General Relativity: An Introduction for Physicists · Problem 1.2

The spacetime of special relativity — Hobson, Efstathiou & Lasenby

Physics · spacetime-interval-invariance

Problem

Two events

A

and

B

have coordinates

(t_A, x_A, y_A, z_A)

and

(t_B, x_B, y_B, z_B)

respectively. Show that both the time difference

\Delta t = t_B - t_A

and the quantity

\Delta r^2 = \Delta x^2 + \Delta y^2 + \Delta z^2

are separately invariant under any Galilean transformation, whereas the quantity

\Delta s^2 = c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2

is invariant under any Lorentz transformation.