Time Functions as Utilities |
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Authors: | E Minguzzi |
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Institution: | (1) Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA;(2) Center for Simulation of Advanced Rockets, Computational Science and Engineering Program, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA;(3) Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA;(4) Center for Geometric and Biological Computing, Department of Computer Science, Duke University, Durham, NC, USA |
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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. |
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