In Newton's world, time is a simple concept—a universal time exists by which all events can be measured. However, special and general relativity state that the time measurements becomes dependent on how you relate to the process you wish to time. The classical example is that of someone on a rapidly moving train would measure time at a different rate than a person standing still on the station platform. The equations that describe such phenomena are well understood as they apply to deterministic (non-random) systems, but it is not known how to describe stochastic (random) processes or motion in a relativistic sense.

One of the most well known examples of a stochastic process in physics is Brownian motion. This was originally used to describe the random, jiggling motion of particles suspending in a fluid medium. While relativistic issues with colloids in water are pretty much nonexistent, the same mechanics describes thermalization in quark-gluon plasmas and other complex, high-energy processes that take place in our Universe. One problem that arises is how to properly measure time in these complex systems.

In relativity, there are two definitions of time: the coordinate time and the proper time. Coordinate time can be described as what a stationary (inertial) clock, infinitely removed from an event, would measure. Proper time, on the other hand, is the time measured by a clock that exists at the same point as the event taking place—this means that it depends not only on the nature of the event, but on the path taken through space by the clock as the event occurs. When one has a process that involves random motion over time, which time should be used?

The time evolution of a stochastic process is defined by a Langevin equation, which is a differential equation similar to those that describe the time evolution of motion in classical mechanics, but with an extra term that captures the random nature of the process at any given time. In a paper set to be published in an upcoming issue of Physical Review Letters, a trio of theoretical physicists derive the mathematical relation between the proper time of a stochastically moving particle and the coordinate time of the same particle. In doing so, the trio derive a relationship between the two types of time for a relativistic stochastic process.

Through computer simulations, they demonstrated the difference between the two measurement types. They plotted the 'stationary' probability density function, or PDF—a measurement of the probability of finding a particle at a particular time, position, and momentum—and compared it to the absolute momentum of a particle undergoing a type of stochastic process. The PDF normally takes a shape similar to a normal bell curve distribution. When the proper time is used, it has a lower peak at zero momentum and is "smeared out," having a larger width at half the maximum height than it does when the coordinate times are used.

With an understanding of the relationship between proper and coordinate time in this class of stochastic processes, the researchers look at the problem of how to describe a stochastic process that is moving relative to an observer. Working through the mathematics, they derive how a stochastic process will appear when it undergoes a Lorentz transformation—that is, what it will look like when it moves by a stationary observer at a high rate of speed.

They conclude the paper by pointing out that the current work renders "the special relativistic Langevin theory [...] as complete as the classical theories of nonrelativistic Brownian motions and deterministic relativistic motions, respectively, both of which are included as special limit cases." They point out one critical underlying issue: one must be careful about describing the "stationarity" of stochastic processes in special relativity. The coordinate parameterization of time is better suited when describing diffusion processes, but the proper time parameterization is better for events where particle creation/annihilation is occurring, since a particle's lifetime is described by its proper time. They finish by pointing out that the paper lays the groundwork for describing stochastic processes in general relativity.

Physical Review Letters, 2009. DOI: upcoming (Pre-print available though arXiv)