Robust Singularity Theorem

TL;DR

This work extends the Penrose-Wall singularity theorem to the full semiclassical gravity regime by introducing wedges and a generalized max-entropy framework. By replacing classical area with the generalized entropy $S_{\rm gen}$ and defining discrete expansions (PNE/PNC) for wedges, the authors prove a semiclassical singularity theorem that excludes bounces inside black holes and in broad cosmologies even at finite $cG\hbar$. The approach circumvents the limitations of the touching lemma in the strict $G\hbar\to0$ limit, leveraging the Generalized Second Law (GSL), the Quantum Focusing Conjecture (QFC), and related entropy bounds. The result broadens the domain where singularity formation is guaranteed under quantum corrections, incorporating Hawking evaporation and entanglement islands while preserving the predictive power of semiclassical gravity for both black holes and cosmological models.

Abstract

We prove the Penrose-Wall singularity theorem in the full semiclassical gravity regime, significantly expanding its range of validity. To accomplish this, we modify the definition of quantum-trapped surfaces without affecting their genericity. Our theorem excludes controlled "bounces" in the interior of a black hole and in a large class of cosmologies.

Figures (3)

• Figure 1: Spatial projection onto a Cauchy slice. The edges of the regions $a\subset b$ touch at the point $p$ (left figure). Classically, the outward expansion of $a$ is obviously no smaller than that of $b$ at $p$. But the difference in quantum expansions is unconstrained, because a null deformation of $a$ at $p$ (light green) will necessarily access a different region than any null deformation of $b$ at $p$ (light blue) -- except if the edges of $a$ and $b$ overlap in an entire open set (black ellipse, right figure). Only the latter situation arises in the new proof given here.
• Figure 2: Penrose diagram of an evaporating black hole; the event horizon is blue and the future singularity is orange. The nonexpanding null directions of spheres are shown in green Bousso:1999cb. (For the green dotted line see Ref. Bousso:2022tdb.) The pink lines denote the Hawking radiation that has arrived at $\mathscr{I}^+$ (thick) and its causal past (thin). The red dot marks the Quantum Extremal Surface associated with the radiation, at which the classical null expansions are canceled by an equal and opposite quantum term. The two terms remain comparable at the sphere marked by the blue dot; therefore a strict $cG\hbar \to 0$ limit cannot be taken while holding the overall geometry fixed. The blue sphere is quantum trapped, even after it is slightly perturbed to make its shape generic. The blue geodesic is incomplete, but the proof of the Penrose-Wall singularity theorem does not apply because it requires $cG\hbar\to 0$. Here we prove a theorem that applies at fixed nonzero $cG\hbar$.
• Figure 3: Wedges involved in the proof of the singularity theorem. Top left: conformal spacetime diagram. Bottom right: only the Cauchy slice $\Sigma$ and its subregions are shown. $a$ is the blue shaded region, assumed to be noncompact and trapped (PNC). Its future null deformation $b$ is the exterior of the red circle. Assuming geodesic completeness (no singularity), the null geodesic $\gamma$ defines a causal horizon $\dot I^-(\gamma)$. The orange curve is $\dot I^-(\gamma)\cap \Sigma$, and the wedge $c$ lies to its right. The wedge $d=a\Cup c$ (blue and pink regions combined) must be PNE on the edge portion $\eth d\setminus \eth a$ (the arc where the orange and purple curves coincide), in contradiction with $a$ being PNC.

Theorems & Definitions (1)

• proof