General Relativity: An Introduction for Physicists · Problem 1.12

The spacetime of special relativity — Hobson, Efstathiou & Lasenby

Physics · spacetime-diagrams

Problem

Draw a spacetime diagram illustrating the coordinate transformation corresponding to two inertial frames

S

and

S'

in standard configuration (i.e. where

S'

moves at a speed

v

along the positive

x

-direction and the two frames coincide at

t = t' = 0

). Show that the angle between the

x

- and

x'

- axes is the same as that between the

t

- and

t'

- axes and has the value

\tan^{-1}(v/c)

Intuitive explanation

A spacetime diagram reveals how moving observers carve up spacetime differently: events that are simultaneous in $S$ lie along horizontal lines, while events simultaneous in $S'$ tilt at angle $\theta = \tan^{-1}(v/c)$ . The key insight is that the $x'$ -axis (where $t'=0$ ) and $t'$ -axis (where $x'=0$ ) must satisfy the Lorentz transformation equations, and their symmetric tilt around the $45°$ light-cone bisector reflects the invariance of the spacetime interval. This geometric symmetry—equal angles for both axes—emerges directly from how relativity treats space and time as unified dimensions.

Intuitive explanation

Related theory

spacetime-diagrams

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