Generalized Second Law for Cosmology

TL;DR

The paper addresses the lack of a general second law in cosmology by introducing Q-screens, quantum-corrected holographic screens that rely on the generalized entropy $S_ ext{gen}$ to track thermodynamic behavior locally in arbitrary spacetimes. It defines $S_ ext{gen}$ and quantum expansion $\Theta_k$, constructs quantum marginal surfaces and Q-screens, and posits a New Generalized Second Law (GSL) stating that $S_ ext{gen}$ increases monotonically outside past or future Q-screens, with a proof structure contingent on the Quantum Focussing Conjecture $\text{QFC}$. The paper illustrates the law with evaporating black holes and cosmological models, showing that Hawking radiation can compensate for area loss and that area growth dominates in cosmology, respectively, thereby validating the GSL in these settings. Altogether, it extends thermodynamic principles to general spacetimes, linking geometry, quantum information, and semiclassical gravity in a cosmological context.

Abstract

We conjecture a novel Generalized Second Law that can be applied in cosmology, regardless of whether an event horizon is present: the generalized entropy increases monotonically outside of certain hypersurfaces we call past Q-screens. A past Q-screen is foliated by surfaces whose generalized entropy (sum of area and entanglement entropy) is stationary along one future null direction and increasing along the other. We prove that our Generalized Second Law holds in spacetimes obeying the Quantum Focussing Conjecture. An analogous law applies to future Q-screens, which appear inside evaporating black holes and in collapsing regions.

Figures (5)

• Figure 1: The generalized entropy $S_\mathrm{gen}$ is the area $A$ (in Planck units) of a surface that splits a Cauchy surface, plus the von Neumann entropy $S_\mathrm{out}$ of the quantum fields on one side $\Sigma_\mathrm{out}$. The quantum expansion $\Theta_k$ is the rate at which $S_\mathrm{gen}$ changes as the splitting surface is varied in the orthogonal null direction $k^a$.
• Figure 2: Black hole formed by dust collapse. The thin green lines are future light cones which form a null foliation of the spacetime. (a) No Hawking radiation. A dot indicates the marginal surface on each light cone. The area of the classical holographic screen increases towards the exterior and past (arrow). (b) Hawking radiation included. A solid (hollow) dot marks the quantum marginal (marginal) surface(s) on each light cone. The classical screen (short dashed) now lies outside the event horizon (long dashed) during evaporation. A future Q-screen lies inside the black hole. Its area decreases during evaporation. But due to the production of Hawking radiation, the generalized entropy outside the Q-screen increases monotonically, as demanded by our conjecture.
• Figure 3: Radiation dominated expanding universe; dots and lines as in Fig. 2b. The classical and Q-screen nearly coincide; the area and generalized entropy both grow monotonically to the future.
• Figure 4: (a) A surface $\sigma$ that splits a Cauchy surface defines a partition of the entire spacetime into four regions, given by the past or future domains of dependence and the chronological future or past of the two partial Cauchy surfaces $\Sigma^\pm$. (b) The pairwise unions $K^\pm$ depend only on $\sigma$, not on the choice of Cauchy surface. $K^\pm$ share a boundary $N = N^+\cup N^- \cup \sigma$ generated by light-rays orthogonal to $\sigma$. (c) If a hypersurface $\cal H$ foliated by $\sigma(r)$ has $\alpha<0$ everywhere (see text for definition), then the sets $K^+(r)$ are monotonic under inclusion, and the sets $N(r)$ define a null foliation of spacetime.
• Figure 5: The flow from leaf to leaf along a Q-screen can be decomposed as a sequence of infinitesimal motions in the $k^a$ and $l^a$ null directions. In the $\pm k^a$ direction, the generalized entropy is locally stationary by definition of the Q-screen, $\Theta_k=0$. Because $\alpha<0$ by Theorem \ref{['thm:structure']}, the motion is always towards $-l^a$, along which the generalized entropy increases since $\Theta_l<0$. Hence the generalized entropy increases along the flow.

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Conjecture 2.8
• Lemma 4.1
• Lemma 4.2