Précis of epistemology/Spatio-temporal measurement devices

A rigid ruler, or a compass, makes it possible to measure the distance between its extremities by comparing it to other distances. A rigid ruler is therefore a device for measuring a length, or a spatial interval. The important point is that the ruler is rigid, so that the measured distance is always the same. Spatio-temporal measurement devices can be designed on the same model. A simple clock is such a device. It allows to point two events which are always separated by the same duration. One can also use two clocks fixed to the same rigid support, each being used to point a certain event. If the device that triggers the two clocks is regular, the spatio-temporal interval between the two pointed events can always be the same. Such devices make it possible to measure spacetime in the same way that rigid rulers make it possible to measure space.

An interval is timelike if it is on the path of a massive material point. It is lightlike if it is on the path of a ray of light in vacuum. All other intervals are spacelike. When an interval is spacelike, there is always an inertial frame for which its extremities are simultaneous events. Spatio-temporal measurement devices measure the three kinds of intervals. A single clock is sufficient to measure timelike intervals. Two clocks fixed on a rigid ruler and suitably synchronized make it possible to measure spacelike or lightlike intervals.

All lightlike intervals are equal

The theory makes it possible to assign a real number to all spatio-temporal intervals, timelike, lightlike and spacelike, or more precisely to their square. Timelike intervals in particular are all equal to zero and therefore all equal to each other. This is a priori surprising. This means, for example, that the spatio-temporal interval between the emission of a photon and its reception three meters away is equal to the interval between its emission and its reception three light-years away, as if the advancing photon never departed from its starting point. Isn't it wonderland?

Spatio-temporal measurement devices and the slowing down of clocks allow to understand this counterintuitive result. Imagine a rocket launched towards a star three light-years from Earth. A photon is emitted at the rear of the rocket and received at the front, three meters away, from the point of view of the rocket, so a tiny fraction of a second later. But because of the slowing down of clocks, what lasts a tiny fraction of a second from the point of view of the rocket can last three years from the point of view of the Earth, provided that the rocket is fast enough. The same spatio-temporal device measures both a lightlike interval of three light-years and a lightlike interval of three meters. It thus establishes their equality.

Relativistic wheels