---
title: The Lorentz Group and Rapidity
module: Spacetime and the Lorentz Group
moduleNumber: 2
lessonNumber: 3
order: 203
summary: >
  The Lorentz transformations are the linear maps that preserve the Minkowski
  metric, and they form the group O(1,3). Boosts are hyperbolic rotations
  parametrized by rapidity, which adds along a line where velocity does not. The
  boost and rotation generators fix the group's local structure; its four
  disconnected components are set by two signs; and two non-collinear boosts
  compose into a boost plus a rotation, the Wigner rotation behind Thomas
  precession.
topics: [Spacetime and the Lorentz Group]
draft: false
sources:
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §1 — Lorentz transformations as matrices"
  - book: Taylor & Wheeler
    ref: "Spacetime Physics, Ch. 8 — The velocity parameter (rapidity)"
---

A Lorentz transformation is any linear change of inertial coordinates that leaves
the [interval](/relativity/spacetime-and-the-lorentz-group/minkowski-spacetime-and-the-interval)
$\d s^2 = \eta_{\mu\nu}\,\d x^\mu\,\d x^\nu$ unchanged. The set of all such maps is
a group, and its structure organizes every frame change in the theory. Boosts
turn out to be rotations through an imaginary-like angle in a time–space plane,
and reparametrizing them by rapidity makes the composition of collinear boosts as
simple as adding numbers.

## The defining condition

Let $\Lambda^\mu{}_\nu$ be the matrix of a coordinate change $x'^\mu = \Lambda^\mu{}_\nu x^\nu$.
Invariance of the interval requires
$\eta_{\mu\nu}\,\d x'^\mu\,\d x'^\nu = \eta_{\alpha\beta}\,\d x^\alpha\,\d x^\beta$
for all displacements, which forces

$$
\eta_{\alpha\beta} = \eta_{\mu\nu}\,\Lambda^\mu{}_\alpha\,\Lambda^\nu{}_\beta.
$$

In matrix form, with $\eta = \operatorname{diag}(-1,+1,+1,+1)$, this is
$\Lambda^{\mathsf T}\eta\,\Lambda = \eta$. Any matrix satisfying it is a Lorentz
transformation.

> **Definition (Lorentz group).** The Lorentz group $O(1,3)$ is the set of all
> real $4\times4$ matrices $\Lambda$ obeying $\Lambda^{\mathsf T}\eta\,\Lambda = \eta$.
> It is closed under matrix multiplication, contains the identity, and every
> element has an inverse (also Lorentz), so it is a group. It is the metric's
> symmetry group: the isometries of Minkowski spacetime that fix the origin.

Two immediate consequences follow from the defining condition. Taking the
determinant of $\Lambda^{\mathsf T}\eta\,\Lambda = \eta$ and using
$\det\eta \ne 0$ gives $(\det\Lambda)^2 = 1$, so $\det\Lambda = \pm 1$. Reading off
the $\alpha = \beta = 0$ component gives
$-1 = -(\Lambda^0{}_0)^2 + \sum_i (\Lambda^i{}_0)^2$, hence
$(\Lambda^0{}_0)^2 = 1 + \sum_i (\Lambda^i{}_0)^2 \ge 1$, so either
$\Lambda^0{}_0 \ge 1$ or $\Lambda^0{}_0 \le -1$. These two signs partition the
group; the transformations continuously connected to the identity have
$\det\Lambda = +1$ and $\Lambda^0{}_0 \ge 1$.

## Boosts as hyperbolic rotations

Compare a spatial rotation in the $x$–$y$ plane with a boost in the $t$–$x$ plane.
The rotation preserves $x^2 + y^2$ and mixes the two coordinates through circular
functions,

$$
\begin{pmatrix} x' \\ y' \end{pmatrix}
= \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}.
$$

A boost must instead preserve $-c^2t^2 + x^2$, and the functions that keep a
hyperbola fixed are the hyperbolic ones. Writing $x^0 = ct$ and introducing a
parameter $\phi$,

$$
\begin{pmatrix} x'^0 \\ x'^1 \end{pmatrix}
= \begin{pmatrix} \cosh\phi & -\sinh\phi \\ -\sinh\phi & \cosh\phi \end{pmatrix}
\begin{pmatrix} x^0 \\ x^1 \end{pmatrix},
$$

and the identity $\cosh^2\phi - \sinh^2\phi = 1$ guarantees
$-x'^0{}^2 + x'^1{}^2 = -x^0{}^2 + x^1{}^2$. Matching this to the standard boost
$\Lambda^0{}_0 = \gamma$, $\Lambda^0{}_1 = -\gamma\beta$ identifies the parameter.

> **Definition (Rapidity).** The rapidity $\phi$ of a boost of speed $v = \beta c$
> is defined by
> $$
> \tanh\phi = \beta, \qquad \cosh\phi = \gamma, \qquad \sinh\phi = \gamma\beta.
> $$
> A boost is a hyperbolic rotation through the rapidity angle $\phi$ in the
> relevant time–space plane. As $\beta \to 1$, $\phi \to \infty$: no finite
> rapidity reaches the speed of light.

The circular angle of a rotation is bounded and periodic; the rapidity is
unbounded and runs over the whole real line as $\beta$ runs over $(-1, 1)$. A
boost is a rotation whose "angle" can grow without limit, which is why speeds pile
up against $c$ without ever reaching it.

$$
% caption: A boost hyperbolically rotates the spacetime grid. Coordinate lines of
% the moving frame shear toward the light line at forty-five degrees while every
% invariant hyperbola of constant interval maps to itself.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.3,0) -- (4.6,0) node[right] {x};
  \draw[->, black] (0,-0.3) -- (0,4.6) node[above] {ct};
  \draw[black, dashed] (0,0) -- (4.3,4.3);
  % original grid lines (light)
  \foreach \y in {1,2,3} \draw[black] (0,\y) -- (4.2,\y);
  \foreach \x in {1,2,3} \draw[black] (\x,0) -- (\x,4.2);
  % boosted axes beta=0.5
  \draw[->, acc, thick] (0,0) -- (2.0,4.0) node[above] {ct'};
  \draw[->, acc, thick] (0,0) -- (4.0,2.0) node[right] {x'};
  % invariant hyperbola
  \draw[black!70, thick] (0,1.4) .. controls (0.7,1.45) and (1.4,1.75) .. (2.0,2.25)
     .. controls (2.5,2.65) and (3.0,3.2) .. (3.35,3.6);
  \node[black!70, anchor=west] at (2.3,2.7) {invariant hyperbola};
\end{tikzpicture}
$$

## Rapidity adds

The decisive advantage of the rapidity is its behavior under composition. Two
collinear boosts multiply as hyperbolic rotations, and hyperbolic rotations add
their angles:

$$
\begin{pmatrix} \cosh\phi_2 & -\sinh\phi_2 \\ -\sinh\phi_2 & \cosh\phi_2 \end{pmatrix}
\begin{pmatrix} \cosh\phi_1 & -\sinh\phi_1 \\ -\sinh\phi_1 & \cosh\phi_1 \end{pmatrix}
= \begin{pmatrix} \cosh(\phi_1+\phi_2) & -\sinh(\phi_1+\phi_2) \\ -\sinh(\phi_1+\phi_2) & \cosh(\phi_1+\phi_2) \end{pmatrix},
$$

using the hyperbolic addition formulas. Collinear boosts therefore compose by
adding rapidities,

$$
\phi = \phi_1 + \phi_2.
$$

The nonlinear [velocity-addition rule](/relativity/foundations/lorentz-transformation-spacetime)
is the image of this linear law under $\beta = \tanh\phi$. Applying the
$\tanh$ addition formula,

$$
\beta = \tanh(\phi_1 + \phi_2)
= \frac{\tanh\phi_1 + \tanh\phi_2}{1 + \tanh\phi_1\tanh\phi_2}
= \frac{\beta_1 + \beta_2}{1 + \beta_1\beta_2}.
$$

Rapidity is the coordinate in which composition is trivial; velocity is the
nonlinear function of it that saturates at $c$. Because $\abs{\tanh\phi} < 1$ for
every finite $\phi$, no sum of rapidities produces a speed at or above $c$.

$$
% caption: Rapidity is additive; velocity is not. On the rapidity line two boosts
% add end to end, while the corresponding velocities combine through the
% relativistic sum and stay below the ceiling at the speed of light.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % rapidity number line
  \draw[->, black] (0,2.4) -- (6.2,2.4) node[right] {rapidity};
  \foreach \x in {0,1,2,3,4,5} \draw[black] (\x,2.32) -- (\x,2.48);
  \draw[acc, very thick, ->] (0,2.4) -- (2,2.4);
  \draw[acc, very thick, ->] (2,2.4) -- (5,2.4);
  \node[acc, anchor=south] at (1.0,2.5) {phi one};
  \node[acc, anchor=south] at (3.5,2.5) {phi two};
  \node[black, anchor=north] at (2.5,2.28) {added end to end};
  % velocity curve tanh
  \draw[->, black] (0,-0.3) -- (0,1.9) node[above] {beta};
  \draw[->, black] (0,0) -- (6.2,0) node[right] {rapidity};
  \draw[black, dashed] (0,1.5) -- (6.0,1.5) node[right, black] {ceiling c};
  \draw[very thick] (0,0) .. controls (1.4,0.95) and (2.6,1.35) .. (4.2,1.45)
     .. controls (5.0,1.48) and (5.6,1.49) .. (6.0,1.495);
\end{tikzpicture}
$$

## Generators: boosts and rotations

Near the identity, a Lorentz transformation is $\Lambda = I + \omega$ with
$\omega$ infinitesimal. The defining condition $\Lambda^{\mathsf T}\eta\Lambda = \eta$
to first order requires $\omega^{\mathsf T}\eta + \eta\omega = 0$, i.e. the
lowered-index array $\omega_{\mu\nu} = \eta_{\mu\alpha}\omega^\alpha{}_\nu$ is
antisymmetric. An antisymmetric $4\times4$ array has six independent entries, so
the Lorentz group is six-dimensional:

- **Three rotation generators** $J_x, J_y, J_z$, from the antisymmetric spatial
  blocks $\omega_{ij}$. These generate ordinary rotations of the three space
  axes and close among themselves.
- **Three boost generators** $K_x, K_y, K_z$, from the mixed time–space entries
  $\omega_{0i}$. Each generates a boost along one axis.

The commutators fix the local structure. Rotations close, $[J_i, J_j] = \epsilon_{ijk}J_k$;
boosts and rotations mix, $[J_i, K_j] = \epsilon_{ijk}K_k$; and, the fact that
drives the rest of the lesson, two boosts do not close:

$$
[K_i, K_j] = -\epsilon_{ijk} J_k.
$$

The commutator of two boosts is a rotation, not a boost. This single sign — the
minus that distinguishes the boost algebra from a set of commuting translations —
is the origin of the Wigner rotation and Thomas precession below.

## The four components of O(1,3)

The two invariant signs, $\det\Lambda = \pm 1$ and $\operatorname{sgn}\Lambda^0{}_0 = \pm 1$,
cannot change continuously, so they split $O(1,3)$ into four disconnected pieces.
No path of Lorentz transformations connects two of them; each is reached from the
identity component only by composing with a discrete flip.

| Component | $\det\Lambda$ | $\Lambda^0{}_0$ | Reached by |
| --- | --- | --- | --- |
| Proper orthochronous | $+1$ | $\ge +1$ | identity, boosts, rotations |
| Time reversal | $-1$ | $\le -1$ | $\times\, T$ |
| Parity | $-1$ | $\ge +1$ | $\times\, P$ |
| Parity–time | $+1$ | $\le -1$ | $\times\, PT$ |

The **proper orthochronous** component, written $SO^+(1,3)$, contains the
identity and every transformation built continuously from boosts and rotations.
It preserves both the orientation of space ($\det = +1$) and the direction of
time ($\Lambda^0{}_0 \ge 1$, "orthochronous"). The other three are obtained by
composing with parity $P = \operatorname{diag}(1,-1,-1,-1)$, time reversal
$T = \operatorname{diag}(-1,1,1,1)$, or their product. Physics that respects the
continuous symmetries need not respect the discrete flips, and the weak
interaction famously violates $P$ and $T$ separately.

$$
% caption: The Lorentz group splits into four disconnected components fixed by two
% signs: the determinant and the sign of the time-time entry. Only the proper
% orthochronous block is continuously connected to the identity.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black] (-2.6,0) -- (2.6,0);
  \draw[black] (0,-2.6) -- (0,2.6);
  \node[black, anchor=south west] at (0.1,2.35) {time direction kept};
  \node[black, anchor=north west] at (0.1,-2.35) {time direction reversed};
  \node[black, rotate=90, anchor=south] at (-2.5,0) {space orientation};
  \node[draw=acc, very thick, fill=acc!12, minimum width=2.3cm, minimum height=1.0cm] at (1.35,1.3) {proper ortho};
  \node[draw=black, fill=black!5, minimum width=2.3cm, minimum height=1.0cm] at (-1.35,1.3) {parity};
  \node[draw=black, fill=black!5, minimum width=2.3cm, minimum height=1.0cm] at (-1.35,-1.3) {time reversal};
  \node[draw=black, fill=black!5, minimum width=2.3cm, minimum height=1.0cm] at (1.35,-1.3) {PT};
\end{tikzpicture}
$$

## Wigner rotation and Thomas precession

Because $[K_i, K_j] = -\epsilon_{ijk}J_k \ne 0$, the composition of two boosts in
different directions is not a pure boost. It equals a boost times a rotation,

$$
\Lambda(\vec{v}_2)\,\Lambda(\vec{v}_1) = R(\Omega)\,\Lambda(\vec{v}_{\text{tot}}),
$$

where $R(\Omega)$ is a spatial rotation through the **Wigner angle** $\Omega$. The
rotation is a genuine physical effect, not a coordinate artifact: after two
successive non-collinear boosts, the axes of the final rest frame are rotated
relative to the original, even though no rotation was applied at any step. For two
perpendicular boosts of rapidities $\phi_1, \phi_2$ the Wigner angle satisfies

$$
\tan\Omega = \frac{\sinh\phi_1\,\sinh\phi_2}{\cosh\phi_1 + \cosh\phi_2},
$$

which is second order in the velocities and vanishes only when the boosts are
collinear.

$$
% caption: Two perpendicular boosts compose to a boost plus a rotation. Starting
% axes boosted first along x and then along y arrive rotated by the Wigner angle
% relative to a single direct boost, with no rotation applied at any stage.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % starting frame axes
  \draw[->, black, thick] (0,0) -- (2.0,0);
  \draw[->, black, thick] (0,0) -- (0,2.0);
  \node[black, anchor=north] at (2.0,-0.05) {start};
  % rotated final axes
  \draw[->, acc, very thick] (0,0) -- (1.88,0.68);
  \draw[->, acc, very thick] (0,0) -- (-0.68,1.88);
  \node[acc, anchor=west] at (1.9,0.7) {after two boosts};
  % small arc showing Wigner angle
  \draw[black, ->] (1.4,0) to[bend left=20] (1.32,0.48);
  \node[black, anchor=west] at (1.35,0.28) {Wigner angle};
\end{tikzpicture}
$$

The Wigner rotation accumulates continuously along a curved worldline. A gyroscope
carried around a closed path — an electron orbiting a nucleus, or a spin
transported along an accelerated trajectory — undergoes an infinite succession of
infinitesimal boosts, and the residual rotations add up to a steady precession of
its spin axis. This is **Thomas precession**, and its angular rate for a particle
with acceleration $\vec{a}$ and velocity $\vec{v}$ is

$$
\vec{\omega}_{\text{T}} = \frac{\gamma^2}{\gamma + 1}\,\frac{\vec{a}\times\vec{v}}{c^2}.
$$

In the hydrogen atom the electron's spin–orbit coupling picks up exactly this
kinematic factor: the naive spin–orbit energy is halved by the Thomas factor
$\gamma^2/(\gamma+1) \to \tfrac12$ in the slow-motion limit, and the corrected
value matches the observed fine structure. A purely kinematic consequence of the
non-commuting boost generators shows up as a measurable shift in atomic spectra.

The Lorentz group is the exact symmetry the [dynamics module](/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion)
demands of every physical law. The next lesson applies boosts to the one
four-vector every observer can see directly — the photon's — to derive the
relativistic Doppler shift, aberration, and how a fast object actually looks.

[^carroll-lorentz]: **Carroll**, _Lecture Notes on General Relativity_, §1 — Lorentz transformations as the matrices preserving $\eta_{\mu\nu}$, the boost written with hyperbolic functions, and the generators of the Lorentz group with their commutation relations.

[^tw-rapidity]: **Taylor & Wheeler**, _Spacetime Physics_, Ch. 8 — the velocity parameter (rapidity), its additivity for collinear boosts, and the recovery of the velocity-addition law through $\beta = \tanh\phi$.