Time functions as utilities (Q625479)

This work imports the idea of utility function, originally developed in the field of microeconomics, to the relativistic space-time framework and proves some important results which relate this concept with the causality properties of space-time. Let us give some definition: We denote, as usual in relativistic physics, the causal and chronological relations by \(\leq\) and \(\ll\). A semi-time function is defined to be a continuous map \(t:M\to \mathbb{R}\) such that \(x\ll y \Rightarrow t(x)<t(y)\). A time function is defined to be a continuous map \(t:M\to \mathbb{R}\) such that \(x<y \Rightarrow t(x)<t(y)\). A temporal function is defined to be a \(C^1\) time function with a past directed time-like gradient. The Seifert relation is \(J^{+}_S = \bigcap_{g'>g} J^{+}_{g'}\), where \(g'>g\) means that the time-like cone of \(g'\) contains the causal cone of \(g\), and \(J^{+}_{g'} \subset M \times M\) is the causal relation for the space-time \((M,g')\). The Seifert relation is closed, reflexive and transitive, and furthermore, it is antisymmetric if and only if the space-time is stably causal [\textit{E. Minguzzi}, Classical Quantum Gravity 25, No.~1, Article ID 015010 (2008; Zbl 1132.83003)]. The \(K^{+}\) relation is instead the smallest closed and transitive relation which contains the causal relation [\textit{R. D. Sorkin} and \textit{E. Woolgar}, Classical Quantum Gravity 13, No.~7, 1971--1993 (1996; Zbl 0966.83022)], as a consequence \(J^{+}\subset K^+\subset J^{+}_S\). A space-time is \(K\)-causal if the relation \(K^{+}\) is antisymmetric. It must be mentioned that, in a previous work, the author proved the equivalence between \(K\)-causality and stable causality, and the equality \(K^+ = J^+_S\) in stably causal space-times [\textit{E. Minguzzi}, Commun. Math. Phys. 290, No.~1, 239--248 (2009; Zbl 1182.53065)] (this result is not used in this work where he offers a different proof). A preorder \(R\) on a set \(X\) is defined to be a reflexive and transitive relation on \(X\). An antisymmetric preorder is a partial order. Totality means \(x\leq_R y\) or \(y\leq_R x\). A utility on set \(X\) endowed with a preorder \(R\) is defined to be a function \(u : X\to \mathbb{R}\) with the strictly isotone property \[ \text{``}x \sim_R y \Rightarrow u(x) = u(y)\text{''}\text{ and }\text{``}x <_R y \Rightarrow u(x)< u(y).\text{''} \] As a first step, the author relates the concept of utility with that of time function by proving the following: Theorem. In a \(K\)-causal space-time the family of continuous \(K^{+}\)-utilities coincides with the family of time functions. Then the author takes advantage of Levin's theorem [\textit{V. L. Levin}, Sov. Math., Dokl. 28, 715--718 (1983); translation from Dokl. Akad. Nauk SSSR 273, 800--804 (1983; Zbl 0554.90012)], which concerns the existence of continuous utility functions on a topological space, to prove the following: Theorem. A space-time is \(K\)-causal if and only if it admits a time function (as a consequence time functions are always \(K^{+}\)-utilities). In this case, denoting by \(\mathcal{A}\) the set of time functions, the partial order \(K^{+}\) can be recovered from the time functions, that is \[ (x,y) \in K^{+}\Leftrightarrow\forall t \in \mathcal{A}, \;t(x)\leq t(y). \] In particular every \(K\)-causal space-time admits a time function. Using the equivalence between the existence of time functions and stable causality, he reobtains that \(K\)-causality coincides with stable causality. From the previous theorem, he also obtains that a chronological space-time in which \(\overline{J^{+}}\) is transitive admits a semi-time function. The author ends the paper by introducing, for every time (temporal) function \(t\), a total preorder, on \(M\) given by \[ T^{+}[t] = \{(p,q)\in M\times M: t(p)\leq t(q)\}, \] and proving the following result, which shows that the Seifert relation can be recovered from the set of time or temporal functions allowed by the space-time: Theorem. Let \(\mathcal{A}\) and \(\mathcal{B}\) be respectively the set of time functions and the set of temporal functions allowed by a space-time. In every space-time, \[ K^{+}\subset J^{+}_S\subset \bigcap_{t\in \,\mathcal{A}}\, T^{+}[t] \subset \bigcap_{t\in \,\mathcal{B}}\, T^{+}[t]. \] In a stably causal space-time, \[ K^{+}= J^{+}_S= \bigcap_{t\in \,\mathcal{A}}\, T^{+}[t] = \bigcap_{t\in \,\mathcal{B}}\, T^{+}[t]. \] Then, the author argues that since, in general, the identity \(J^{+}_S=J^+\) does not hold (but it holds in causally simple space-times, and globally hyperbolic space-times that are causally simple), the causal relation \(J^+_S\) is physically more significant than \(J^+\).