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Proving the Achronal Averaged Null Energy Condition from the Generalized Second Law

Aron C. Wall

TL;DR

This work shows that the achronal ANEC on null lines can be derived from the generalized second law applied to causal horizons in semiclassical gravity, given CPT and a viable renormalization of generalized entropy. By perturbing a classical background with quantum fields and employing horizon thermodynamics, the author derives the ANEC at leading order via the linearized Raychaudhuri equation and weak monotonicity. The analysis further extends to gravitational fluctuations, introducing a shear-inclusive ANEC to maintain causality and energy positivity, with discussion of renormalization and horizon response. Overall, the GSL is presented as a foundational mechanism underpinning energy conditions in semiclassical gravity, contingent on horizon stability and appropriate regularization.

Abstract

A null line is a complete achronal null geodesic. It is proven that for any quantum fields minimally coupled to semiclassical Einstein gravity, the averaged null energy condition (ANEC) on null lines is a consequence of the generalized second law of thermodynamics for causal horizons. Auxiliary assumptions include CPT and the existence of a suitable renormalization scheme for the generalized entropy. Although the ANEC can be violated on general geodesics in curved spacetimes, as long as the ANEC holds on null lines there exist theorems showing that semiclassical gravity should satisfy positivity of energy, topological censorship, and should not admit closed timelike curves. It is pointed out that these theorems fail once the linearized graviton field is quantized, because then the renormalized shear squared term in the Raychaudhuri equation can be negative. A "shear-inclusive" generalization of the ANEC is proposed to remedy this, and is proven under an additional assumption about perturbations to horizons in classical general relativity.

Proving the Achronal Averaged Null Energy Condition from the Generalized Second Law

TL;DR

This work shows that the achronal ANEC on null lines can be derived from the generalized second law applied to causal horizons in semiclassical gravity, given CPT and a viable renormalization of generalized entropy. By perturbing a classical background with quantum fields and employing horizon thermodynamics, the author derives the ANEC at leading order via the linearized Raychaudhuri equation and weak monotonicity. The analysis further extends to gravitational fluctuations, introducing a shear-inclusive ANEC to maintain causality and energy positivity, with discussion of renormalization and horizon response. Overall, the GSL is presented as a foundational mechanism underpinning energy conditions in semiclassical gravity, contingent on horizon stability and appropriate regularization.

Abstract

A null line is a complete achronal null geodesic. It is proven that for any quantum fields minimally coupled to semiclassical Einstein gravity, the averaged null energy condition (ANEC) on null lines is a consequence of the generalized second law of thermodynamics for causal horizons. Auxiliary assumptions include CPT and the existence of a suitable renormalization scheme for the generalized entropy. Although the ANEC can be violated on general geodesics in curved spacetimes, as long as the ANEC holds on null lines there exist theorems showing that semiclassical gravity should satisfy positivity of energy, topological censorship, and should not admit closed timelike curves. It is pointed out that these theorems fail once the linearized graviton field is quantized, because then the renormalized shear squared term in the Raychaudhuri equation can be negative. A "shear-inclusive" generalization of the ANEC is proposed to remedy this, and is proven under an additional assumption about perturbations to horizons in classical general relativity.

Paper Structure

This paper contains 12 sections, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The GSL and its Time Reverse
  3. The Structure of the Classical Background
  4. The Perturbation Expansion
  5. Zeroth Order
  6. Half Order
  7. First Order
  8. Higher Orders
  9. Proof of the ANEC
  10. Renormalization
  11. Gravitational Fluctuations
  12. Conclusion

Figures (1)

  • Figure 1: A Penrose diagram of the unperturbed classical spacetime. $H$ is the horizon, shown here as a segment reaching from a past timelike infinite point $H^-$ to a future timelike infinite point $H^+$. $P = \mathcal{I}^-(H)$ is the open region to the chronological past of $H$, and $F = \mathcal{I}^+(H)$ is the open region to the chronological future of $H$. There may also be a closed region $O$ which is spacelike to $H$. The slices $T_1$ and $T_2$, which coincide in $O$, are also shown. $\Delta T$ is the spacetime region between the two slices. Some specific intersections of regions important to the proof are also shown. The boundary of the spacetime is not shown, except for the past and future of the horizon $H$.