Lorentz Transformations and the Gamma Factor

To go from the spacetime coordinates (x, ct) of one inertial frame to those (x', ct') of another frame moving at velocity v along the x axis, classical mechanics used the Galilean transformations, simple but incompatible with the constancy of c. Einstein replaced them with the Lorentz transformations, which intimately mix space and time.

These transformations are written: x' = gamma × (x − v t) and t' = gamma × (t − v x / c²), where gamma is the Lorentz factor, defined by gamma = 1 / sqrt(1 − v²/c²). This factor measures the strength of relativistic effects: as long as v remains small compared with c, gamma is almost 1 and we recover Galileo's formulas; but as v approaches c, gamma grows without bound.

Let us check a numerical case: for v = 0.8 c, we compute v²/c² = 0.64, so 1 − v²/c² = 0.36, whose square root is 0.6. The gamma factor is then 1 / 0.6, or about 1.67. It is precisely this factor that multiplies the proper time to give the duration measured from another frame, and that divides the proper length to give the contracted length.

Geometrically, on a Minkowski diagram, switching to the moving frame does not turn the x' and ct' axes by the same angle as an ordinary rotation would: the two axes tilt toward each other, symmetrically with respect to the lines of the light cone, which themselves remain unchanged, in accordance with the second postulate.

        ct   ct'
         ^   /
         |  /
         | /       x'
         |/    _.-'
         X_.-'
        /|
       / |
      /  |
     /   |------------> x
    /    |

(the x' axis is tilted by an angle whose slope is v/c, written beta)
(the ct' axis is tilted symmetrically, with respect to the 45-degree lines)
(the lines of the light cone, however, remain unchanged in both frames)
(gamma = 1 / sqrt(1 - v^2/c^2) gives the scale factor between the two systems)

The diagram above superimposes the two sets of axes. It shows how the same event gets different coordinates depending on the chosen frame, while keeping one essential quantity invariant: the spacetime interval s² = (ct)² − x², which has the same numerical value in all inertial frames. This invariance is the true mathematical heart of the theory.