The Minkowski invariant

In Galilean relativity the time interval $\Delta t$ and the spatial distance $\Delta s = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$ between two events were separately invariant, i.e. the same in every reference frame. Time dilation has already shown us that in special relativity this is no longer true: neither $\Delta t$ nor $\Delta s$ is conserved when passing from one RF to another. And yet not everything is relative — there exists a precise combination of space and time that all observers measure identically.

Principle — Space-time invariant

If two events $E_1$ and $E_2$ have, in some RF, a time difference $\Delta t$ and a spatial difference $\Delta s$, the quantity $\ev{(\Delta\tau)^2 = (\Delta t)^2 - (\Delta s/c)^2}$ has the same value in all inertial reference frames. This is the Minkowski invariant.

The minus sign is what distinguishes space-time from ordinary geometry: it is not Pythagoras’ theorem (where the squares are added), but a “hyperbolic” version of it in which time and space enter with opposite sign. The sign of the result classifies the relationship between the two events.

If $(\Delta\tau)^2 > 0$ the interval is timelike: there exists an RF in which the two events are co-spatial (occur at the same point), and in it $\Delta\tau$ is precisely their proper time. These are events that can be causally connected, because a signal slower than light can go from one to the other.

If $(\Delta\tau)^2 < 0$ the interval is spacelike: there exists an RF in which the two events are simultaneous (but not co-spatial), and their spatial distance in that RF is the proper length. These are events too far apart in space and too close together in time for light to have time to connect them: neither can influence the other.

Links

Topics: Special relativity Concepts: Space-time invariant · Time dilation Methods: Minkowski diagram

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