Semiclassical Einstein equations from holography and boundary dynamics

In this paper, we consider how to formulate semiclassical problems in the context of AdS/CFT correspondence, based on proposal Compere and Marolf. Our prescription involves effective action with self-action term for boundary dynamical fields, which can be viewed as imposing mixed conditions gravity dual. We derive Einstein equations sourced by CFT stress-energy tensor. Analyzing perturbations holographic equations, find a universal parameter $\gamma_d$ controls contribution from CFTs specifies dynamics AdS boundary. As simple example, examine $3$-dimensions $4$-dimensional dual, show that BTZ black hole vanishing expectation value tensor becomes unstable due backreaction quantum when exceeds certain critical value.