Unbounded entropy in spacetimes with positive cosmological constant

TL;DR

The paper challenges the notion that a positive cosmological constant Λ alone determines the gravity sector dual to a finite-dimensional Hilbert space by constructing flux-supported product spacetimes of the form $K_p\times M_q$ with $M_q=S^q$. Through a Kaluza–Klein stability analysis, it identifies stability windows and tachyonic instabilities across AdS$_p\times S^q$ and dS$_p\times S^q$ backgrounds, showing that some stable solutions possess observable entropy exceeding the proposed bound $N=S_0$ with $S_0=\frac{\Omega_{D-2}}{4}\left[\frac{(D-1)(D-2)}{2\Lambda}\right]^{\frac{D-2}{2}}$ (reducing to $N=\frac{3\pi}{\Lambda}$ in $D=4$). Consequently, the simple $\ ext{all}(\Lambda(N))$ class is not adequate to capture finite-entropy quantum-gravity sectors, indicating that Λ must be supplemented by additional parameters (e.g., flux content) to define consistent gravity duals. The results imply that a true Λ–N correspondence may not exist in higher dimensions and accentuate the need for refined criteria when seeking finite-Hilbert-space descriptions of quantum gravity. In 4D there is an exception, but the broader conclusion remains that a positive cosmological constant alone does not suffice to bound observable entropy.

Abstract

In theories of gravity with a positive cosmological constant, we consider product solutions with flux, of the form (A)dS_p x S^q. Most solutions are shown to be perturbatively unstable, including all uncharged dS_p x S^q spacetimes. For dimensions greater than four, the stable class includes universes whose entropy exceeds that of de Sitter space, in violation of the conjectured "N-bound". Hence, if quantum gravity theories with finite-dimensional Hilbert space exist, the specification of a positive cosmological constant will not suffice to characterize the class of spacetimes they describe.