Energy conditions in general relativity and quantum field theory

TL;DR

This review analyzes how energy conditions constrain gravitating matter in general relativity and quantum field theory, contrasting classical pointwise conditions such as $T_{ab} t^a t^b \ge 0$ and $R_{ab} k^a k^b \ge 0$ with their quantum counterparts. It surveys quantum energy inequalities (QEIs), averaged energy conditions, and the achronal averaged null energy condition (AANEC), elucidating their interrelations through the classical-quantum correspondence and renormalization formalisms involving Hadamard states. The authors discuss how QEIs—including state-dependent and absolute forms—impose stability constraints and can, in suitable limits, recover classical energy conditions and averaged forms, with the Casimir effect illustrating quantum violations. Applications to singularity theorems, black hole physics, and the viability of exotic spacetimes reveal both the power and the limits of energy conditions in semiclassical gravity, while pointing toward possible deep connections to quantum gravity via AANEC. The outlook highlights open questions about the universality of AANEC in curved and interacting quantum field theories and the quest for a general, theory-independent energy condition emerging from a complete theory of quantum gravity.

Abstract

This review summarizes the current status of the energy conditions in general relativity and quantum field theory. We provide a historical review and a summary of technical results and applications, complemented with a few new derivations and discussions. We pay special attention to the role of the equations of motion and to the relation between classical and quantum theories. Pointwise energy conditions were first introduced as physically reasonable restrictions on matter in the context of general relativity. They aim to express e.g. the positivity of mass or the attractiveness of gravity. Perhaps more importantly, they have been used as assumptions in mathematical relativity to prove singularity theorems and the non-existence of wormholes and similar exotic phenomena. However, the delicate balance between conceptual simplicity, general validity and strong results has faced serious challenges, because all pointwise energy conditions are systematically violated by quantum fields and also by some rather simple classical fields. In response to these challenges, weaker statements were introduced, such as quantum energy inequalities and averaged energy conditions. These have a larger range of validity and may still suffice to prove at least some of the earlier results. One of these conditions, the achronal averaged null energy condition, has recently received increased attention. It is expected to be a universal property of the dynamics of all gravitating physical matter, even in the context of semiclassical or quantum gravity.

Figures (6)

• Figure 1: The main energy conditions for perfect fluids represented as regions of allowed energy density and pressure.
• Figure 2: Implications of energy conditions. In the figure an arrow from $a$ to $b$ means $a$ implies $b$.
• Figure 3: "Wedge" removal of Minkowski space.
• Figure 4: Spacetime diagram of light rays in Schwarzschild spacetime (in color online). We use Eddington-Finkelstein coordinates with $-\left( 1-\frac{2M}{r} \right) \mathrm{d}v^2+2\mathrm{d}v \mathrm{d}r=0 \,,$for radial null geodesics and we set $\tilde{t}=v-r$. For $v=$const. we have the ingoing and for $v \neq$ const. the outgoing light rays. However, inside the horizon $r<r_s \equiv 2M$ both ingoing (blue) and outgoing (red) radial null geodesics are directed towards $r=0$.
• Figure 5: Penrose diagram of an asymptotically flat spacetime with a black hole. Here $\mathscr{I}^{\pm}$ denote future and past null infinity, $i^0$ is spacelike infinity and $i^-$ is past infinity. The black hole region is indicated in gray with event horizon $H$. $\Sigma$ is a Cauchy surface and the shaded region indicates matter collapsing to a black hole.

Theorems & Definitions (20)

• Proposition 2.1
• proof
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 3.1
• Theorem 3.2
• proof
• Definition 3.3
• Theorem 4.1