Generalized geodesic deviation equations and an entanglement first law for rotating BTZ black holes

The change in holographic entanglement entropy (HEE) for small fluctuations about pure anti–de Sitter is given by a perturbative expansion of the area functional in terms of the change in the bulk metric and the embedded extremal surface. However, it is known that changes in the embedding appear at second order or higher. In this paper we show that these changes in the embedding can be systematically calculated in the ($2+1$)-dimensional case by accounting for the deviation of the spacelike geodesics between a spacetime and perturbations over it. Here we consider rotating BTZ as a perturbation over ${\mathrm{AdS}}_3$ and study deviations of spacelike geodesics in them. We argue that these deviations arise naturally as solutions of a “generalized geodesic deviation equation.” Using this we perturbatively calculate the changes in HEE up to second order for rotating BTZ. This expression matches with the small system size expansion of the change in HEE obtained by the proposal of Hubeny, Rangamani, and Takayanagi for rotating BTZ. We also write an alternative form of the entanglement first law for rotating BTZ. To do this one needs to go beyond the leading order in the perturbation series discussed above. That is precisely the reason we consider finding a systematic way to calculate it. To put our result on a firm footing we further show that it is this alternative first law that approaches the thermal first law in the large subsystem size limit.

Figure 1

The mapping of the geodesics.

Figure 2

Plot of $T_E$ vs ‘$l$’ for different black hole temperatures. As ‘$l$’ increases, the temperature asymptotes to thermal value.

Figure 3

Plot of ${\mathrm{\Omega }}_E$ vs ‘$l$’ for different black hole angular velocities. As ‘$l$’ increases, the angular velocity asymptotes to thermal value.

Figure 4

Plot of $\frac{2G\pi \mathrm{\Delta }{S}_E}{l}\mathrm{vs}l$ for different Bekenstein Hawking entropies. As ‘$l$’ increases $\frac{2G\pi \mathrm{\Delta }{S}_E}{l}$ asymptotes to thermal value.