Mathematics.

relativity

The Lorentz Group

Mathematical Physics90 minDifficulty9 out of 10

You should know: lie groups, special relativity mathematics

Overview

The Lorentz group O(1,3) is the group of linear transformations of Minkowski spacetime ℝ^{1,3} that preserve the Lorentz inner product η(x,x) = -c²t² + x² + y² + z². It encodes the symmetry of special relativity: all inertial observers related by Lorentz transformations agree on the laws of physics. Its structure — rotations, boosts, and their compositions — underlies the classification of elementary particles via Wigner's representation theory.

Intuition

Rotations in 3D space preserve the Euclidean distance x² + y² + z². Lorentz transformations do the same job in 4D spacetime, but with a metric that has one negative sign: they preserve -c²t² + x² + y² + z². A 'boost' is a hyperbolic rotation mixing the time and one spatial axis, just as an ordinary rotation mixes two spatial axes — but replacing sines and cosines with hyperbolic sines and cosines.

Formal Definition

Definition

Let η = diag(-1, 1, 1, 1) be the Minkowski metric. The Lorentz group is the set of real 4×4 matrices preserving η.

O(1,3)={Λ∈GL(4,R):ΛTη Λ=η}O(1,3) = \{ \Lambda \in \mathrm{GL}(4,\mathbb{R}) : \Lambda^T \eta\, \Lambda = \eta \}
Definition of the Lorentz group
det⁡Λ=±1,∣Λ00∣≥1\det \Lambda = \pm 1, \quad |\Lambda^0{}_0| \geq 1
Consequences for the determinant and time component
SO+(1,3)={Λ∈O(1,3):det⁡Λ=+1,  Λ00≥1}SO^+(1,3) = \{ \Lambda \in O(1,3) : \det\Lambda = +1,\; \Lambda^0{}_0 \geq 1 \}
Proper orthochronous Lorentz group
Λ(β)μν=(γ−γβ00−γβγ0000100001),γ=(1−β2)−1/2\Lambda(\beta)^\mu{}_\nu = \begin{pmatrix} \gamma & -\gamma\beta & 0 & 0 \\ -\gamma\beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \quad \gamma = (1-\beta^2)^{-1/2}
Boost in the x-direction

Notation

NotationMeaning
O(1,3)O(1,3)Full Lorentz group (4 connected components)
SO+(1,3)SO^+(1,3)Proper orthochronous Lorentz group (identity component)
ημν\eta_{\mu\nu}Minkowski metric tensor, diag(-1,+1,+1,+1)
γ=(1−v2/c2)−1/2\gamma = (1-v^2/c^2)^{-1/2}Lorentz factor

Properties

Four connected components

O(1,3)=SO+(1,3)∪T⋅SO+(1,3)∪P⋅SO+(1,3)∪PT⋅SO+(1,3)O(1,3) = SO^+(1,3) \cup T\cdot SO^+(1,3) \cup P\cdot SO^+(1,3) \cup PT\cdot SO^+(1,3)

Lie algebra

so(1,3)≅sl(2,C)R\mathfrak{so}(1,3) \cong \mathfrak{sl}(2,\mathbb{C})_{\mathbb{R}}

Double cover

Spin(1,3)≅SL(2,C)→2:1SO+(1,3)\mathrm{Spin}(1,3) \cong SL(2,\mathbb{C}) \xrightarrow{2:1} SO^+(1,3)

Worked Examples

  1. 1

    Under Λ(β): ct' = γ(ct - βx), x' = γ(x - β ct).

    ct′=γ(ct−βx),x′=γ(x−βct)ct' = \gamma(ct - \beta x),\quad x' = \gamma(x - \beta ct)
  2. 2

    Compute -(ct')² + (x')².

    −(ct′)2+(x′)2=γ2[−(ct−βx)2+(x−βct)2]-(ct')^2 + (x')^2 = \gamma^2[-(ct-\beta x)^2 + (x-\beta ct)^2]
  3. 3

    Expand: γ²[-(c²t²-2βctx+β²x²)+(x²-2βctx+β²c²t²)] = γ²(β²-1)(c²t²-x²) = -(c²t²-x²).

    =γ2(β2−1)(c2t2−x2)=−(c2t2−x2)= \gamma^2(\beta^2 - 1)(c^2 t^2 - x^2) = -(c^2 t^2 - x^2)

✓ Answer

The invariant -(ct)² + x² is preserved, confirming Λ(β) ∈ O(1,3).

Practice Problems

Hardproof writing

Prove that the composition of two boosts in the same direction is again a boost, and find the combined rapidity.

Hardfree response

Explain why the (1/2, 0) and (0, 1/2) representations of SL(2,ℂ) correspond to left-handed and right-handed Weyl spinors.

Common Mistakes

Common Mistake

Confusing the Lorentz group with the Poincaré group

The Lorentz group is the homogeneous part (linear transformations preserving the origin). The Poincaré group also includes spacetime translations and is the full symmetry group of special relativity.

Common Mistake

Assuming boosts in different directions commute

Two boosts in different directions do not commute; their composition involves a spatial rotation called a Wigner rotation or Thomas rotation.

Quiz

How many connected components does the full Lorentz group O(1,3) have?
The universal double cover of SO⁺(1,3) is:

Historical Background

Lorentz introduced the transformations bearing his name in 1904 while trying to explain the Michelson–Morley null result. Poincaré in 1905 recognised that these form a group and named it after Lorentz. Einstein's special relativity paper (1905) provided the physical foundation. Wigner's 1939 analysis of the unitary representations of the (inhomogeneous) Lorentz group — the Poincaré group — showed that particles are classified by mass and spin.

  1. 1904

    Lorentz derives the transformation equations for electrodynamics

    Hendrik Lorentz

  2. 1905

    Poincaré names the group; Einstein provides physical interpretation

    Henri Poincaré, Albert Einstein

  3. 1939

    Wigner classifies unitary representations of the Poincaré group

    Eugene Wigner

Summary

  • O(1,3) is the group of 4×4 real matrices preserving the Minkowski metric; it has four connected components.
  • The physically relevant component is SO⁺(1,3) (proper orthochronous), which has SL(2,ℂ) as its universal double cover.
  • The complexified Lie algebra decomposes as sl(2,ℂ) ⊕ sl(2,ℂ), labelling representations by pairs (j₁, j₂).
  • Wigner's theorem classifies elementary particles by their mass and spin as irreducible representations of the Poincaré group.

References

  1. BookWigner, E.P. — On Unitary Representations of the Inhomogeneous Lorentz Group, Ann. Math. 40 (1939), 149–204
  2. BookWald, R.M. — General Relativity (1984), University of Chicago Press, Appendix B
  3. WebsiteWikipedia — Lorentz group
  4. WebsiteMathWorld — Lorentz Group
  5. WebsitenLab — Lorentz group