What Are the Lorentz Transformation Equations?

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with

where the Lorentz factor is

Here, $c$ is the speed of light in a vacuum. The inverse transformations are obtained by replacing $v$ with (-v):

Why They Are Needed

The equations follow from the two postulates of special relativity: the laws of physics are identical in all inertial reference frames, and every inertial observer measures the same value of $c$.

Unlike the Galilean relation $t'=t$, the Lorentz transformation mixes space and time. Consequently, observers in relative motion can disagree about elapsed time, measured length, and whether spatially separated events are simultaneous.

For example, let $v=0.60c$, so $\gamma=1.25$. For an event at $x=300\ \text{m}$ and $t=2.0\ \mu\text{s}$, $x'=1.25[300-(0.60c)(2.0\times10^{-6})]=-75\ \text{m}$. Also, $t'=1.25[2.0\times10^{-6}-(0.60c)(300)/c^2]=1.75\times10^{-6}\ \text{s}$. The negative (x') is a coordinate, not a negative distance. Both results describe the same event as measured in different frames. This calculation shows that both the measured position and time coordinate change; neither frame is preferred.

At low speeds, $v\ll c$, $\gamma\approx1$ and $vx/c^2$ is negligible. The equations then reduce approximately to $x'=x-vt$ and $t'=t$, recovering Galilean relativity.

Common misconception: $x$ and (x') are not positions of two different objects. They are coordinates assigned to the same event by different frames.

Exam Technique

For a Calculate question, define the frame direction, convert all quantities to SI units, calculate $\gamma$, and then substitute into both the position and time equations. Keep the sign convention consistent: reversing which frame moves requires changing the sign of $v$.