Identification of a gravitational arrow of time (1409.0917v1)

Abstract: It is widely believed that special initial conditions must be imposed on any time-symmetric law if its solutions are to exhibit behavior of any kind that defines an arrow of time'. We show that this is not so. The simplest non-trivial time-symmetric law that can be used to model a dynamically closed universe is the Newtonian $N$-body problem with vanishing total energy and angular momentum. Because of special properties of this system (likely to be shared by any law of the Universe), its typical solutions all divide at a uniquely defined point into two halves. In each a well-defined measure of shape complexity fluctuates but grows irreversibly between rising bounds from that point. Structures that store dynamical information are created as the complexity grows and act asrecords'. Each solution can be viewed as having a single past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and will only be aware of the records of one branch and deduce a unique past and future direction from inspection of the available records.

• The paper shows a gravitational model produces an arrow of time naturally from time-symmetric laws, challenging the need for special initial conditions.
• The paper uses shape space analysis and introduces a complexity measure that increases irreversibly, acting as a proxy for temporal direction in these systems.
• This work suggests that time's arrow can be an emergent property of gravity, offering new perspectives for understanding spacetime, cosmology, and quantum gravity.

Overview of "Identification of a Gravitational Arrow of Time"

The paper "Identification of a Gravitational Arrow of Time" by Julian Barbour, Tim Koslowski, and Flavio Mercati addresses a fundamental question in theoretical physics: the emergence of the arrow of time from time-symmetric physical laws. The authors challenge the widely held belief that special initial conditions are necessary for an arrow of time to manifest in the solutions to time-symmetric laws of physics. Instead, they demonstrate that a gravitational model based on the Newtonian $N$-body problem, with vanishing total energy and angular momentum, inherently produces time-asymmetric behavior in the absence of special initial conditions.

Key Contributions

1. Reformulation of the Newtonian $N$-Body Problem: The paper posits that the Newtonian $N$-body problem with zero total energy and angular momentum, when interpreted on a relational basis, naturally divides solutions into two distinct halves. Each half sees complexity increasing irreversibly from a common point, which acts as a turning point.
2. Complexity as a Measure of Temporal Direction: A novel contribution of the paper is the introduction of a complexity measure, defined as a dimensionless ratio of two characteristic lengths of the system—root-mean-square separation and mean harmonic separation. This complexity acts as a proxy for temporal directionality, growing irreversibly in systems with these properties.
3. Shape Space and Dynamical Similarity: By eliminating the roles of absolute scale and orientation, the authors utilize a shape space description to analyze dynamics. They reinterpret traditional Newtonian dynamics in terms of shape degrees of freedom, reducing extraneous variables and focusing on an intrinsic description of evolution.
4. Friction-Like Effects in Shape Dynamics: The paper introduces a friction-like effect in the Hamiltonian dynamics of shape variables, which leads to the spontaneous growth of complexity over time. This mechanism captures how systems evolve towards greater structure and order without needing predefined initial conditions.
5. Implications for Gravity and Geometrogenesis: The work suggests that gravitationally bound systems naturally exhibit an arrow of time as they move towards more complex configurations. The mechanism can predict the emergence of Newtonian physics and spacetime concepts from relational variables late in the universe's history.

Implications and Future Directions

The results of this paper have significant theoretical implications. They propose that rather than relying on a "past hypothesis," time's arrow could be an emergent property of the universe's fundamental laws, influenced by gravity's tendency to generate anisotropy and complexity. While demonstrated within a Newtonian framework, the authors argue that similar properties would extend to general relativity, potentially leading to a relational foundation for cosmology under shape dynamics.

Practically, understanding time's arrow in terms of shape dynamics could influence approaches to quantum gravity and the understanding of spacetime's nature. Further research could extend these ideas to full general relativity models or other scale-invariant theories, exploring whether such mechanisms could unify various arrows of time observed in different physical domains, such as thermodynamics or quantum mechanics.

In summary, this paper provides a novel perspective on how the arrow of time could arise naturally from gravitational laws without relying on fixed initial conditions, offering a pathway towards reconciling the observed temporal asymmetry with fundamental time-symmetric physics.