For the <null coordinates in two-dimensional Minkowski spacetime>, direct substitution gives
$$
u'=x'+y'=e^\varphi u,
\qquad
v'=x'-y'=e^{-\varphi}v.
$$
Thus a boost dilates one null direction and contracts the other by the reciprocal factor. It preserves
$$
uv=x^2-y^2.
$$
The invariant curves are therefore the level sets $x^2-y^2=c$: the branches of hyperbolas for $c\ne0$, together with the two null lines when $c=0$. Each connected branch is preserved by the identity component.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 1 / c / Solution

{title2=$u=x+y,\ v=x-y$}

A boost of rapidity $\varphi$ acts on null coordinates by $u\mapsto e^\varphi u$ and $v\mapsto e^{-\varphi}v$, preserving $uv=x^2-y^2$.


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 Incoming links: Null coordinates in two-dimensional Minkowski spacetime