The Generalized Second Law implies a Quantum Singularity Theorem

TL;DR

This work develops a quantum generalization of Penrose-type singularity theorems by replacing classical energy conditions with the fine-grained generalized second law (GSL) of horizon thermodynamics. It proves monotonicity properties of the generalized entropy $S_{ ext{gen}}$ and introduces quantum trapped surfaces, leading to quantum-corrected singularity results and no-go theorems for baby universes, traversable wormholes, warp drives, time machines, and inflation restart in asymptotically flat/AdS spacetimes. The results are established in a semiclassical regime and argued to plausibly extend to full quantum gravity, contingent on horizon-based causality concepts and global hyperbolicity. The framework has wide-ranging implications for black hole physics, cosmology, and the fundamental limits on exotic spacetime geometries, suggesting horizon thermodynamics could play a foundational role in quantum gravity.

Abstract

The generalized second law can be used to prove a singularity theorem, by generalizing the notion of a trapped surface to quantum situations. Like Penrose's original singularity theorem, it implies that spacetime is null geodesically incomplete inside black holes, and to the past of spatially infinite Friedmann--Robertson--Walker cosmologies. If space is finite instead, the generalized second law requires that there only be a finite amount of entropy producing processes in the past, unless there is a reversal of the arrow of time. In asymptotically flat spacetime, the generalized second law also rules out traversable wormholes, negative masses, and other forms of faster-than-light travel between asymptotic regions, as well as closed timelike curves. Furthermore it is impossible to form baby universes which eventually become independent of the mother universe, or to restart inflation. Since the semiclassical approximation is used only in regions with low curvature, it is argued that the results may hold in full quantum gravity. An introductory section describes the second law and its time-reverse, in ordinary and generalized thermodynamics, using either the fine-grained or the coarse-grained entropy. (The fine-grained version is used in all results except those relating to the arrow of time.) A proof of the coarse-grained ordinary second law is given.

Figures (8)

• Figure 1: The logical dependencies of the main hypotheses, theorems, and results in this article. If a proposition has one or more arrows pointing to it, then the conjunction of all propositions pointing to it is used in the proof of that proposition. $g_0$ is a particular point in the spacetime where the semiclassical approximation must be valid; its location depends on the particular result being considered (cf. section \ref{['app']}). The no-time-machines result uses the assumption that $J^+(p)\,\cap\,J^{-}(q)$ is compact for any points $p$ and $q$; since this is weaker than global hyperbolicity it is shown as following from it, although going directly from global hyperbolicity to no-time-machines is of course trivial.
• Figure 2: Black hole horizons, Rindler horizons, and de Sitter horizons are all special cases of "causal horizons". The future causal horizon $H_\mathrm{fut}$ is defined as the part of the spacetime which is causally visible to some future-infinite timelike worldline $W_\mathrm{fut}$, shown as a thick line. The GSL declares that the entropy is increasing with time on complete spatial slices outside of $H_\mathrm{fut}$ (shown as red lines).
• Figure 3: Two null surfaces $N$ and $M$ are pictured as they appear at one time, on the slice $\Sigma$. $N$ is nowhere inside of $M$, and coincides with $M$ at $g_0$. (i) the generating null vectors $k^a$, projected onto the surface $\Sigma$, must be normal to the null surfaces. Because $M$ can only bend inwards relative to $N$ at $g_0$, it is expanding faster than $N$ there (Lemma A). (ii) $f$ is the proper distance between the two null surfaces $N$ and $M$, viewed as a function of $M$. Near the point $g_0$, $f$ is very gently sloped, and thus points on $N$ and $M$ may be identified. Integration of $\nabla^2 f$ shows that it is always possible to find a point $X$ near $g_0$ at which $M$ is expanding faster than $N$, unless (iii) the surfaces coincide exactly in a neighborhood of $g_0$ (Lemma B).
• Figure 4: (i) Two null surfaces $M$ and $N$ drawn on a time slice $\Sigma$, where $N$ is nowhere inside of $M$. They coincide in a neighborhood of $g_0$. This is the same situation as Fig. \ref{['fronts']}, illustrated with a different choice of $M$ and $N$. The dotted line is the spatial cross-section used to make (ii) a spacetime diagram of the same situation. $N$ and $M$ move outwards at the speed of light. The time slice $\Sigma$ is evolved forwards in time to a new time slice $\Sigma^\prime$ near $g_0$. All the information in $\mathrm{Ext}(M)\,\cap\,\Sigma$ is contained in three regions: $B$, $C$ and $D$. Removal of the region $D$ can only decrease the amount of entanglement between $B$ and $C$, which can be used to show that the entropy outside of $M$ increases faster than the entropy outside of $N$.
• Figure 5: The region $R$ is divided by a null surface $N$ into a past region $P$ and a future region $F$. A time slice $\Sigma$ of $R$ is evolved forwards in time to $\Sigma^\prime$. All information in $R$ is stored in the three regions $B$, $C$, and $D$. Weak monotonicity implies that the generalized entropy of $F$ is increasing faster than the generalized entropy of $P$.