General Relativity: An Introduction for Physicists · Problem 1.5

The spacetime of special relativity — Hobson, Efstathiou & Lasenby

Physics · general-lorentz-boost

Problem

An inertial frame

S'

is related to

S

by a boost

\vec{v}

whose components in

S

are

(v_x, v_y, v_z)

. Show that the coordinates

(ct', x', y', z')

and

(ct, x, y, z)

of an event are related by

\begin{pmatrix} ct' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma\beta_x & -\gamma\beta_y & -\gamma\beta_z \\ -\gamma\beta_x & 1+\alpha\beta_x^2 & \alpha\beta_x\beta_y & \alpha\beta_x\beta_z \\ -\gamma\beta_y & \alpha\beta_y\beta_x & 1+\alpha\beta_y^2 & \alpha\beta_y\beta_z \\ -\gamma\beta_z & \alpha\beta_z\beta_x & \alpha\beta_z\beta_y & 1+\alpha\beta_z^2 \end{pmatrix} \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix},

where

\vec{\beta} = \vec{v}/c

\gamma = (1-|\vec{\beta}|^2)^{-1/2}

and

\alpha = (\gamma-1)/|\vec{\beta}|^2

. Hint: The transformation must take the same form if both

S

and

S'

undergo the same spatial rotation.