Time functions as utilities

Abstract

Every time function on spacetime gives a (continuous) total preordering of the spacetime events which respects the notion of causal precedence. The problem of the existence of a (semi-)time function on spacetime and the problem of recovering the causal structure starting from the set of time functions are studied. It is pointed out that these problems have an analog in the field of microeconomics known as utility theory. In a chronological spacetime the semi-time functions correspond to the utilities for the chronological relation, while in a K-causal (stably causal) spacetime the time functions correspond to the utilities for the K^+ relation (Seifert's relation). By exploiting this analogy, we are able to import some mathematical results, most notably Peleg's and Levin's theorems, to the spacetime framework. As a consequence, we prove that a K-causal (i.e. stably causal) spacetime admits a time function and that the time or temporal functions can be used to recover the K^+ (or Seifert) relation which indeed turns out to be the intersection of the time or temporal orderings. This result tells us in which circumstances it is possible to recover the chronological or causal relation starting from the set of time or temporal functions allowed by the spacetime. Moreover, it is proved that a chronological spacetime in which the closure of the causal relation is transitive (for instance a reflective spacetime) admits a semi-time function. Along the way a new proof avoiding smoothing techniques is given that the existence of a time function implies stable causality, and a new short proof of the equivalence between K-causality and stable causality is given which takes advantage of Levin's theorem and smoothing techniques.

15 pages, 1 figure. Replaces 0905.0274 "From time to causality" which is now mostly included in section 4

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Cited by in corpus (20)

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• Normally preordered spaces and utilities
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• On differentiability of volume time functions
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• Globally hyperbolic spacetimes: slicings, boundaries and counterexamples
• Convexity and quasi-uniformizability of closed preordered spaces
• Global Hyperbolicity through the Eyes of the Null Distance
• Time functions on Lorentzian length spaces
• Causality theory of spacetimes with continuous Lorentzian metrics revisited
• Quasi-pseudo-metrization of topological preordered spaces
• The classification of preordered spaces in terms of monotones: complexity and optimization
• The disappearance of causality at small scale in almost-commutative manifolds
• Generally covariant $N$-particle dynamics
• Representing preorders with injective monotones
• Gluing of Lorentzian length spaces and the causal ladder
• Further observations on the definition of global hyperbolicity under low regularity
• Time functions and $K$-causality between measures
• The representation of spacetime through time functions