Historical research guide · 0575 · 2013
The generalized second law implies a quantum singularity theorem
Aron C. Wall
In one minute
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.
Connection
Chapter 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.
From the source
Every 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
Wall 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
The 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
Wall 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
Wall 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
The 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
The way in
https://doi.org/10.1088/0264-9381/30/16/165003How to cite it
Aron C. Wall (2013) The generalized second law implies a quantum singularity theorem doi:10.1088/0264-9381/30/16/165003