Entropic Counterpart of Perturbative Einstein Equation

TL;DR

This work formulates an entropic counterpart to the perturbative Einstein equations within the AdS/CFT framework by expressing the first-order change in holographic entanglement entropy $\Delta S_A$ as a gauge-invariant boundary consequence of bulk metric fluctuations. Using linearized perturbations about BTZ, AdS$_4$, and AdS$_4$-BH backgrounds, the authors derive precise differential constraints on $\Delta S_A$ for ball or disk subsystems, revealing a wave-like equation in AdS$_3$ and a hyperbolic constraint in AdS$_4$ that govern the entropic response. They demonstrate that in AdS$_4$-BH backgrounds the leading large-$l$ behavior of $\Delta S_A$ satisfies a clean constraint equation and that subleading time dependence is fixed by IR boundary conditions, linking entanglement dynamics to quasinormal modes. To access all metric components, boosted subsystems are introduced, yielding a complete basis of entropic observables $\Delta S_A^{(\beta)}$ with a generalized constraint that remains valid across boosts. Overall, the paper provides a gauge-invariant, entropic route to encodings of bulk gravity dynamics and offers a framework for reconstructing bulk information from boundary entanglement data, with potential extensions to nonlinear regimes and action formulations involving $\Delta S_A$ as fundamental fields.

Abstract

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.

Figures (4)

• Figure 1: The schematic picture of the interpretation of constraint equation. The HEE becomes non-trivial only if the boundary $\partial A$ of the subsystem $A$ intersects with locally excited regions. The brown, green and blue circles are three examples of the boundaries $\partial A$ for which the HEE becomes non-trivial.
• Figure 2: A sketch of the minimal surface in the AdS BH when the size $l$ of the subsystem is very large.
• Figure 3: The plot of $r(z)/l$ for various choices of $z_*$ (or equally $l$). We set $z_H=1$. The blue, red and yellow curve correspond to $(1-z_*,l)=(10^{-3},2.739), (10^{-4},3.461)$ and $(10^{-5},4.173)$, respectively.
• Figure 4: Plot of $z_{\star}(l)$ as a function of $l$. Here we have again chosen $z_H=1$. The blue dots are numerically evaluated points, while the dotted red line is a best fit to the data with a function of the form $z_H - \alpha e^{-\beta l}$. The best fit values are $\alpha = 4.106, \ \beta = 3.008 \approx 3$.