTZ black hole after the coordinate transformation and the identification of periodicity. The geometric entropy and time-like entanglement entropy are reproduced by the identification. New Lorentzian entanglement growth is defined by the coefficient of linear growth of time-like entanglement entropy.