Thermodynamics of a spherically symmetric causal diamond in Minkowski spacetime
Abstract
We compute a gravitational on-shell action of a finite, spherically symmetric causal diamond in (d + 2)-dimensional Minkowski spacetime, finding it is proportional to the area of the bifurcate horizon A_B. We then identify the on-shell action with the saddle point of the Euclidean gravitational path integral, which is naturally interpreted as a partition function. This partition function is thermal with respect to a modular Hamiltonian K. Consequently, we determine, from the on-shell action using standard thermodynamic identities, both the mean and variance of the modular Hamiltonian, finding〈K〉=〈(∆K)2〉= A_B/4G_N. Finally, we show that modular fluctuations give rise to fluctuations in the geometry, and compute the associated phase shift of massless particles traversing the diamond under such fluctuations.
Copyright and License
© The Authors. Article funded by SCOAP3. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Acknowledgement
We would like to thank Ning Bao, Mathew Bub, Clifford Cheung, Luca Ciambelli, Bartlomiej Czech, Laurent Freidel, Marc Klinger, Prahar Mitra, Hirosi Ooguri, and Julian Sonner for helpful discussions. K.F., T.H., and K.Z. are supported by the Heising-Simons Foundation “Observational Signatures of Quantum Gravity” collaboration grant 2021-2817, the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and the Walter Burke Institute for Theoretical Physics. T.H. would like to thank the Galileo Galilei Institute for Theoretical Physics for their hospitality and the INFN for partial support during the completion of this work. K.Z. is also supported by a Simons Investigator award.
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Additional details
Related works
- Is new version of
- Discussion Paper: arXiv:2507.22977 (arXiv)
Funding
- Heising-Simons Foundation
- 2021-2817
- United States Department of Energy
- DE-SC0011632
- California Institute of Technology
- Walter Burke Institute for Theoretical Physics -
- Istituto Nazionale di Fisica Nucleare
- Simons Foundation
- SCOAP3
Dates
- Submitted
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2025-08-12
- Accepted
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2025-11-16
- Available
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2025-12-17Version of record