the Lorentz factor
/ LOR-ents /
The Lorentz factor is the single number that measures 'how relativistic' a situation is — how strongly time dilation, length contraction, and the other effects kick in at a given speed. Written with the Greek letter gamma, it is the dial that turns from 'ordinary Newtonian physics' up to 'full-blown relativity' as an object's speed climbs toward that of light. At a walking pace it is essentially 1 (no effect); as speed approaches c it shoots off toward infinity.
Precisely: gamma = 1 / sqrt(1 - v^2/c^2), where v is the object's speed and c is the speed of light. When v is small compared with c, the fraction v^2/c^2 is tiny, so gamma is almost exactly 1 and relativity is invisible. As v grows, v^2/c^2 approaches 1, the square root approaches 0, and gamma balloons: gamma = 1.005 at 10% of c, 1.15 at half c, 2 at about 87% of c, 7 at 99% of c. Because you cannot take the square root of a negative number, v can never reach or exceed c — that ceiling is built right into the formula.
Gamma is the workhorse symbol of special relativity. A moving clock's ticks are stretched by gamma (delta-t = gamma times tau); a moving object's length is squeezed by dividing by gamma (L = L_0 / gamma); relativistic momentum is p = gamma m v and total energy is E = gamma m c^2. Learn to read gamma and you can read the whole theory: it tells you, at a glance, how far from everyday physics you have travelled.
At v = 0.6c: v^2/c^2 = 0.36, so 1 - 0.36 = 0.64, sqrt(0.64) = 0.8, and gamma = 1 / 0.8 = 1.25. Clocks run 1.25 times slow and lengths shrink to 1/1.25 = 0.8 of their proper value. At everyday 100 m/s, gamma differs from 1 by about five parts in 10^14 — utterly undetectable.
gamma climbs slowly at first, then explodes toward infinity as v nears c — and never lets v reach c.
The Lorentz factor is named after Hendrik Lorentz, who wrote the transformation equations before Einstein explained what they meant. Gamma is always greater than or equal to 1, never less — so relativistic effects only ever stretch time and shrink length, never the reverse.