The generalized second law implies a quantum singularity theorem
Aron C. Wall
Abstract and summary · read the original at the source
In one page
Aron Wall, then at the University of Maryland, asks what becomes of the singularity theorems once quantum fields are allowed to do what they really do. Penrose’s classic theorem assumes the energy density stays positive, and quantum field theory breaks that assumption locally all the time — which is why people have hoped a dose of energy below the vacuum level could let spacetime bounce and dodge the singularity. Wall swaps the assumption out. In its place he puts the generalized second law: horizon area plus the entropy outside it never decreases. From that alone he builds a quantum version of a trapped surface and recovers Penrose’s conclusion — black holes and spatially infinite expanding cosmologies still run into geodesic incompleteness. The same machinery reaches further, forbidding negative masses, traversable wormholes between distant regions, closed timelike curves, and warp drives of one specific kind: those that advance a light ray by a finite time over an infinite distance. Wall marks the boundaries of his own result carefully, and says exactly where they lie.
Why it matters hereChapter 4 needs to know precisely which door the physics closes and which it leaves standing open, and this is the paper that draws that line most sharply. Wall’s no-go is about asymptotic warp drives measured out to null infinity; in his own sentence, it does not apply where the speed-up happens over a finite distance — which is the regime every buildable design works in.
What it claims
01Every classical singularity theorem rests on a positivity condition on the stress-energy tensor, and all such conditions can be violated locally in quantum field theory — so Wall replaces the energy condition with the fine-grained generalized second law of horizon thermodynamics, chosen because it is widely believed to follow from the statistical mechanics of quantum gravitational degrees of freedom and so to survive into a full theory of quantum gravity.Section 1, first two paragraphs
Published and peer-reviewed02Wall generalizes the trapped surface to quantum situations: a compact surface counts as quantum trapped when the fine-grained generalized entropy of the null surface shot outwards and to the future from it is decreasing at every one of its points. Given a globally hyperbolic spacetime containing such a surface, with the semiclassical approximation valid near it but nowhere else required, the generalized second law forces the spacetime to be null geodesically incomplete.Section 3.2, Theorem 3 with Eq. 34, and Theorem 4
Published and peer-reviewed03The consequences track Penrose’s: 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 only a finite amount of entropy-producing process in the past, unless the arrow of time reverses at some moment of lowest entropy.Abstract; Sections 4.1 and 4.2; Section 6, third paragraph
Published and peer-reviewed04Wall defines an asymptotic warp drive as a compact spacetime region through which some null curve reaches future null infinity a finite time ahead of any curve that goes around it, and shows that in asymptotically flat, globally hyperbolic spacetime either no such region exists or the fine-grained generalized second law is violated. The same argument forbids negative ADM masses, because a negative mass turns the Shapiro delay into an advance, and forbids traversable wormholes between two distant asymptotic regions.Section 4.3, the warp drive definition and the Positive Energy Theorem paragraph; the Traversable Wormholes paragraph in Section 4.1
Published and peer-reviewed05Wall states the scope of that result himself: it applies only to warp drives producing a finite advance for lightrays travelling over an infinite distance, and in his words the result does not apply to cases where there is a speed up only over a finite distance. He also calls the four-dimensional asymptotically Schwarzschild case somewhat trivial, since the logarithmically divergent Shapiro delay from the exterior field already swamps any finite advance produced inside — the result bites harder in five dimensions or more, and in anti-de Sitter space.Section 4.3, the comments following the warp drive definition and the Positive Energy Theorem discussion
What to watch06The premise is not yet a settled law: Wall reiterates that the generalized second law has only been proven in limited regimes, that other reasonable formulations of it may exist, and that the indications these results carry over into full quantum gravity are necessarily speculative — the argument being that the semiclassical approximation was used only in nearly classical regions far from high curvature.Abstract, penultimate sentence; Section 6, first and second paragraphs
What to watch
Read it · abstract
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.
The introduction 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.)
The way in
https://doi.org/10.1088/0264-9381/30/16/165003Published as Classical and Quantum Gravity 30, 165003 (2013). The preprint is on arXiv as 1010.5513, posted under the arXiv.org perpetual non-exclusive licence, which does not grant redistribution — so this page carries the summary, the claims and the author’s own abstract, and sends the reader to the source. The claims here are read against arXiv version 5, dated 6 December 2016, which carries the same results with corrected section numbering; written at the Maryland Center for Fundamental Physics, University of Maryland.
How to cite it
Aron C. Wall (2013) The generalized second law implies a quantum singularity theorem. doi:10.1088/0264-9381/30/16/165003
Where it sits in the curriculum
The metric, warp drives and wormholesWhat the vacuum isThe unified picture