Geometry of small causal diamonds

Abstract

The geometry of small causal diamonds is systematically studied, based on three distinct constructions that are common in the literature, namely the geodesic ball, the Alexandrov interval and the lightcone cut. The causal diamond geometry is calculated perturbatively using Riemann normal coordinate expansion up to the leading order in both vacuum and non-vacuum. We provide a collection of results including the area of the codimension-two edge, the maximal hypersurface volume and their isoperimetric ratio for each construction, which will be useful for any applications involving the quantitative properties of causal diamonds. In particular, by solving the dynamical equations of the expansion and the shear on the lightcone, we find that intriguingly only the light-cone cut construction yields an area deficit proportional to the Bel-Robinson superenergy density W in four dimensional spacetime, but such a direct connection fails to hold in any other dimension. We also compute the volume of the Alexandrov interval causal diamond in vacuum, which we believe is important but missing from the literatures. Our work complements and extends the earlier works on the causal diamond geometry by Gibbons and Solodukhin, Jacobson, Senovilla and Speranza and others. Some potential applications of our results in mathematical general relativity and quantum gravity are discussed.

arXiv comment14 pages, 1 figure, published version

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Cited by in corpus (14)

• New perspective on thermodynamics of spacetime: The emergence of unimodular gravity and the equivalence of entropies
• Quantum phenomenological gravitational dynamics: A general view from thermodynamics of spacetime
• On the continuum limit of Benincasa-Dowker-Glaser causal set action
• Thermodynamics as a tool for (quantum) gravitational dynamics
• The small sphere limit of quasilocal energy in higher dimensions along lightcone cuts
• Path integral derivation of the thermofield double state in causal diamonds
• From spacetime thermodynamics to Weyl transverse gravity
• Emergence of quadratic gravity from entanglement equilibrium
• Applications of the Hawking Energy in Inhomogeneous Cosmology
• Gravity from thermodynamics in vacuum: Lorentz invariance and the role of Bel-Robinson tensor
• Gravity from equilibrium thermodynamics of stretched light cones
• Local gravitational energy in higher dimensions
• Thermodynamics of spacetime from minimal area
• Local foliations by critical surfaces of the Hawking energy and small sphere limit