Determinism and Asymmetry in General Relativity

Abstract

This paper concerns the question of which collections of general relativistic spacetimes are deterministic relative to which definitions. We begin by considering a series of three definitions of increasing strength due to Belot (1995). The strongest of these definitions is particularly interesting for spacetime theories because it involves an asymmetry condition called ``rigidity'' that has been studied previously in a different context (Geroch 1969; Halvorson and Manchak 2022; Dewar 2024). We go on to explore other (stronger) asymmetry conditions that give rise to other (stronger) forms of determinism. We introduce a number of definitions of this type and clarify the relationships between them and the three considered by Belot. We go on to show that there are collections of general relativistic spacetimes that satisfy much stronger forms of determinism than previously known. We also highlight a number of open questions.

Figures (2)

• Figure 1: Two copies of the spacetime $(M,g)$ from Example 1. The time translation isometry $\varphi$ from the initial segment $U$ to the initial segment $U'$ does not extend to a global isometry because of the $t=0$ 'edge' of the spacetime (dotted line).
• Figure 2: The spacetime $(M',g)$ from Example 3. Because of the 'missing' region (above the dotted line), the only global isometry is the identity map. But there is a non-trivial reflection isometry from the initial segment $U$ to itself.