Study of masses interacting via gravity challenges the idea that special initial conditions are needed to give time a direction (PDF)

The fundamental laws of physics, we believe, do not depend on the direction of time. Why, then, is the future so different from the past? The origin of this “arrow of time” has puzzled physicists and philosophers for more than a century, and it remains one of the fundamental conceptual problems of modern physics [1]. Although a preferred direction of time can occur in models of physical systems, this typically happens only if one inserts very special initial conditions. Julian Barbour at the University of Oxford and his colleagues [2] have now shown this tinkering isn’t necessary to produce an arrow of time in a system of masses interacting via Newtonian gravity. They demonstrate that the evolution of this surprisingly simple system almost always contains a unique moment of lowest “complexity,” a point they identify as a “past” from which two distinct (and more complex) “futures” emerge.

In their gravitational model, Barbour and his colleagues find a state of “low complexity” that is analogous to Boltzmann’s low-entropy fluctuation. But in their case, no rare statistical fluctuation is necessary to explain this state; instead, it arises naturally out of simple physical laws that have no explicit dependence on the direction of time. The authors study one of the simplest possible systems: a collection of N point particles interacting through Newtonian gravity. Their only assumptions are that the total energy (potential plus kinetic) and the total angular momentum of the system are zero. From earlier numerical simulations and analytic analysis, it is known that in the distant future, such a system tends to break up into weakly interacting subsystems—typically, pairs of masses in Keplerian orbits [8]. Starting with such a dispersed system and running time backwards, one might expect that it would coalesce in the past into a state of high density. Barbour and his coauthors show analytically that this expectation is right: for almost every initial configuration of masses, there is a unique moment of minimum size and maximum uniformity. From this point, the system expands outward, approximately symmetrically in both directions of time (Fig. 1). The system is therefore globally symmetric in time, as the equations dictate, and yet has a local arrow of time.

As a key step in their argument, the authors analyze the evolution of the masses in “shape space,” a space of observables that describe the shape of the system but are independent of its size and orientation. Three bodies, for instance, form a triangle, and their shape space is the space of similar triangles. Shape space contains a natural dimensionless measure of complexity, denoted 𝒞S, which is determined by the moment of inertia and the total Newtonian gravitational potential. 𝒞S describes the degree of nonuniformity and clustering; it has a minimum at the moment of minimum size and grows approximately monotonically from that minimum in both directions of time. Barbour and his colleagues provide a fairly simple and intuitive explanation for this behavior by showing that the dynamics of the N-body system in shape space has an effective friction term, which provides a sort of dissipation even though the underlying equations of motion are symmetric in time. (Credit: Steven Carlip)