Arrow(s) of Time without a Past Hypothesis (1809.04646v3)

Abstract: The paper discusses recent proposals by Carroll and Chen, as well as Barbour, Koslowski, and Mercati to explain the (thermodynamic) arrow of time without a Past Hypothesis, i.e., the assumption of a special (low-entropy) initial state of the universe. After discussing the role of the Past Hypothesis and the controversy about its status, we explain why Carroll's model - which establishes an arrow of time as typical - can ground sensible predictions and retrodictions without assuming something akin to a Past Hypothesis. We then propose a definition of a Boltzmann entropy for a classical $N$-particle system with gravity, suggesting that a Newtonian gravitating universe might provide a relevant example of Carroll's entropy model. This invites comparison with the work of Barbour, Koslowski, and Mercati that identifies typical arrows of time in a relational formulation of classical gravity on shape space. We clarify the difference between this gravitational arrow in terms of shape complexity and the entropic arrow in absolute spacetime and work out the key advantages of the relationalist theory. We end by pointing out why the entropy concept relies on absolute scales and is thus not relational.

• The paper explores non-standard approaches, like the Carroll-Chen and Barbour relational models, that aim to derive the thermodynamic arrow of time from physical laws without assuming a low-entropy Past Hypothesis.
• The Carroll-Chen model proposes a universe where entropy increases infinitely in both time directions, yielding a natural arrow from a minimal entropy point.
• The relationalist approach by Barbour et al. derives a gravitational arrow of time from the growth of "shape complexity" on a compact shape space, avoiding issues present in other models and enabling statistical analysis.

Arrow(s) of Time Without a Past Hypothesis

The paper by Dustin Lazarovici and Paula Reichert tackles the complex issue of time's directionality in thermodynamic systems without invoking a Past Hypothesis, traditionally assumed to explain the low-entropy start of the universe. The authors analyze recent theoretical models to explore whether the arrow of time can be accounted for as a typical feature of physics, rather than relying on the premise of a special initial condition.

Summary of Key Arguments and Models

1. Critique of the Past Hypothesis: The paper starts by examining the conventional approach to the arrow of time, which often relies on the assumption of a low-entropy initial state, encapsulated in the Past Hypothesis. This assumption is debated regarding its necessity and the demand for further explanation of why the universe began in such a special state.
2. Carroll and Chen's Model: A notable portion of the paper is dedicated to evaluating Sean Carroll and Jennifer Chen's proposition of a universe where entropy can increase infinitely in both temporal directions. Their model discards the idea of a finite equilibrium, suggesting instead a "U-shaped" entropy curve that naturally provides time's arrow by allowing systems to typically evolve away from a point of minimal entropy towards higher entropy states over time.
3. A Gravitating Universe as a Test Case: Newtonian gravity is proposed as a viable realization of Carroll's entropy model. The authors argue that a classical gravitating system could serve as a simple yet physically relevant example of a system that fits this framework, where the expansion of such a system leads naturally to typical entropy growth without needing artificially imposed boundary conditions.
4. Relationalist Approach by Barbour, Koslowski, and Mercati: Parallel to the Carroll-Chen model, Julian Barbour and collaborators propose a relational framework in which complex systems evolve on "shape space," neutralizing Newtonian absolute space notions. They show that the growth of what they term "shape complexity" can align with entropy’s increase, providing a natural gravitational arrow of time within Newtonian gravitating universes.
5. Statistical Mechanics on Shape Space: By eliminating redundant degrees of freedom, the relational approach solves the issue of non-normalizable phase space measures inherent in the Carroll model. With a compact, finite shape space, Barbour and his collaborators can conduct a statistical analysis typically impossible in standard frameworks, showing the potential for a uniform arrow of time derived from relational mechanics.

Implications and Future Directions

This paper provides substantial insights into non-standard approaches to explaining the thermodynamic arrow of time. It opens speculative but mathematically grounded avenues for interpreting the universe's initial conditions not as an extraordinary boundary that requires external imposition but as emergent properties of universal laws.

In a broad sense, the analysis contributes to ongoing discussions in statistical mechanics and cosmology regarding initial conditions' roles versus dynamic laws' implications. Additionally, the relationalist reformation of classical mechanics might appeal to theoretical physicists focused on foundational aspects of time, space, and gravitation, suggesting further interdisciplinary connections.

Future research could extend these ideas to other forces and scales beyond Newtonian mechanics, exploring quantum or relativistic regimes. Moreover, identifying concrete empirical consequences of these theoretical insights could guide observational cosmology in testing the validity of these innovative perspectives on temporal asymmetry and entropy.