Parabolic metric on space-time

The parabolic metric on $\mathbb R^n\times\mathbb R$ is

d_p((x,t),(y,s))=\max\{|x-y|,\sqrt{|t-s|}\}.

The triangle inequality follows from the Euclidean triangle inequality and $\sqrt{a+b}\le\sqrt a+\sqrt b$ . The map $(x,t)\mapsto(\lambda x,\lambda^2t)$ multiplies this distance by $\lambda>0$ .

Balls and scaling

A ball of radius $r$ is $B_r(x)\times(t-r^2,t+r^2)$ . Its space-time Lebesgue volume is a dimensional constant times $r^{n+2}$ . A backward cylinder uses $B_r(x)\times(t-r^2,t)$ ; it respects the same space-time scaling but is not a metric ball centered at $(x,t)$ .