The Entropy of a Vacuum: What Does the Covariant Entropy Count?

TL;DR

The paper tackles the origin of black hole entropy in quantum gravity by proposing that the Bekenstein-Hawking entropy counts the entropy of the vacuum, $S_{ m vac} \\approx {\\cal A}/(4 l_{ m P}^2)$, i.e., the logarithm of the number of independent vacuum realizations $|\\psi_k\\rangle$ that can support emergent quantum field theories on a fixed background. Each vacuum realizes an emergent QFT via an Unruh–Israel construction, with stretched-horizon degrees of freedom providing the second exterior region and ensuring a smooth horizon; unitarity is preserved through information flow from collapsing matter to the vacuum sector and then to final Hawking radiation. The framework extends to de Sitter and other spacetimes, positing a universal Hilbert-space structure ${\\cal H} = \\bigoplus_{\\partial{\\cal M}} {\\cal H}_{\\partial{\\cal M}}$ and arguing that the covariant entropy bound saturates chiefly through horizon (vacuum) degrees of freedom, while matter/radiation contributions are subleading. Collectively, these ideas offer a coherent microscopic account of horizon entropy, complementarity, and information transfer that is consistent with locality outside horizons and preserves horizon smoothness, thereby addressing firewall concerns and informing our understanding of cosmological horizons as well.

Abstract

We argue that a unitary description of the formation and evaporation of a black hole implies that the Bekenstein-Hawking entropy is the "entropy of a vacuum": the logarithm of the number of possible independent ways in which quantum field theory on a fixed classical spacetime background can emerge in a full quantum theory of gravity. In many cases, the covariant entropy counts this entropy--the degeneracy of emergent quantum field theories in full quantum gravity--with the entropy of particle excitations in each quantum field theory giving only a tiny perturbation. In the Rindler description of a (black hole) horizon, the relevant vacuum degrees of freedom manifest themselves as an extra hidden quantum number carried by the states representing the second exterior region; this quantum number is invisible in the emergent quantum field theory. In a distant picture, these states arise as exponentially degenerate ground and excited states of the intrinsically quantum gravitational degrees of freedom on the stretched horizon. The formation and evaporation of a black hole involve processes in which the entropy of collapsing matter is transformed into that of a vacuum and then to that of final-state Hawking radiation. In the intermediate stage of this evolution, entanglement between the vacuum and (early) Hawking radiation develops, which is transferred to the entanglement among final-state Hawking quanta through the evaporation process. The horizon is kept smooth throughout the evolution; in particular, no firewall develops. Similar considerations also apply for cosmological horizons, for example for the horizon of a meta-stable de-Sitter space.