Abstract
We study causal diamonds in Minkowski, Schwarzschild, (anti) de Sitter, and Schwarzschild-de Sitter spacetimes using Euclidean methods. The null boundaries of causal diamonds are shown to map to isolated punctures in the Euclidean continuation of the parent manifold. Boundary terms around these punctures decrease the Euclidean action by $A_\diamond/4$, where $A_\diamond$ is the area of the holographic screen around the diamond. We identify these boundary contributions with the maximal entropy of gravitational degrees of freedom associated with the diamond.
Highlights
It has become increasingly clear over the past few years that the key to understanding how Einstein’s theory of general relativity fits into the framework of quantum mechanics is the relationship between quantum information and spacetime geometry
Much of the work in this area has focused on the anti–de Sitter (AdS)=conformal field theory (CFT) correspondence, where the Ryu-Takayanagi formula [1] connects precise calculations in quantum field theory (QFT) to the areas of spacetime submanifolds of the dual geometry
Evidence mounts [8,9,10] that Euclidean path integrals over geometries can reproduce features of quantum gravity that go beyond entropies of subsystems
Summary
It has become increasingly clear over the past few years that the key to understanding how Einstein’s theory of general relativity fits into the framework of quantum mechanics is the relationship between quantum information and spacetime geometry. Our focus is on the Euclidean path integral, we would like to note the interesting fact that one can obtain a boundary contribution to the Lorentzian action over a finite region bounded by a light sheet equal to A=4 by careful consideration of the null surface boundary term [22,23,24]. The connection of this very general Lorentzian result to the Euclidean path integral and its thermodynamic interpretation bears further investigation
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