Lorentz Transformations

The postulates demand that all inertial frames agree on the interval $d{s}^2={\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$. The linear transformations $x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$ that preserve $\eta$ are the Lorentz transformations; with translations they form the Poincaré group. This page derives the boost, introduces rapidity, treats composition and Thomas–Wigner rotation, and records the matrix/tensor form. The group-theoretic structure is developed in Lorentz & Poincaré groups. Units: $c=1$ unless restored; $\eta =\mathrm{diag}(+,-,-,-)$.

Definition

A Lorentz transformation is a linear map preserving the metric:

$${\eta }_{\mu \nu }{\Lambda }^{\mu }{}_{\rho }{\Lambda }^{\nu }{}_{\sigma }={\eta }_{\rho \sigma },\text{equivalently }{\Lambda }^{\mathsf{T}}\eta \Lambda =\eta .$$

This guarantees $d{s}^{\prime 2}=d{s}^2$. Taking determinants gives $(\det \Lambda {)}^2=1$; the $00$-component obeys $|{\Lambda }^0{}_0|\ge 1$. The four sign choices split $O(1,3)$ into four components; physics lives in the proper orthochronous part $S{O}^{+}(1,3)$ ($\det =+1$, ${\Lambda }^0{}_0\ge 1$), connected to the identity — boosts and rotations.

Boost along $x$

For relative velocity $v$ along $x$, write $\beta =v/c$ and the Lorentz factor

$$\gamma =\frac{1}{\sqrt{1-{\beta }^2}}\ge 1.$$

The coordinates transform as

$$\begin{array}{rlrl}c{t}^{\prime }& =\gamma (ct-\beta x),& x^{\prime }& =\gamma (x-\beta ct),\\ y^{\prime }& =y,& z^{\prime }& =z.\end{array}\text{\hspace{1em}(LT1)}$$

In matrix form $x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$ with

$$\Lambda (\beta )=\left(\begin{array}{cccc}\gamma & -\gamma \beta & 0& 0\\ -\gamma \beta & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).$$

The inverse is $\Lambda (-\beta )$: the unprimed frame moves at $-v$ relative to the primed,

$$ct=\gamma (c{t}^{\prime }+\beta x^{\prime }),x=\gamma (x^{\prime }+\beta c{t}^{\prime }).\text{\hspace{1em}(LT2)}$$

The two frames and an event

An event $P$ carries coordinates $(ct,x)$ in $S$ and $(c{t}^{\prime },x^{\prime })$ in $S^{\prime }$; the boost relates them. On a Minkowski diagram the $S^{\prime }$ axes tilt symmetrically toward the light line $ct=x$ (the $c{t}^{\prime }$ axis is $x=\beta ct$, the $x^{\prime }$ axis is $ct=\beta x$): both observers locate the same point $P$ and merely project it onto differently-tilted axes, which is the whole content of $x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$. Greater $\beta$ tilts the primed axes closer to the $45^{\circ }$ light line.

▶ Interactive diagram — drag the event $P$ and slide $\beta$ to watch both coordinate readings change while $s^2=(ct{)}^2-x^2$ stays fixed.

Derivation from the postulates

The boost is fixed almost completely by symmetry. Work in the $tx$ plane (the boost leaves $y,z$ untouched) and seek the map from frame $S$ to $S^{\prime }$ moving at velocity $v$ along $x$.

Step 1 — Linearity from homogeneity. Spacetime is homogeneous: no event is special, so a free particle (straight worldline) in $S$ must map to a free particle in $S^{\prime }$. Maps taking all straight lines to straight lines are affine; fixing the common origin makes them linear. Hence $$c{t}^{\prime }=Act+Bx,x^{\prime }=Cct+Dx,$$ where $A,B,C,D$ are real dimensionless coefficients depending only on the relative velocity $v$ (homogeneity forbids dependence on $x,t$; using natural $c=1$ keeps them dimensionless). Four unknowns remain; the next three steps fix them.

Step 2 — Motion of the origin. The spatial origin of $S^{\prime }$ ($x^{\prime }=0$) moves at $v$ in $S$, i.e. $x=\beta ct$. Then $0=Cct+D\beta ct$ gives $C=-\beta D$. Writing $D\equiv \gamma$, $$c{t}^{\prime }=Act+Bx,x^{\prime }=\gamma (x-\beta ct).$$

Step 3 — Reciprocity and isotropy. By the relativity postulate plus spatial isotropy, the inverse boost is the same with $v\to -v$, so $S$ moves at $-v$ in $S^{\prime }$. This symmetry forces $A=\gamma$ and $B=-\gamma \beta$: $$c{t}^{\prime }=\gamma (ct-\beta x),x^{\prime }=\gamma (x-\beta ct).$$ Only $\gamma (\beta )$ remains undetermined.

Step 4 — Invariant speed fixes $\gamma$. The second postulate makes the interval invariant: $c^2{t}^{\prime 2}-x^{\prime 2}=c^2{t}^2-x^2$. Substituting, $${\gamma }^2(ct-\beta x{)}^2-{\gamma }^2(x-\beta ct{)}^2={\gamma }^2(1-{\beta }^2)(c^2{t}^2-x^2),$$ so invariance requires ${\gamma }^2(1-{\beta }^2)=1$, i.e. $$\gamma =\frac{1}{\sqrt{1-{\beta }^2}}.$$ This is the unique solution preserving the interval, completing the boost. As $c\to \infty$ ($\beta \to 0$, $\gamma \to 1$) it collapses to the Galilean $t^{\prime }=t$, $x^{\prime }=x-vt$.

Conversely, once $\gamma$ is fixed the same algebra runs forward and proves invariance: $c^2{t}^{\prime 2}-x^{\prime 2}={\gamma }^2(1-{\beta }^2)(c^2{t}^2-x^2)=c^2{t}^2-x^2$. So interval invariance and the boost are equivalent statements of ${\Lambda }^{\mathsf{T}}\eta \Lambda =\eta$; the direct proof of invariance from the postulates is the other direction.

Rapidity

Write $\beta =\tanh \varphi$; then $\gamma =\cosh \varphi$, $\gamma \beta =\sinh \varphi$, and the boost is a hyperbolic rotation by rapidity $\varphi$:

$$\Lambda (\varphi )=\left(\begin{array}{cc}\cosh \varphi & -\sinh \varphi \\ -\sinh \varphi & \cosh \varphi \end{array}\right)\text{ (in the tx block).}$$

Rapidities add for collinear boosts: ${\varphi }_{\text{tot}}={\varphi }_1+{\varphi }_2$, the cleanest statement of velocity addition (see kinematics). Recovering velocities: $\tanh ({\varphi }_1+{\varphi }_2)=({\beta }_1+{\beta }_2)/(1+{\beta }_1{\beta }_2)$.

Composition and Thomas–Wigner rotation

Collinear boosts compose into a single boost. Non-collinear boosts do not: $\Lambda ({\mathbf{v}}_1)\Lambda ({\mathbf{v}}_2)$ equals a boost times a spatial rotation, the Thomas–Wigner rotation. This is why boosts alone are not a subgroup, and the precession is responsible for the Thomas factor in spin–orbit coupling. The full closure is the Lorentz group.

General transformation and the Poincaré group

A general element is a rotation $R\in SO(3)$ times a boost; with spacetime translations $a^{\mu }$ it becomes Poincaré,

$$x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{x}^{\nu }+a^{\mu },$$

the full symmetry group of SR. Six parameters (3 boosts + 3 rotations) plus 4 translations give the 10-parameter group whose conserved charges are energy, momentum, and angular momentum. Details: group/lorentz-poincare.md and math/group-theory/00-README.md.

Next

Read off the kinematic consequences (dilation, contraction, Doppler) and build four-vectors that transform with $\Lambda$. Dynamics follows in relativistic dynamics.