Lorentz Factor

Every result in special relativity funnels through this one number. Moving clocks run slow by $\gamma$; moving objects contract along their motion by $\gamma$; momentum is $\gamma m v$ rather than $mv$; total energy is $\gamma m c^2$. It is written in terms of $\beta = v/c$, the speed as a fraction of light speed, which keeps it dimensionless and makes its behaviour obvious. And it is not a correction factor pasted onto Newtonian mechanics — it is geometry. In ordinary space the quantity that all observers agree on is $\sqrt{x^2 + y^2}$. In spacetime the agreed quantity carries a minus sign, $\sqrt{(ct)^2 - x^2}$, and $\gamma$ is simply what that minus sign does to the arithmetic when you change from one observer's coordinates to another's.

The interesting thing about $\gamma$ is its shape. It clings to 1 with great determination and then leaves abruptly. At $\beta = 0.1$ it is 1.005. At half the speed of light it is only 1.155. It reaches 2 at $\beta = 0.866$, 7.09 at 99%, and 22.4 at 99.9%. Protons in the LHC at 6.8 TeV have $\gamma \approx 7250$, meaning their onboard clocks run more than seven thousand times slow relative to the tunnel. The clean classroom case is $\beta = 0.6$, which gives $\gamma = 1/\sqrt{1 - 0.36} = 1/0.8 = 1.25$ exactly — worth remembering, because it makes a whole family of textbook problems come out in round numbers.

The name is Hendrik Lorentz's and the interpretation is not. Between 1892 and 1904 Lorentz assembled these transformations to explain why Michelson and Morley, working in Cleveland in 1887 with an interferometer sensitive enough to detect a fraction of the expected effect, had found no sign of the Earth's motion through the luminiferous aether. George FitzGerald had already suggested in 1889 that moving objects physically shrink; Lorentz developed the mathematics on the same assumption, that this was a dynamical effect of the aether on matter. Einstein in 1905 derived exactly the same factor from two postulates — the laws of physics are the same in all inertial frames, and light has one speed for everyone — with no aether at all and no contracting mechanism required. The equations kept Lorentz's name and lost his explanation. Einstein's own account was that the Michelson–Morley result played little part in his thinking; what bothered him was the asymmetry in how Maxwell's theory described a magnet moving past a conductor versus a conductor moving past a magnet, when only the relative motion can matter.

Four things to watch. $\beta$ is a fraction, not a speed: enter 0.6, not $1.8 \times 10^{8}$ m/s, and divide by $c$ yourself if that is what you have. $\gamma$ is never less than 1 and equals 1 exactly at rest, so a value below 1 means something has been inverted. At ordinary speeds the useful approximation is $\gamma - 1 \approx \beta^2/2$, because a calculator will hand back a flat 1.000000 for anything terrestrial — an airliner at 250 m/s has $\gamma - 1 \approx 3.5 \times 10^{-13}$, which is real but below the display. And the conceptual trap: asking whose clock is really running slow. The relation is symmetric, each observer measures the other's clock as the slow one, and both are correct. There is no fact of the matter until the two are brought together again, and bringing them together requires that one of them accelerate.