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Newtonian Time from Timeless Dynamics

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Abstract

The role of a time parameter is vital to physics, yet time is often taken for granted. Newtonian physics assumes an idealized time variable that is unmeasurable. In this paper, we answer calls for clearer accounts of the emergence of Newtonian time from timeless mechanical systems (Smolin 2013). We consider a three-particle system in the timeless formalism of Jacobi. Time is abstracted from paths in configuration space in accord with Mach’s principle of universal inertial reference frames. This paper makes three contributions to the conversation around time. First, our physical demonstration of how Newtonian time is constructed completes Mach’s arguments about clocks and time in classical, non-relativistic systems. Second, this paper discusses the integration of these ideas regarding time into quantum mechanics. Finally, we answer the question, ‘what do clocks measure?’ using these simple physical systems and use that answer as a foundation for engagement with current metaphysical debates about time.

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Notes

  1. Many thanks to David Albert, Michael Dickson and our journal reviewers for their helpful comments.

  2. Our daily experiences of the apparent passing of time is sometimes associated with absolute, or Newtonian, time, called ‘manifest time’ (see Sellars 2017 and Callender 2017, among others).

  3. Newton 1995The Principia—all quotes of Newton in this section are from The Principia in the Scholium to the Definitions; all the translations are taken from Mach’s The Science of Mechanics, 222–223.

  4. Mach (1919, 24). First, Mach is making a point related to Aristotle’s claim that time is inseparable from change and that time is unknowable without change (Physics III). Second, Mach proceeds to make a similar argument about space (Mach, 1919, 226-238). It would be helpful to develop a parallel case study to this paper where the focus is on demonstrating the development of abstract space as a metric for articulating relations and features of a system. Third, in this paper we have taken the concept of time to be less fundamental than space and have offered the detailed example of how it is developed in a physical system, but our starting assumption should not be taken as a denial of Mach’s further and similar point about space.

  5. There remain some metaphysical questions about the conceptual need for time (for more on the metaphysical debates see Crisp 2003; Deasy 2015; Maudlin 2007; Prior 2008; Tallant 2010 and Tallant 2019, among others). Whether metaphysicians choose to dither and insist on the logical need for real time, be that in the form of numerical and physical sequence or some more robust notion of time, abstract time is beyond the requirements of a physical system and therefore beyond the scope of this paper as well. The need to sequence events and physical configurations is one motivation for positing an abstract time. However, it is worth noticing that sequencing of events is not necessarily a need in physical spaces so much as in the perceptual and psychological spaces.

  6. Some philosophers of time already press for this sort of understanding of time (see Barbour 2001; Corish 2009; Gentry 2021; Rovelli 2018, among others).

  7. Generalizations such as adding dimensions to the configuration space, using a non-Euclidean configuration space, or allowing differing masses for the particles are straightforward and do not bring any conceptually new features into play.

  8. While we have done away with time, we certainly have not done away with space. The configuration space is simply assumed to be Euclidean, although any configuration space geometry could be used. This assumption of a fixed spatial geometry is not essential, but allows us to focus on issues associated with time alone. A class of models which generalize the Jacobi-type models and, in particular, have no ‘absolute space’, have been provided by Barbour and Bertotti (1982).

  9. The parametrization allows us to discuss ‘motion’ using differential equations.

  10. Here a dot denotes a derivative with respect to the arbitrary time .

  11. This is a well-known feature of timeless dynamical systems (Sundermeyer 1982).

  12. We emphasize that the unparametrized paths are unambiguously determined; a parametrization of the paths is completely arbitrary.

  13. The motion of particle 3 provides a clock only in regions of phase space where . For motion which includes , one should select some other dynamical variable to serve as a clock.

  14. A similar quantization strategy has been proposed by Rovelli in the context of the Barbour-Bertotti model (Rovelli 1991b).

  15. The precise definition and algebraic properties of these operators are somewhat figurative pending a precise delineation of the operator domains.

  16. The conditional probabilities must be integrated over c; see Gambini et al. (2009) for details.

  17. Here as in discussions about other sorts of measurement, we might have questions about circularity where the thing measuring is also being measured, although such worries have epistemic responses. Hasok Chang (1995; 2004) deals extensively with the problem of circularity in measurement with respect to temperature and thermometer development.

  18. Philosophers like Tim Maudlin argue that the physical and mathematical descriptions in theories like General Relativity demonstrate the reality of time and put relationalists on their heals (Maudlin 2012). However, at least in some cases, Maudlin seems to treat clocks as if they were outside of the physical system they measure (Maudlin 2012) and mathematics itself as having temporal properties (Maudlin 1993 and 2002; Maudlin et al. 2020). At the very least, future work would want to reconsider the logical and metaphysical claims that have been built off of some of these assumptions.

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Authors and Affiliations

  1. Utah State University, Logan, USA

    Brittany Gentry, Charles Torre & Zach Zito

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Correspondence to Brittany Gentry.

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Gentry, B., Torre, C. & Zito, Z. Newtonian Time from Timeless Dynamics. J Gen Philos Sci (2025). https://doi.org/10.1007/s10838-024-09695-4

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  • DOI https://doi.org/10.1007/s10838-024-09695-4

Keywords

  • Philosophy of Physics
  • Time
  • Epistemology
  • Metaphysics

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