Jb Manchak

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.

We examine whether the “privileged coordinates” of a geometric space encode its “amount of structure.” In doing so, we compare this coordinate approach to comparing amounts of structure to the more familiar automorphism approach. We first show that on a natural understanding of the former, it faces one of the same well-known problems as the latter. We then capture a precise sense in which the two approaches are closely related to one another, and we conclude by discussing whether they might still prove useful in cases of philosophical interest, despite their shortcomings.

Determinism is the thesis that the past determines the future, but efforts to define it precisely have exposed deep methodological disagreements. Standard possible-worlds formulations of determinism presuppose an "agreement" relation between worlds, but this relation can be understood in multiple ways -- none of which is particularly clear. We critically examine the proliferation of definitions of determinism in the recent literature, arguing that these definitions fail to deliver clear verdicts about actual scientific theories. We advocate a return to a formal approach, in the logical tradition of Carnap, that treats determinism as a property of scientific theories, rather than an elusive metaphysical doctrine. We highlight two key distinctions: (1) the difference between qualitative and "full" determinism, as emphasized in recent discussions of physics and metaphysics, and (2) the distinction between weak and strong formal conditions on the uniqueness of world extensions. We argue that defining determinism in terms of metaphysical notions such as haecceities is unhelpful, whereas rigorous formal criteria -- such as Belot's D1 and D3 -- offer a tractable and scientifically relevant account. By clarifying what it means for a theory to be deterministic, we set the stage for a fruitful interaction between physics and metaphysics.

This book concerns the modal structure of spacetime within the context of Einstein’s general relativity. The aim is to expose a rich set of philosophical issues somewhat informally and from a bird’s eye view. No familiarity with general relativity is presupposed. A large number of examples (worked out in coordinates) and corresponding diagrams (over 130) will play a central role in illustrating the ideas involved. One of the intended readers is a non-expert. She is perhaps a philosophy graduate student just getting started with an interest in better understanding cosmic possibility delimited by Einstein’s theory. Under her belt, she has little more than some basic set theory and a passing acquaintance with calculus and vector spaces. The other intended reader is the expert. She will find several interconnecting lines of current research mapped out and a clear thesis defended. The investigation will focus on the maximality property of spacetime -- the requirement that (a model of) the universe must be “as large as it can be.” The maximality of spacetime is something of a dogma within the context of generality relativity. In the background, it is often presupposed by practitioners without comment in order to go on to investigate a number of other foundational topics (e.g. determinism). Here, I do not defend or oppose the case for spacetime maximality. Instead, I simply put forward a thesis that nonetheless runs counter to the prevailing orthodoxy: it is not at all clear that the universe is as large as it can be. To shed light on the situation, 120 precise questions are identified and most are settled (see the final page). My hope is that, by drawing attention to these issues, even more progress can be made by others.

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.

Within the context of general relativity, the Heraclitus asymmetry property requires that no distinct pair of spacetime events have the same local structure Manchak and Barrett (2023). Here, we explore Heraclitus-maximal worlds – those which are “as large as they can be” with respect to the Heraclitus property. Using Zorn’s lemma, we prove that such worlds exist and highlight a number of their properties. If attention is restricted to Heraclitus-maximal worlds, we show senses in which observers have the epistemic resources to know which world they inhabit.

A supertask is a task that consists in infinitely many component steps, but which in some sense is completed in a finite amount of time. Supertasks were studied by the pre-Socratics and continue to be objects of interest to modern philosophers, logicians and physicists. The term “super-task” itself was coined by J.F. Thomson (1954). Here we begin with an overview of the analysis of supertasks and their mechanics. We then discuss the possibility of supertasks from the perspective of general relativity.

Wilhelm (Forthcom Synth 199:6357–6369, 2021) has recently defended a criterion for comparing structure of mathematical objects, which he calls Subgroup. He argues that Subgroup is better than SYM \(^*\), another widely adopted criterion. We argue that this is mistaken; Subgroup is strictly worse than SYM \(^*\). We then formulate a new criterion that improves on both SYM \(^*\) and Subgroup, answering Wilhelm’s criticisms of SYM \(^*\) along the way. We conclude by arguing that no criterion that looks only to the automorphisms of mathematical objects to compare their structure can be fully satisfactory.

One usually identifies a particular collection of geometric objects with the models of general relativity. But within this standard collection lurk ‘physically unreasonable’ models of spacetime. If such models are ruled out, attention can be restricted to some sub-collection of ‘physically reasonable’ models which can be considered a variant theory of general relativity. Since we have yet to identify a privileged sub-collection of ‘physically reasonable’ models, it is helpful to think of ‘general relativity’ in a pluralistic way; we can study a collection of such ‘physically reasonable’ collections of models.

I investigate the principle *anything goes* within the context of general relativity. After a few preliminaries, I show a sense in which the universe is unknowable from within this context; I suggest that we 'keep our options open' with respect to competing models of it. Given the state of affairs, proceeding counter-inductively seems to be especially appropriate; I use this method to blur some of the usual lines between 'reasonable' and 'unreasonable' models of the universe. Along the way, one is led to a useful collection of variant theories of general relativity -- each theory incompatible with the standard formulation. One may contrast one variant theory with another in order to understand foundational questions within 'general relativity' in a more nuanced way. I close by sketching some of the work ahead if we are to embrace such a pluralistic methodology.

We consider various curious features of general relativity, and relativistic field theory, in two spacetime dimensions. In particular, we discuss: the vanishing of the Einstein tensor; the failure of an initial-value formulation for vacuum spacetimes; the status of singularity theorems; the non-existence of a Newtonian limit; the status of the cosmological constant; and the character of matter fields, including perfect fluids and electromagnetic fields. We conclude with a discussion of what constrains our understanding of physics in different dimensions.

This exploration of the global structure of spacetime within the context of general relativity examines the causal and singular structures of spacetime, revealing some of the curious possibilities that are compatible with the theory, such as `time travel' and `holes' of various types. Investigations into the epistemic and modal structures of spacetime highlight the difficulties in ruling out such possibilities, unlikely as they may seem at first. The upshot seems to be that what counts as a `physically reasonable' spacetime structure in modern physics is far from clear.