Lorentz Transformations

A Lorentz transformation is a linear change of inertial coordinates that preserves the spacetime interval. Rotations change spatial directions; boosts mix space and time. Their composition governs how relativistic wave equations and their solutions compare between inertial observers. This page constructs the transformations; Four-Vectors explains the physical objects on which they act.

Required background. Metric and Units supplies the metric and index notation.

The Lorentz condition

For $x'^\mu=\Lambda^\mu{}_{\nu}x^\nu$, preserving $x^\mu\eta_{\mu\nu}x^\nu$ for every $x$ requires

$$\Lambda^{\mathsf T}\eta\Lambda=\eta, \qquad \Lambda^{-1}=\eta^{-1}\Lambda^{\mathsf T}\eta.$$

Taking determinants gives $(\det\Lambda)^2=1$. A Lorentz matrix is generally not orthogonal with respect to the Euclidean inner product; replacing $\Lambda^{-1}$ by $\Lambda^{\mathsf T}$ is incorrect for boosts.

A spatial rotation has the block form $\operatorname{diag}(1,R)$ with $R^{\mathsf T}R=I_3$. Proper spatial rotations also obey $\det R=1$. They preserve time components and rotate spatial components in the usual way.

A boost and its rapidity

Use a passive convention: the primed frame moves at velocity $v\hat{\mathbf x}$ relative to the unprimed frame, and the origins coincide at $t=t'=0$. Then

$$\begin{pmatrix}ct'\\x'\end{pmatrix} = \begin{pmatrix}\gamma&-\gamma\beta\\-\gamma\beta&\gamma\end{pmatrix} \begin{pmatrix}ct\\x\end{pmatrix}, \qquad y'=y,\quad z'=z,$$

where $\beta=v/c$ and $\gamma=(1-\beta^2)^{-1/2}$. The worldline $x=vt$ indeed gives $x'=0$, which checks the physical meaning of the signs. The inverse transformation sends $\beta$ to $-\beta$.

Define the rapidity $\xi$ by

$$\beta=\tanh\xi,\qquad \gamma=\cosh\xi,\qquad \gamma\beta=\sinh\xi.$$

The boost is a hyperbolic rotation,

$$B_x(\xi)= \begin{pmatrix}\cosh\xi&-\sinh\xi\\-\sinh\xi&\cosh\xi\end{pmatrix},$$

in its $ct,x$ block. Multiplying two such blocks and using the hyperbolic addition formulas gives

$$B_x(\xi_2)B_x(\xi_1)=B_x(\xi_1+\xi_2), \qquad \beta_{12}=\frac{\beta_1+\beta_2}{1+\beta_1\beta_2}.$$

Rapidity, rather than velocity, adds for collinear boosts. Two successive boosts with $\beta_1=\beta_2=3/5$ have combined $\beta_{12}=15/17$, less than one; their combined Lorentz factor is $17/8$.

For a frame velocity $\boldsymbol\beta c$ in an arbitrary direction, resolve the displacement parallel and perpendicular to that direction:

$$\begin{aligned} ct'&=\gamma(ct-\boldsymbol\beta\cdot\mathbf x),\\ \mathbf x'&=\mathbf x+ \frac{\gamma-1}{\beta^2} (\boldsymbol\beta\cdot\mathbf x)\boldsymbol\beta -\gamma\boldsymbol\beta ct. \end{aligned}$$

The continuous $\beta\to0$ limit is the identity. This form makes it explicit that the perpendicular component is unaffected by a pure boost.

Which transformations preserve the future?

The Lorentz condition implies $|\Lambda^0{}_{0}|\geq1$. Thus two discrete choices distinguish four connected components: $\det\Lambda=\pm1$ and the sign of $\Lambda^0{}_{0}$. The identity component is $SO^+(1,3)$, the proper orthochronous Lorentz group. It contains ordinary rotations and finite boosts connected continuously to the identity.

Parity $P=\operatorname{diag}(1,-1,-1,-1)$ reverses spatial orientation. Time reflection $T=\operatorname{diag}(-1,1,1,1)$ reverses the time orientation. Neither can be reached continuously from the identity within the Lorentz group. Their action on quantum states requires additional structure; a classical time-reflection matrix does not itself specify the antiunitary quantum time-reversal operator.

For a future-directed on-shell momentum, $E>0$ and $E\geq c|\mathbf p|$ (with $E>c|\mathbf p|$ for positive mass), a finite boost gives

$$E'=\gamma(E-\mathbf v\cdot\mathbf p)>0.$$

Hence a change between proper orthochronous frames cannot turn a positive energy into a negative energy. The two sheets of the massive mass shell are not two ordinary observers’ descriptions of one positive-energy particle.

Quantum phases and noncommuting boosts

Transforming both $p$ and $x$ with the same Lorentz matrix leaves $p\cdot x$ unchanged, so the phase $e^{-ip\cdot x/\hbar}$ is invariant. For a photon moving along $+x$, an $x$ boost gives

$$E'=E\gamma(1-\beta)=Ee^{-\xi}.$$

The Doppler shift changes the separate frequency and wavelength, while the phase assigned to the same event remains unchanged. A scalar amplitude satisfies $\phi'(x')=\phi(x)$; spinors have an additional transformation on their components, developed in the covariant Dirac equation.

Collinear rapidity addition does not extend to arbitrary boost directions. For an infinitesimal example, let $b_i$ be the real four-vector matrix with entries $(b_i)^0{}_i=(b_i)^i{}_0=-1$, and all other entries zero. Then $[b_x,b_y]$ has spatial entries $(x,y)=1$ and $(y,x)=-1$: it generates a rotation. Non-collinear boost composition consequently contains a rotation, which is the kinematic source of Wigner rotations. These real matrices should not be confused with Hermitian generators acting on a quantum Hilbert space.

Exercises

1. Show directly that the $ct,x$ boost preserves $(ct)^2-x^2$.

Solution

Expanding the difference of squares cancels the mixed terms and leaves $\gamma^2(1-\beta^2)[(ct)^2-x^2]=(ct)^2-x^2$.

2. A primed observer measures a velocity $u'_x$. Derive its unprimed value.

Solution

Differentiate the inverse boost: $dx=\gamma(dx'+vdt')$ and $dt=\gamma(dt'+v\,dx'/c^2)$. Their ratio gives $u_x=(u'_x+v)/(1+vu'_x/c^2)$. In particular $u'_x=c$ implies $u_x=c$.

3. For a spacelike displacement with $\Delta x>c\Delta t>0$, find a boost for which the two events are simultaneous. Can a larger subluminal boost reverse their time ordering?

Solution

$\Delta t'=\gamma(\Delta t-v\Delta x/c^2)$ vanishes at $v=c^2\Delta t/\Delta x<c$. Any $v$ between this value and $c$ makes $\Delta t'<0$. No causal signal can connect such spacelike events.

References

• J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1998, chapter 11 — boosts and the Lorentz group.
• W. Rindler, Introduction to Special Relativity, 2nd ed., Oxford University Press, 1991 — rapidity and relativistic kinematics.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, section 2.3 — Lorentz transformations and quantum representations.