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Lorentz transformations

The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities.

Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity.

The Lorentz factor appears in the Lorentz transformations:

The inverse Lorentz transformations are:

When v ≪ c and x is small enough, the v / c and vx / c terms approach zero, and the Lorentz transformations approximate to the Galilean transformations.

t ′ = γ (t − v x / c 2), {\displaystyle t'=\gamma (t-vx/c^{2}),} x ′ = γ (x − v t) {\displaystyle x'=\gamma (x-vt)} etc., most often really mean Δ t ′ = γ (Δ t − v Δ x / c 2), {\displaystyle \Delta t'=\gamma (\Delta t-v\Delta x/c^{2}),} Δ x ′ = γ (Δ x − v Δ t) {\displaystyle \Delta x'=\gamma (\Delta x-v\Delta t)} etc. Although for brevity the Lorentz transformation equations are written without deltas, x means Δ x, etc. We are, in general, always concerned with the space and time differences between events.

Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the S frame can only be moving forwards or reverse with respect to S ′. So inverting the equations simply entails switching the primed and unprimed variables and replacing v with − v.

Example: Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. At time t = t ′ = 0, Stella's spaceship accelerates instantaneously to a speed of 0.5 c. The distance from Earth to Mars is 300 light-seconds (about 90.0 × 10 km). Terence observes Stella crossing the finish-line clock at t = 600.00 s. But Stella observes the time on her ship chronometer to be t ′ = γ (t − v x / c 2) = 519.62 s {\displaystyle t^{\prime }=\gamma \left(t-vx/c^{2}\right)=519.62\ {\text{s}}} as she passes the finish line, and she calculates the distance between the starting and finish lines, as measured in her frame, to be 259.81 light-seconds (about 77.9 × 10 km). 1).