2019
DOI: 10.1103/physrevd.100.064020
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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 e… Show more

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Cited by 14 publications
(14 citation statements)
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“…However, the vacuum limit (12) is not proportional to the BR superenergy Q in any dimensions n > 4. Note that the area of the lightcone cut S l itself has a deficit due to curvature that is proportional to Q 0 in four dimensions [15], but it is not proportional to Q in higher dimensions [31]. Here, we see the behaviour of the QLM is in line with the area deficit.…”
mentioning
confidence: 53%
“…However, the vacuum limit (12) is not proportional to the BR superenergy Q in any dimensions n > 4. Note that the area of the lightcone cut S l itself has a deficit due to curvature that is proportional to Q 0 in four dimensions [15], but it is not proportional to Q in higher dimensions [31]. Here, we see the behaviour of the QLM is in line with the area deficit.…”
mentioning
confidence: 53%
“…The presence of inhomogeneities is reflected in the appearance of shear terms in (21) and (24). It is stressed again, that, due to the conservation law (25), the so generated 1-parameter family of density and equationof-state parameters (ρ(z), w(z)) does not describe the evolution of a FLRW-fluid with redshift in general, but rather reinterprets the given inhomogeneous geometry with a RW-bias. Thirdly, the monotonicity of the Hawking energy along the lightcone is applied to FLRW-spacetimes to derive bounds (28) & (29) on w(z) and ρ(z), which are in particular relevant for redshifts beyond the turnaround of the area distance.…”
Section: Discussionmentioning
confidence: 99%
“…The differential equation ( 18) links the area to the matter content in an exact way, in contrast to the expansions of the area about Minkowski space, for instance about the lightcone vertex or within the causal diamond construction as discussed in [24,25]. However, these expansions are recovered from (18) for a general affine parameter λ by solving it in a series expansion in the respective regime.…”
Section: A Link Between Area and Matter Contentmentioning
confidence: 99%
“…1 for a deeper analysis of GLCDs, see, e.g., [29][30][31]. 2 We remember that this interpretation, as are all interpretations, is not free of controversy.…”
Section: Conflicts Of Interestmentioning
confidence: 95%