  Entanglement entropy in a field theory, with a holographic dual, may be
viewed as a quantity which encodes the diffeomorphism invariant bulk gravity
dynamics. This, in particular, indicates that the bulk Einstein equations would
imply some constraints for the boundary entanglement entropy. In this paper we
focus on the change in entanglement entropy, for small but arbitrary
fluctuations about a given state, and analyze the constraints imposed on it by
the perturbative Einstein equations, linearized about the corresponding bulk
state. Specifically, we consider linear fluctuations about BTZ black hole in 3
dimension, pure AdS and AdS Schwarzschild black holes in 4 dimensions and
obtain a diffeomorphism invariant reformulation of linearized Einstein equation
in terms of holographic entanglement entropy. We will also show that
entanglement entropy for boosted subsystems provides the information about all
the components of the metric with a time index.
