Thermodynamics of a spherically symmetric causal diamond in Minkowski spacetime

A bstract 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 $$ {A}_{\mathcal{B}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:math> . 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 〉 = $$ \frac{A_{\mathcal{B}}}{4{G}_N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> </mml:msub> <mml:mrow> <mml:mn>4</mml:mn> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> </mml:math> . 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.