Lorentz Transformation

A Lorentz transformation is a linear map of spacetime that preserves the relativistic interval between events. In special relativity, it relates measurements made in inertial reference frames with coincident origins, allowing for relative motion and differently oriented spatial axes. Unlike transformations in Newtonian physics, it can mix space and time coordinates. Its defining invariant incorporates the vacuum speed of light, $c$, so observers related by a Lorentz transformation agree on that speed even when their measurements of distances and durations differ. (feynmanlectures.caltech.edu)

Historical and physical foundations

The transformation is named after Hendrik Lorentz, whose work on moving charged bodies and electromagnetism included a formulation published in 1904. It emerged from efforts to reconcile electromagnetic theory with experiments that did not reveal motion relative to a supposed light-carrying ether. In 1905, Albert Einstein derived the coordinate transformation from the relativity principle and the constancy of light speed, without requiring an ether or an absolutely stationary reference frame. (nobelprize.org)

The physical motivation involves Maxwell’s equations, whose form is compatible with Lorentz transformations rather than the Galilean transformations of classical mechanics. Einstein’s interpretation treated the transformed time as the time actually measured by synchronized clocks in the moving frame, rather than merely an auxiliary mathematical variable. (feynmanlectures.caltech.edu)

Standard boost along one axis

Let $S'$ move with constant velocity $v$ along the positive $x$-axis of $S$. Assume parallel spatial axes, origins that coincide at $t=t'=0$, and clocks synchronized by Einstein’s light-signal convention. The coordinates of the same event are related by

Here $\gamma$ is the Lorentz factor, and $|v|<c$. This rotation-free transformation is called a boost. Its inverse is obtained by exchanging primed and unprimed coordinates and replacing $v$ with $-v$. These equations transform the coordinates assigned to an event; they do not physically transport the event elsewhere. (physics.umd.edu)

For velocities much smaller than $c$, $\gamma$ approaches one and the equations approach $x'=x-vt$, $t'=t$, with unchanged transverse coordinates. Galilean kinematics therefore appears as the low-speed limit. The position-dependent term in $t'$, absent from the Galilean transformation, is essential to relativistic synchronization. (feynmanlectures.caltech.edu)

Invariant interval and matrix form

For two events, define the spacetime interval using the sign convention

Every Lorentz transformation preserves this quantity. Positive, zero, and negative values describe timelike, lightlike, and spacelike separations, respectively. Thus the transformation preserves the light cone and the classification of separations, although it generally changes individual coordinate differences. The geometry described by this interval is Minkowski space. (feynmanlectures.caltech.edu)

Writing $X=(ct,x,y,z)^{\mathsf T}$, the transformation becomes $X'=\Lambda X$, where $\Lambda$ is a real $4\times4$ matrix. If $\eta=\operatorname{diag}(1,-1,-1,-1)$ is the metric tensor, interval preservation requires

The standard boost has the form

Unlike an ordinary Euclidean rotation, this transformation preserves an indefinite quadratic form rather than a positive-definite squared distance. (damtp.cam.ac.uk)

Measurable consequences

Time dilation. For two ticks of a clock at rest in $S'$, $\Delta x'=0$. Their separation in $S$ satisfies $\Delta t=\gamma\Delta\tau$, where $\Delta\tau$ is the clock’s proper time. The moving clock accumulates less time than the coordinate-time interval measured in $S$. (physics.umd.edu)

Length contraction. A rod at rest in $S'$, aligned with the motion and having rest length $L_0$, has length $L=L_0/\gamma$ in $S$. Measuring that length requires recording its endpoints simultaneously in $S$; those endpoint events are not simultaneous in $S'$. (feynmanlectures.caltech.edu)

Relativity of simultaneity. If two separated events satisfy $\Delta t=0$, then $\Delta t'=-\gamma v\Delta x/c^2$. Simultaneity therefore depends on the reference frame. Proper, orthochronous transformations nevertheless preserve the temporal order of causally connected events. (feynmanlectures.caltech.edu)

Differentiating the coordinate equations gives the longitudinal velocity transformation,

Substitution of $u_x=c$ gives $u'_x=c$, directly demonstrating light-speed invariance. (physics.umd.edu)

Group structure and rapidity

Lorentz transformations form the Lorentz group, $O(1,3)$. The full group includes spatial reflections and time reversal. Its proper, orthochronous subgroup consists of transformations continuously connected to the identity, preserves spatial orientation and the direction of time, and has six continuous parameters: three for rotations and three for boosts. Adding spacetime translations produces the Poincaré group, which also permits different coordinate origins. (damtp.cam.ac.uk)

A boost can be parametrized by rapidity, $\phi$, with

These hyperbolic functions express a boost as a hyperbolic rotation. Rapidities add for successive boosts along the same axis, whereas the corresponding velocities combine through the relativistic velocity-addition law. (damtp.cam.ac.uk)

Transformation of physical quantities

Lorentz transformations apply not only to event coordinates but also to four-vectors and tensors. For example, energy and momentum combine into four-momentum, $P=(E/c,p_x,p_y,p_z)$. Under the standard boost, $E'=\gamma(E-vp_x)$ and $p'_x=\gamma(p_x-vE/c^2)$. Electric and magnetic fields likewise transform together as components of the electromagnetic field tensor. This framework enables frame-independent formulations of electromagnetic theory and quantum field theory. (feynmanlectures.caltech.edu)