The Lorentz Transformation — Question 5

Question 5

A Lorentz matrix has unit determinant and a metric inverse

Write a one-dimensional boost as $X\prime=\Lambda X$, with $X=(ct,x)^{\mathsf T}$ and $\Lambda=\gamma\begin{pmatrix}1&-\beta\\-\beta&1\end{pmatrix}$. Let $\eta=\operatorname{diag}(1,-1)$.

1. Find the determinant and inverse of $\Lambda$.

2. Prove $\Lambda^{\mathsf T}\eta\Lambda=\eta$ and state the preserved quadratic form.

Question 5 - Solution

A Lorentz matrix has unit determinant and a metric inverse

(a) Method and calculation

$$\boxed{\det\Lambda=\gamma^2(1-\beta^2)=1}.$$$$\boxed{\Lambda^{-1}=\gamma\begin{pmatrix}1&\beta\\\beta&1\end{pmatrix}=\Lambda(-\beta)}.$$

(b) Interpretation and further calculation

$$\Lambda^{\mathsf T}\eta\Lambda=\gamma^2\begin{pmatrix}1-\beta^2&0\\0&-(1-\beta^2)\end{pmatrix}=\eta.$$$$\boxed{X\prime^{\mathsf T}\eta X\prime=X^{\mathsf T}\eta X=c^2t^2-x^2}.$$

Physical check

The preserved metric has opposite time and space signs. This is not an ordinary Euclidean rotation.