The inverse triangle inequality

In Euclidean geometry the well-known triangle inequality holds: in a triangle the sum of two sides is always greater than the third. Equivalently, a broken line is always longer than the straight line joining the same endpoints. It is the geometric statement that “the shortest path is the straight one”.

In special relativity the exact opposite happens. For intervals of proper time $\Delta\tau$, measured along the motion of a traveller, a broken trajectory is always shorter than a straight one. The reason lies in the minus sign of the invariant: whereas Euclidean distance sums the squares, proper time subtracts them,

$(\Delta\tau)^2 = (\Delta t)^2 - (\Delta s)^2,$

so that every stretch of spatial motion $\Delta s$ removes proper time rather than adding to it. The relativistic invariant behaves like a “negative distance”, and the triangle inequality is reversed: the broken path loses proper time at every deviation.

Concretely: if a traveller goes from an event $E_A$ to an event $E_B$ (connected by a time-type interval), they can do so in a straight line or by breaking the path into intermediate stages. Summing the proper times of the individual legs — each computed with the invariant — always gives a total smaller than the proper time of the direct trajectory. The straight line maximises; every broken path shortens it.

Principle — Maximisation of proper time

Between two events $E_A$ and $E_B$ connected by a time-type interval, the uniform straight-line trajectory is the one that maximises proper time. Any deviation from it — that is, any change of reference frame — shortens the proper time experienced by the traveller.

This is the deep reason behind the twin paradox. Of two twins, the one who follows a broken trajectory (leaves, then reverses course, i.e. changes reference frame) ages less than the one who stays at rest in a single inertial frame. Hence the golden rule: “the twin who stays put is always the older one”. Whoever never changes reference frame lives the maximum possible proper time between two events; every acceleration, every deviation, is life-time subtracted.

Links

Topics: Special relativity Concepts: Spacetime invariant · Time dilation Skills: Changing reference frame Methods: Minkowski diagram

Related exercises: Mr Rossi goes to the theatre · Problem — The reference frame of minimum time · The rabbit race