Deriving the Lorentz transformation

Transforming between inertial frames

We want equations connecting coordinates in two inertial frames. Let frame $S'$ move at speed $v$ in the positive x-direction relative to frame $S$. The origins coincide at $t=t'=0$.

Classically, one would write $x'=x-vt$ and $t'=t$. Special relativity requires a new transformation that preserves light speed for all inertial observers.

Linearity

Because space and time are homogeneous, the transformation between inertial frames should be linear. Uniform motion in one frame should remain uniform in another. Therefore $x'$ and $t'$ should be linear combinations of $x$ and $t$.

The standard Lorentz transformation is

$x'=gamma(x-vt)$

$y'=y$

$z'=z$

ight)$$ where $$gamma=rac{1}{sqrt{1-v^2/c^2}}$$ ## Inverse transformation The inverse transformation is obtained by replacing $v$ with $-v$: $$x=gamma(x'+vt')$$ $$t=gammaleft(t'+rac{vx'}{c^2} ight)$$ This symmetry reflects the principle of relativity. If $S'$ moves at $+v$ relative to $S$, then $S$ moves at $-v$ relative to $S'$. ## Preserving light speed Consider a light pulse moving in the positive x-direction. In frame $S$, it satisfies $$x=ct$$ Transforming, $$x'=gamma(ct-vt)=gamma t(c-v)$$ and $$t'=gammaleft(t-rac{vct}{c^2} ight)=gamma tleft(1-rac{v}{c} ight)$$ The ratio is $$rac{x'}{t'}=c$$ So light also moves at speed $c$ in $S'$. ## Galilean limit When $vll c$, the Lorentz factor approaches 1: $$gammaapprox1$$ and the time transformation becomes approximately $$t'approx t$$ while $$x'approx x-vt$$ Thus the Lorentz transformation reduces to the Galilean transformation at low speeds. ## Mixing of space and time The Lorentz transformation shows that space and time coordinates mix. The time coordinate $t'$ depends on both $t$ and $x$. This is the mathematical source of relativity of simultaneity. Similarly, the space coordinate $x'$ depends on both $x$ and $t$. Measurements of length and time are frame-dependent projections of spacetime. ## Events, not objects Lorentz transformations apply to events. An event has coordinates $(x,t)$ in one frame and $(x',t')$ in another. To analyze rods, clocks, or particles, first identify the relevant events, then transform their coordinates. Many mistakes in relativity come from transforming objects vaguely instead of transforming clearly defined events. ## The big idea The Lorentz transformation is the coordinate transformation required by Einstein's postulates. It preserves light speed, reduces to the Galilean transformation at low speeds, and mixes space and time. It is the mathematical backbone of special relativity.