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How to count states in gravity

Vijay Balasubramanian1,2,3,* and Tom Yildirim4,†

  • *Contact author: vijay@physics.upenn.edu
  • †Contact author: tom.yildirim@physics.ox.ac.uk
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Phys. Rev. D 114, 026030 – Published 23 July, 2026

DOI: https://doi.org/10.1103/9v3t-91qh

Abstract

Gibbons and Hawking proposed that the Euclidean gravity path integral with periodic boundary conditions in time computes the thermal partition sum of gravity. As a corollary, they argued that a derivative of the associated free energy with respect to the Euclidean time period computes gravitational entropy. Why is this interpretation correct? That is, why does this path integral compute a trace over the Hilbert space? Here, we show that the quantity computed by the Gibbons-Hawking path integral is equal to an a priori different object—an explicit thermal trace over the Hilbert space spanned by states produced by the Euclidean gravity path integral. This follows in two ways; (a) if the Hilbert space with two boundaries factorizes into a product of two single boundary Hilbert spaces, as we have previously shown, and (b) via explicit resolution of the trace by a spanning basis of states. We similarly show how a replicated Euclidean gravity path integral with a single periodic boundary computes a Hilbert space trace of powers of the density matrix, explaining why this approach computes the entropy of states entangled between two universes.

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Phys. Rev. D 114, 026030– Published 23 July, 2026
  • Received 5 March 2026
  • Accepted 26 June 2026
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