Relativity Could Lead to Lightning Fast Number Crunching

April 5, 2007

Jean Luc Vay, a physicist with Lawrence Berkeley National Laboratory's Accelerator and Fusion Research Division has found a way to use the Lorentz transformation — the set of equations that forms the mathematical basis for Albert Einstein's theory of special relativity — to reduce by many orders of magnitude the number of computations needed to solve highly complex mathematical problems, especially those in physics.

The equations in the Lorentz transformation, named for the Dutch physicist Hendrik Lorentz, describe the relationship between the space and time coordinates of two systems moving at a constant speed relative to one another. Often these equations are calculated from an "at rest" frame of reference; that is, the coordinates are described from the view of a stationary observer in the laboratory.

For modeling interactions between particles or between particles and radiation, for instance, calculating the interactions from a stationary reference frame can require hours or even months on powerful supercomputers. And that's often by approximating the fundamental equations.

The interaction of relativistic objects, it turns out — counterintuitively — need not be viewed from a frame of reference at rest in the laboratory. Instead, it can be viewed from any one of an infinite number of moving frames of reference.

According to Vay, within this infinite number of frames of reference there will be one that minimizes the range of space and time scales on which the objects interact. Finding this optimal reference frame, Vay says, reduces the number of calculations required to model the interactions.

The key to Vay's success is that, contrary to conventional thinking, which holds that under a Lorentz transformation the "complexity" of a system is invariant, when a system is observed from a moving frame of reference, it actually becomes — from a certain point of view — less coomplex. For very complicated reasons, this reduced complexity makes it possible to reduce the number of computations required to describe such systems.

This conclusion, despite being a direct consequence of special relativity, is not widely known, even by many physicists, according to Vay. As a result, says Vay, "We should be able to tackle mathematical problems that have been unmanageable, and we should also be able to put much more detail into problems for which computer resources have been barely adequate."

Vay's Lorentz transformation discovery is reported in the March 30 issue of Physical Review Letters, in his paper, "Nonvariance of Space - and Time - Scale Ranges Under Lorentz Transformation and the Implications for the Study of Relativistic Interactions."

Sources: Physical Review Focus (http://focus.aps.org/story/v19/st10); Berkeley Lab View, March 16, 2006 (http://www.lbl.gov/Publications/Currents/archive/#9