Energy Conditions and Quantum Information

TL;DR

This article surveys energy conditions in general relativity and their quantum extensions, emphasizing how local and averaged conditions constrain spacetime structure, singularities, and black hole thermodynamics. It then connects these classical ideas to quantum information through modular Hamiltonians, entanglement entropy, and the Bekenstein bound, showing how ANEC and QNEC arise from information-theoretic principles. In the holographic context, the work explains how AdS/CFT and the RT/QES/island program recast spacetime geometry in terms of boundary entanglement, leading to a holographic proof of ANEC and a resolution of the black hole information paradox via islands. The discussion highlights the deep interplay between geometry, energy conditions, and quantum information, with causality and holography enforcing energy positivity in ways that transcend traditional local energy conditions.

Abstract

The concept of energy lies at the foundation of physical science. In general relativity and quantum field theory, the positivity and conservation of energy are encapsulated by the so-called energy-momentum tensor and the energy conditions. In recent efforts to unify fundamental physics with quantum information, the energy conditions have come to play a crucial role in establishing numerous theorems. In this article, we review the basics of energy conditions in general relativity and their applications in gravitational physics, quantum field theory, and the holographic principle. Through these applications, we explore the profound connection between the energy conditions and quantum information

Figures (17)

• Figure 1: The geometric interpretation of the expansion $\theta$, the shear-tensor $\sigma_{\mu \nu}$, and the rotation $\omega_{\mu \nu}$.
• Figure 2: A null geodesic congruence whose expansion takes initially negative value $\theta_0<0$ converges, and within the affine length $\lambda \leqslant 2/|\theta_0|$, the cross-section area $A$ of the congruence vanishes, where by definition $\theta \rightarrow -\infty$.
• Figure 3: Null geodesics emanating from a subset $S$ of $M$. One such null geodesic $\gamma$ is passing through $p$, where its expansion changes sign from positive to negative, and then has a conjugate point $r$ in $J^+(p)$. Beyond this conjugate point, $\gamma$ fails to remain on the boundary $\partial J^+(S)$ and enters the interior $I^+(S)$. Since any point $q$ on $\gamma$ within $I^+(S)$ can be connected to $S$ by a timelike curve, $\gamma$ no longer is achronal beyond the conjugate point $r$. See also Figure \ref{['gammaprime']} below.
• Figure 4: (a) Reverse triangle inequality. (b) A timelike geodesic $\gamma$ from $p$ to $q$ admits a point $r$ conjugate to $p$ between $p$ and $q$. (c) A timelike geodesic $\gamma'$ from $p$ to $q$, constructed by deforming $\gamma$ so that $\gamma'$ has a larger value of the proper time than $\gamma$. Compare Figure \ref{['achronal']}.
• Figure 7: (a) A "wormhole (handle-structure)" exists in the interior of a sphere $\sigma$ within a $3$-dimensional space $\Sigma$. (b) This situation can be constructed by excising two $3$-dimensional balls from the interior of $\sigma$ and identifying their $2$-sphere boundaries.