The Lorentz Group and Rapidity

A Lorentz transformation is any linear change of inertial coordinates that leaves the interval $d s^{2} = η_{μν} d x^{μ} d x^{ν}$ 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 $Λ^{μ}_{ν}$ be the matrix of a coordinate change $x^{' μ} = Λ^{μ}_{ν} x^{ν}$ . Invariance of the interval requires $η_{μν} d x^{' μ} d x^{' ν} = η_{α β} d x^{α} d x^{β}$ for all displacements, which forces

η_{α β} = η_{μν} Λ^{μ}_{α} Λ^{ν}_{β} .

In matrix form, with $η = diag (- 1, + 1, + 1, + 1)$ , this is $Λ^{T} η Λ = η$ . Any matrix satisfying it is a Lorentz transformation.

Two immediate consequences follow from the defining condition. Taking the determinant of $Λ^{T} η Λ = η$ and using $det η \neq = 0$ gives $(det Λ)^{2} = 1$ , so $det Λ = \pm 1$ . Reading off the $α = β = 0$ component gives $- 1 = - (Λ^{0}_{0})^{2} + \sum_{i} (Λ^{i}_{0})^{2}$ , hence $(Λ^{0}_{0})^{2} = 1 + \sum_{i} (Λ^{i}_{0})^{2} \geq 1$ , so either $Λ^{0}_{0} \geq 1$ or $Λ^{0}_{0} \leq - 1$ . These two signs partition the group; the transformations continuously connected to the identity have $det Λ = + 1$ and $Λ^{0}_{0} \geq 1$ .

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,

(x^{'} y^{'}) = (cos θ sin θ - sin θ cos θ) (x y) .

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

(x^{'0} x^{'1}) = (cosh ϕ - sinh ϕ - sinh ϕ cosh ϕ) (x^{0} x^{1}),

and the identity $cosh^{2} ϕ - sinh^{2} ϕ = 1$ guarantees $- x^{'0}^{2} + x^{'1}^{2} = - x^{0}^{2} + x^{1}^{2}$ . Matching this to the standard boost $Λ^{0}_{0} = γ$ , $Λ^{0}_{1} = - γ β$ identifies the parameter.

The circular angle of a rotation is bounded and periodic; the rapidity is unbounded and runs over the whole real line as $β$ 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.

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.

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:

(cosh ϕ_{2} - sinh ϕ_{2} - sinh ϕ_{2} cosh ϕ_{2}) (cosh ϕ_{1} - sinh ϕ_{1} - sinh ϕ_{1} cosh ϕ_{1}) = (cosh (ϕ_{1} + ϕ_{2}) - sinh (ϕ_{1} + ϕ_{2}) - sinh (ϕ_{1} + ϕ_{2}) cosh (ϕ_{1} + ϕ_{2})),

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

ϕ = ϕ_{1} + ϕ_{2} .

The nonlinear velocity-addition rule is the image of this linear law under $β = tanh ϕ$ . Applying the $tanh$ addition formula,

β = tanh (ϕ_{1} + ϕ_{2}) = \frac{tanh ϕ _{1} + tanh ϕ _{2}}{1 + tanh ϕ _{1} tanh ϕ _{2}} = \frac{β _{1} + β _{2}}{1 + β _{1} β _{2}} .

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

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.

Generators: boosts and rotations

Near the identity, a Lorentz transformation is $Λ = I + ω$ with $ω$ infinitesimal. The defining condition $Λ^{T} η Λ = η$ to first order requires $ω^{T} η + η ω = 0$ , i.e. the lowered-index array $ω_{μν} = η_{μα} ω^{α}_{ν}$ is antisymmetric. An antisymmetric $4 \times 4$ 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 $ω_{ij}$ . 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 $ω_{0 i}$ . Each generates a boost along one axis.

The commutators fix the local structure. Rotations close, $[J_{i}, J_{j}] = ϵ_{ij k} J_{k}$ ; boosts and rotations mix, $[J_{i}, K_{j}] = ϵ_{ij k} K_{k}$ ; and, the fact that drives the rest of the lesson, two boosts do not close:

[K_{i}, K_{j}] = - ϵ_{ij k} 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, $det Λ = \pm 1$ and $sgn Λ^{0}_{0} = \pm 1$ , 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	$det Λ$	$Λ^{0}_{0}$	Reached by
Proper orthochronous	$+ 1$	$\geq + 1$	identity, boosts, rotations
Time reversal	$- 1$	$\leq - 1$	$\times T$
Parity	$- 1$	$\geq + 1$	$\times P$
Parity–time	$+ 1$	$\leq - 1$	$\times P T$

The proper orthochronous component, written $S O^{+} (1, 3)$ , 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 ( $Λ^{0}_{0} \geq 1$ , orthochronous). The other three are obtained by composing with parity $P = diag (1, - 1, - 1, - 1)$ , time reversal $T = diag (- 1, 1, 1, 1)$ , 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.

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.

Wigner rotation and Thomas precession

Because $[K_{i}, K_{j}] = - ϵ_{ij k} J_{k} \neq = 0$ , the composition of two boosts in different directions is not a pure boost. It equals a boost times a rotation,

Λ (v_{2}) Λ (v_{1}) = R (Ω) Λ (v_{tot}),

where $R (Ω)$ is a spatial rotation through the Wigner angle $Ω$ . 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 $ϕ_{1}, ϕ_{2}$ the Wigner angle satisfies

tan Ω = \frac{sinh ϕ _{1} sinh ϕ _{2}}{cosh ϕ _{1} + cosh ϕ _{2}},

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

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.

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 $a$ and velocity $v$ is

ω_{T} = \frac{γ ^{2}}{γ + 1} \frac{a \times 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 $γ^{2} / (γ + 1) \to \frac{1}{2}$ 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 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.