Indeterminism in General Relativity

一个永恒的哲学问题是我们的世界是否具有确定性。如果我们相信物理理论告诉我们世界的真实面貌，我们不妨问问这些理论是否具有决定论。如果决定论是理论形式主义的一个特征，那将是可取的——我们只要看看它，就能马上解决这个问题。但事实并非如此，决定论是理论所描述的一个特征，因此我们需要知道理论究竟对我们的物理现实说了些什么，也就是说，我们需要让我们的理论得到解释。就广义相对论而言，乍一看，我们似乎很好地掌握了这一理论对世界的描述（至少比量子理论好得多）。然而，GR的决定论问题尚未得到解决。一方面，时空理论中的决定论概念存在一般性问题。首先，如何找到一个似是而非的决定论定义（Butterfield 1989, Earman 1986, Melia 1999），它既(i)公正地明确（in）决定论的玩具例子（Melia 1999, Cudek 2023a），又（ii）充分独立于我们的形而上学行为（见Earman and Norton 1987）。其次，如何理解理论模型与这些模型所代表的可能情况之间的关系：关于这一点的冲突观点，请参见Weatherall 2018， Fletcher 2020和Belot 2018。这反过来又与对称和结构的广泛主题密切相关（杜瓦2022是最近的概述）。这两个问题将是我论文前两章的重点。另一方面，GR特有的一些问题似乎对任何看似合理的决定论都构成了威胁。首先，有一些众所周知的例子，比如具有封闭类时曲线或裸奇点的时空（更多例子，参见Earman 1995和Doboszewski 2019）。但是，这些异常是否真的在物理上是可能的，从而对决定论构成威胁，这个问题仍然悬而未决（Earman 1995, Manchak 2011）。其次，当代GR前沿研究的一个重要组成部分是研究确定性的某种吸引人的形式标准的形式先决条件（即给定初始数据的最大未来柯西发展的不可扩展性）。最近的结果似乎表明，相当令人惊讶的是，宇宙常数的值在建立这一标准方面发挥了重要作用（见Dafermos和Luk 2017和Kehle 2022）。本文的第三章和第四章将讨论这些结果的哲学意义。在我的博士项目中，我将试图将这些松散的线索收集在一起，并研究在广义相对论背景下统一解释决定论的可能性。关于决定论的定义、决定论与模态形而上学之间的关系以及模型对可能性的表示的部分将扩展我以前的工作（Cudek 2023a, 2023b）。这个项目的结论不仅应该与物理学哲学和我们对相对论的一般理解有关，而且应该与形而上学、认识论和一般科学哲学有关。我解决上述研究问题的方法结合了：(a)涵盖以下问题的哲学文献的专业知识：物理理论中的决定论（重点是时空理论），物理中的模态，以及形式主义与理论解释之间的关系，(b)对研究生水平的广义相对论的技术熟练程度，以及对当代广义相对论数学研究的熟悉程度。因此，我的研究方法将包括阅读和审查论文的标准哲学方法，参加讨论小组和研讨会，以及通过旁听数学或物理系的研究生课程或通过自学来涵盖任何相关的技术材料。

One of the perennial philosophical questions is whether our world is deterministic. If we believe that physical theories tell us what the world is really like, we may just as well ask if these theories are deterministic. It would be desirable if determinism was a feature of the theory's formalism - we could just look at it and settle the issue right away. But this is not so: determinism is a feature of what the theory describes, and so we need to know what exactly the theory says about our physical reality, i.e. we need to have our theories interpreted.In the case of general relativity (GR), it seems at first sight that we have a good grip on what the theory says about the world (at least much better than in the case of quantum theories). Nevertheless, the question of GR's determinism is not yet settled. On the one hand, there are general issues with the notion of determinism in spacetime theories. First, how to find a plausible definition of determinism (Butterfield 1989, Earman 1986, Melia 1999), which both (i) does justice to clearly (in)deterministic toy examples (Melia 1999, Cudek 2023a), and (ii) is sufficiently independent of our metaphysical commitments (see Earman and Norton 1987). Second, how to understand the relation between the theory's models and the possible situations these models represent: for conflicting views on this, see Weatherall 2018, Fletcher 2020 and Belot 2018. This, in turn, is closely related to the vast topic of symmetry and structure (Dewar 2022 is a recent overview). These two issues will preoccupy the first two chapters of my thesis. On the other hand, there are issues specific to GR, which seem to pose a threat to any plausible account of determinism. First, there are well-known examples such as spacetimes with closed time-like curves or naked singularities (for more examples, see Earman 1995 and Doboszewski 2019).But the question of whether these anomalies are genuinely physically possible, and thus a threat to determinism, remains open (Earman 1995, Manchak 2011). Second, a significant strand of contemporary cutting-edge research in GR investigates formal prerequisites for a certain appealing formal criterion for determinism (namely, inextendibility of maximal future Cauchy development of given initial data). Recent results seem to suggest that, quite surprisingly, the value of the cosmological constant plays a significant role in establishing this criterion (see Dafermos and Luk 2017 and Kehle 2022). The third and fourth chapters of my thesis will discuss the philosophical significance of these results.In my doctoral project, I shall seek to gather these loose threads together, and investigate the possibility of a unified account of determinism in the context of GR. The sections on definitions of determinism, the relationship between determinism and modal metaphysics, and the representation of possibilities by models, will expand on my previous work (Cudek 2023a, 2023b). The conclusion of this project should be relevant not only to philosophy of physics and our general understanding of relativity theory, but also to metaphysics, epistemology, and general philosophy of science.My intended method for addressing the research questions set out above combines: (a) expertise in the philosophical literature covering the issues of: determinism in physical theories (with an emphasis on spacetime theories), modality in physics, and the relationship between the formalism and the interpretation of a theory, with (b) technical proficiency in graduate-level GR, and familiarity with contemporary research in mathematics of GR. Thus, my research method will involve both standard philosophical methodology of reading and reviewing papers, attending discussion groups and seminars, as well as covering any relevant technical material by auditing graduate classes in mathematics or physics departments, or through self-study.