Null distance on a spacetime

Christina Sormani; Carlos Vega

doi:10.1088/0264-9381/33/7/085001

paper

Null distance on a spacetime

arXiv:1508.00531 · doi:10.1088/0264-9381/33/7/085001

Abstract

Given a time function $τ$ on a spacetime $M$ , we define a `null distance function', $d^_{τ}$ , built from and closely related to the causal structure of $M$ . In basic models with timelike $\nabla τ$ , we show that 1) $d^_{τ}$ is a definite distance function, which induces the manifold topology, 2) the causal structure of $M$ is completely encoded in $d^_{τ}$ and $τ$ . In general, $d^_{τ}$ is a conformally invariant pseudometric, which may be indefinite. We give an `anti-Lipschitz' condition on $τ$ , which ensures that $d^_{τ}$ is definite, and show this condition to be satisfied whenever $τ$ has gradient vectors $\nabla τ$ almost everywhere, with $\nabla τ$ locally `bounded away from the light cones'. As a consequence, we show that the cosmological time function of [1] is anti-Lipschitz when `regular', and hence induces a definite null distance function. This provides what may be interpreted as a canonical metric space structure on spacetimes which emanate from a common initial singularity, e.g. a `big bang'.

v2: Small changes/improvements, mainly in the Introduction, Section 3.7, and the opening of Section 4, references added, 39 pages, 8 figures. To appear in Classical and Quantum Gravity

Null distance on a spacetime

Abstract

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