Classical Aspects of a Distributional 3+1 Foam Model

TL;DR

This paper models a 3+1 spacetime foam using a distributional approach, demonstrating that certain sequences of causally stable spacetimes exhibit super-polynomial growth in shift vectors and volume expansion, with implications for quantum gravity analogies.

Contribution

It introduces a novel distributional 3+1 foam model based on a fundamental solution to the wave operator, linking it to sequences of globally hyperbolic spacetimes and their geometric properties.

Findings

Shift vectors grow faster than any polynomial in the sequence

Volume expansion exceeds polynomial growth in all directions

Existence of spacetime elements with negligible extrinsic curvature

Abstract

A 3+1 spacetime, with a shift vector that is the unique fundamental solution to the linearized wave operator, is introduced to model an interpretation of Wheeler's layman's analogy of the Quantum foam. To understand the distributional aspects of this model is the guaranteed existence of a sequence of compactly supported shift vectors that converge to the fundamental solution used to introduce a sequence of 3+1 globally hyperbolic spacetimes. Using the sequence of these causally stable spacetimes it is shown that there exists a positive integer such that for all elements in the sequence with a greater index value than this integer and for any Eulerian observer will the shift vector increase more rapidly than any polynomial and the volume expansion is more rapid than a polynomial in all directions. The same conclusion remains valid for the trace of the extrinsic curvature. Nonetheless, it is shown, no matter how volatile the extrinsic curvature is for these elements there also exists elements in the other end of the sequence where the extrinsic curvature is negligible and the spacetime flat.