Gravitational Positive Energy Theorems from Information Inequalities

TL;DR

The paper shows that relative entropy in holographic CFTs maps to a covariantly defined gravitational energy for boundary subregions, yielding an infinite family of positive-energy constraints on bulk geometry. By constructing a suitable bulk vector field and employing the Wald–Iyer formalism, it proves that S(ρ_B||ρ_B^{vac}) equals the difference of the bulk quasi-local energy relative to AdS, and derives both path-integral and geometric arguments for this duality. This leads to concrete geometric constraints, a perturbative framework around AdS, and a generalized Radon-transform relation connecting boundary entanglement data to bulk energy density, with implications for bulk reconstruction and swampland criteria. The work extends the understanding of subregion duality, suggesting that entanglement wedges host well-defined, positive-definite energy dynamics that can be constrained by information-theoretic inequalities and, ultimately, reconstructed from boundary information.

Abstract

In this paper we argue that classical, asymptotically AdS spacetimes that arise as states in consistent ultraviolet completions of Einstein gravity coupled to matter must satisfy an infinite family of positive energy conditions. To each ball-shaped spatial region $B$ of the boundary spacetime, we can associate a bulk spatial region $Σ_B$ between $B$ and the bulk extremal surface $\tilde{B}$ with the same boundary as $B$. We show that there exists a natural notion of a gravitational energy for every such region that is non-negative, and non-increasing as one makes the region smaller. The results follow from identifying this gravitational energy with a quantum relative entropy in the associated dual CFT state. The positivity and monotonicity properties of the gravitational energy are implied by the positivity and monotonicity of relative entropy, which holds universally in all quantum systems.

Figures (9)

• Figure 1: The two Euclidean path-integrals on the left prepare the density matrix of a spherical subsystem in a CFT in vacuum and an arbitrary state, respectively $\sigma$ and $\rho$. The path-integrals appearing in the definition of Rényi relative entropies are of the type on the right.
• Figure 2: The bulk version of the replica trick in figure \ref{['fig1']}. Geometries on the left are dual to vacuum and excited state density matrices, respectively $\sigma$ and $\rho$ . The bulk configuration on the right prepares our quantity of interest in the definition of Rényi relative entropies.
• Figure 3: The analytic continuation of geometries to non-integer $n$ near one.
• Figure 4: For ball-shaped region $B_1$ in the domain of dependence $D$ of ball-shaped region $B_2$, monotonicity of relative entropy implies that the relative entropy for $B_2$ must be larger than or equal to the relative entropy associated with the subsystem $B_1$. Here, the surface $\hat{B}_2$ includes the ball $B_1$ and is a Cauchy surface for the same domain of dependence region $D$ as $B_2$, so it has the same relative entropy as for $B_2$.
• Figure 5: One-to-one correspondence between balls $B$ and pairs of points $(x_+,x_-)$ with $x_+$ in the future of $x_-$. A minimal set of monotonicity constraints is obtained by considering deformations of the ball associated with shifting $x_+$ in a future lightlike direction (red arrow) or $x_-$ in a past lightlike direction. The boundary vector field $\Delta$ generates a conformal transformation that reverses this deformation.