Comments on Jacobson's "Entanglement equilibrium and the Einstein equation"

TL;DR

This paper scrutinizes Jacobson's entanglement-based route to Einstein gravity by performing holographic tests of the key IR-variation assumption for small spheres in a CFT perturbed by a relevant operator ${\cal O}_{\Delta}$. Using AdS/CFT, the authors compute the entanglement entropy variation $\delta S$ to leading order in a small amplitude and organize the result by the operator dimension $\Delta$. They find that for $d/2<\Delta<d$, the IR variation contains the expected $\delta\langle T_{00}\rangle$- and $\delta\langle T^a{}_a\rangle$-dependent piece plus a subleading $R^{2\Delta}$ term proportional to $\delta\langle{\cal O}_{\Delta}\rangle^2$ (with a $1/C_T$ normalization), while for $\Delta<(d/2)$ the $R^{2\Delta}$ term dominates, challenging the Jacobson construction; the special case $\Delta=d/2$ yields logarithmic corrections. The paper also demonstrates alternate quantization can replicate the same structural form, but the dominant behavior in the small-$R$ limit remains problematic, and corroborates the log structure in a concrete $d=2$ free-fermion model. Overall, these results suggest that either the derivation must be restricted to linear variations, spectral content must be limited, or one must modify the gravitational side to absorb or reinterpret these extra contributions.

Abstract

Using holographic calculations, we examine a key assumption made in Jacobson's recent argument for deriving Einstein's equations from vacuum entanglement entropy. Our results involving relevant operators with low conformal dimensions seem to conflict with Jacobson's assumption. However, we discuss ways to circumvent this problem.