Fast, simple and ingenious

Correct use of Einstein's special theory of relativity can simplify computer simulations of relativistic systems

Michal Saaf, Galileo, Issue 105, May 2007

Relativistic systems, in which objects move at speeds close to the speed of light, exist in many different areas of physics, and especially in particle physics. In such systems it is necessary to refer to the special theory of relativity, which describes how factors such as time, space, mass and more will change from the point of view of the moving body. When performing calculations dealing with relative systems, the calculations can be simplified by switching to a more convenient reference system; For example, a collision of two particles may become simpler for human perception (and mathematical calculations) if it is described from the point of view of one of the colliding particles, rather than from the point of view of a stationary observer observing the collision from the side.

Such transitions between reference systems are acceptable and legitimate according to the theory of special relativity, and are carried out by a simple mathematical transformation known as the Lorentz transformation. Using the Lorentz transformation, the process can be described from any reference system moving at a constant speed, and the results can be translated to any other reference system. No information concerning the process is lost when moving from one frame of reference to another, and thus the process can be described from the point of view of one of the colliding particles, then "translate" the information using the Lorentz transformation and know how the process will also look to the eyes of the stationary observer.

When the analysis of relational processes is carried out by a computer, there is a special problem. When a computer analyzes a process numerically, it must divide the longest relevant length (e.g., the length of the track in the particle accelerator) into the shortest relevant segments (e.g., the length of the particle pulse fired in the accelerator), and perform the calculation as it progresses through each move in one segment. However, at relative speeds a difficulty arises due to a phenomenon known as "length contraction": observers moving at different speeds see the same segment as having a different length. A meter rod, as it appears in the laboratory, will appear shorter to an observer passing by at a relative speed, and as the speed increases, so will the difference in the perception of length. Because of the relative changes in the perception of length, different particles in the system may perceive the distances in the system as very different. Therefore, the ratio between the largest length relevant for the purpose of the calculation and the smallest length may be enormous, and running computer simulations of relativistic processes requires a lot of time.

Physicist Jean-Luc Vay (Vay), a researcher at the Accelerator Laboratory in Berkeley, found a simple but brilliant way to reduce the time needed for simulations of relativistic systems: one must calculate and find the reference system from which the ratio between the largest and the smallest length will be reduced. This way, the computer will have to work for a shorter time, since it analyzes a smaller overall length, and divides it into longer segments. Moreover, such a transition sometimes makes it possible to use mathematical approximations, which assume that the system is simpler than it is without detracting significantly from the accuracy of the physical description. Wei used this method for his work, and assumed that scientists all over the world were doing the same, but he was surprised to find that even though the theory of special relativity and the Lorentz transformation have been known to science for over a hundred years, no one had yet applied them in this way. Wei published several examples of calculations summarized in this way; In one of the examples, the calculation time was a thousand times shorter when a more suitable reference system was chosen for it. Scientists were surprised to hear about the simple and effective method, since the calculation time is considered an indication of the complexity of the system, and it is clear that the complexity of the system does not change when moving between reference systems.