[Paper Review] Relativistic Constraints for a Naturalistic Metaphysics of Time

Peter William Evans|arXiv (Cornell University)|Nov 10, 2010

Relativity and Gravitational Theory35 references3 citations

TL;DR

This paper develops a formal framework based on Rovelli's relationalism to analyze time in relativistic physics, showing that both static and dynamic views of time remain logically consistent with special and general relativity when properly constrained. It identifies global hyperbolicity and Cauchy surfaces as key conditions for dynamic time, and argues that constant mean curvature foliations may offer a physical basis for privileged time structures in quantum gravity.

ABSTRACT

The traditional metaphysical debate between static and dynamic views in the philosophy of time is examined in light of considerations concerning the nature of time in physical theory. Adapting the formalism of Rovelli (1995, 2004), I set out a precise framework in which to characterise the formal structure of time that we find in physical theory. This framework is used to provide a new perspective on the relationship between the metaphysics of time and the special theory of relativity by emphasising the dual representations of time that we find in special relativity. I extend this analysis to the general theory of relativity with a view to prescribing the constraints that must be heeded for a metaphysical theory of time to remain within the bounds of a naturalistic metaphysics.

Motivation & Objective

• To resolve the tension between metaphysical views of time (dynamic vs. static) and the formal structure of relativistic physics.
• To clarify why Minkowski spacetime does not definitively rule out objective temporal passage, despite its block-universe appearance.
• To extend the analysis from special to general relativity, identifying physical constraints that must be satisfied for a naturalistic metaphysics of time.
• To assess whether canonical gravity and constant mean curvature (CMC) foliations can provide a physical basis for privileging a global time structure.
• To evaluate whether the Hamiltonian formulation of general relativity supports a dynamic view of time within a naturalistic metaphysical framework.

Proposed method

• Adopts Rovelli's relational formalism to characterize time in physical theories, emphasizing dual representations of time in relativity.
• Uses Minkowski spacetime as a foundation to show that the formal structure does not preclude dynamic time, due to the existence of multiple time-like structures.
• Applies the concept of global hyperbolicity to general relativity, requiring spacetime to be foliable into spacelike Cauchy surfaces for a consistent temporal order.
• Analyzes the Hamiltonian formulation of GR, focusing on the role of the 3-metric and extrinsic curvature in defining spacetime evolution.
• Evaluates the constant mean curvature (CMC) foliation method as a candidate for selecting a unique time slicing in spacetime.
• Assesses whether CMC foliations can provide a physical basis for privileging a preferred time direction, relevant to A-theory metaphysics.

Experimental results

Research questions

• RQ1 Can a dynamic theory of time be consistent with the formal structure of special relativity, given Minkowski spacetime’s block-universe character?
• RQ2 What constraints does general relativity impose on metaphysical theories of time to remain within the bounds of naturalistic metaphysics?
• RQ3 Is global hyperbolicity a necessary condition for the possibility of objective temporal passage in spacetime?
• RQ4 Can the constant mean curvature (CMC) foliation provide a physically grounded basis for a privileged time structure in general relativity?
• RQ5 Does the Hamiltonian formulation of GR support the viability of dynamic time, particularly in the context of quantum gravity?

Key findings

• The formal structure of Minkowski spacetime does not rule out dynamic time due to the dual representation of time, which allows for multiple time-like structures.
• Global hyperbolicity is a necessary condition for a metaphysical theory of dynamic time, as it ensures the existence of Cauchy surfaces and a total temporal order.
• Solutions to Einstein’s field equations that are not globally hyperbolic—such as Gödel’s spacetime—do not necessarily falsify dynamic time, but are physically suspect.
• The Hamiltonian formulation of general relativity requires global hyperbolicity, suggesting that a successful quantum gravity theory may further support a dynamic view.
• The constant mean curvature (CMC) foliation method provides a unique time slicing for a large class of spacetimes, offering a potential physical basis for privileging a global time structure.
• Despite this, Wüthrich (2010) argues that CMC foliations violate the principle of general covariance, undermining their viability for supporting presentism or other A-theoretic views.