Lorentz factor
/ LOR-ents factor; the symbol gamma is said GAM-uh /
Special relativity says that for something moving very fast, time runs slow, lengths shrink, and energy balloons — but by how much? The Lorentz factor is the single number that answers this. Usually written with the Greek letter gamma, it is a stretch-and-shrink multiplier that depends only on how fast you are going compared with light. At a crawl gamma is essentially 1, meaning nothing relativistic happens. As your speed climbs toward the speed of light, gamma climbs toward infinity.
Concretely, gamma equals 1 divided by the square root of (1 minus v-squared over c-squared), where v is the speed and c is the speed of light. Plug in numbers and a pattern appears. At half the speed of light gamma is about 1.15 — a 15 percent effect. At 90 percent of c it is about 2.3. At 99 percent, about 7. At the LHC's 99.9999991 percent, gamma is about 7,500. The factor stays close to 1 for a long time, then shoots up violently near the very end, which is exactly why relativistic effects seem to switch on suddenly.
Gamma is the workhorse of relativistic bookkeeping. A particle's total energy is gamma times its rest energy, and its momentum is gamma times its rest mass times its velocity, so once you know gamma you know nearly everything. It also tells you how much a fast particle's internal clock is slowed: a particle with gamma of 20 ages 20 times slower than one at rest, which is the difference between a short-lived particle decaying inside the beam pipe and living long enough to be tracked across a whole detector.
gamma = 1 / sqrt(1 - v^2/c^2). At v = 0.99c, gamma is about 7; at v = 0.999c, about 22; at v = 0.9999c, about 71.
Adding more 9s to the speed multiplies gamma fast — the factor diverges as v approaches c.
Gamma is always 1 or larger, never less; if a calculation gives gamma below 1 you have made an arithmetic slip, since that would require a speed greater than light.