Fundamentals of Time and Relativity

Lorentz Transformation 4d

A general Lorentz transformation, from one system of spacetime coordinates ${x^{μ}}$ to another system ${{x^{'}}^{μ}}$ , is a linear transformation of the form

5.8

{x^{'}}^{μ} = Λ^{μ}_{ν} x^{ν} + a^{μ} μ, ν = 0, 1, 2, 3

with $Λ$ a 4x4 matrix, independent of the coordinates, and $a^{μ}$ an arbitrary displacement of the origin. The upper index of $Λ$ labels rows, the lower one labels columns.

The fundamental property that distinguishes Lorentz transformations is that they leave invariant the Minkowski distance $d s^{2} = g_{μ ν} d x^{μ} d x^{ν}$ , which means that they are restricted by the condition that the metric is invariant:

5.9

Λ^{μ}_{κ} Λ^{ν}_{λ} g_{μ ν} = g_{κ λ}, g_{μ ν} = diag(1,-1,-1,-1) = g^{μ ν}

The set of all Lorentz transformations of the form (5.8) is called the inhomogeneous Lorentz group or the Poincaré group, and the subset with $a^{μ} = 0$ is called the homogeneous Lorentz group. It has the proper homogeneous Lorentz group as a subgroup; proper because it satisfies the additional requirements $Λ^{0}_{0} \geq 1$ ; $Det Λ = 1$ . The improper Lorentz transformations involve either space inversion or time reversal, or their product.

From the coordinate-independent nature of the 4-vector $x = x^{μ} e_{μ}$ in Minkowski space, see eq. (4.5), one readily derives the transformation rules for the base vectors ${e_{μ}}$ :

5.10

e_{μ}^{'} = e_{ν} {(Λ^{- 1})}^{ν}_{μ} e_{μ} = e_{ν}^{'} Λ^{ν}_{μ}

Hence, the value of the Minkowski inner product (4.7) is invariant under Lorentz transformations as we have from (5.9),(5.10):

5.11

e_{μ}^{'} \cdot e_{ν}^{'} = e_{κ} {(Λ^{- 1})}^{κ}_{μ} \cdot e_{λ} {(Λ^{- 1})}^{λ}_{ν} = (Λ^{- 1})^{κ}_{μ} {(Λ^{- 1})}^{λ}_{ν} g_{κ λ} = g_{μ ν}

Minkowski orthogonality is preserved under Lorentz transformations.
Lorentz transformations are the only non-singular coordinate transformations that leave $d s^{2}$ invariant.