The Best Lorentz Frame for Calculations

Noninvariance of Space and Time Scale Ranges under a Lorentz Transformation and the Implications for the Study of Relativistic Interactions

J.L. Vay

PRL, 130405 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e130405

I don't think this was a particularly well-written article, but the ideas are quite interesting. The basic premise is that you can exploit time dilation and length contraction to find a frame that makes calculations simple.

In most experiments and numerical simulations, there is a hierarchy of length and time scales. For instance, in a carbon nanotube, the nanotube radius is probably the smallest important length scale; the largest might be the length of the nanotube, which can be thousands or millions of tube radii. If I had to carry out simulations that described all length scales, I would have a lot of grid points to worry about.

Vay says, "Well, Jesse, if you boosted to a frame that moves quickly enough, you could end up with a nanotube whose length is equal to or SMALLER than its radius. Einstein tells us that an experiment in this moving frame is just as good as one in the nanotube frame. Why not make it easy on yourself?"

Maybe it wouldn't help me out that much, but for simulations of relativistic beams of electrons, Vay shows that calculation times can be reduced by a factor of a thousand or more.

The first section of the paper is devoted to an example that shows the opposite: that the ratio of the longest to the shortest relevant length (and time) scales can be made extremely large depending on the Lorentz frame you choose for the calculation. Reading the paper a second time, I realized that the point was to demonstrate separation of scales, but since it contradicts the claim of the abstract, I was really confused the first time through.

The three physical examples in the second half of the paper clearly demonstrate the utility of choosing the right Lorentz frame.

To show that this approach is practical, Vay performed the same calculation of the passage of a relativistic beam of protons in a cylinder colliding with an electron gas. In the lab frame, the experiment spans a few kilometers, and the pipe radius is just a centimeter, so the length scales span 5 orders of magnitude.

The lab frame calculation took a week of supercomputer time. The boosted frame calculation took half an hour on the same computer. That's an amazing improvement, but I don't really know when this method will work. I wish Vay had devoted more time to explaining what types of calculations can be improved.