Fundamentals of Time and Relativity

Spacetime Interval

  • Unlike the Euclidean distance function, the distance function of Minkowski space is not positive definite.
  • Spacetime intervals in Minkowski space can be uniquely classified into timelike, spacelike and null (lightlike) vectors.

Relativity theory places all events in the history of the universe in a four-dimensional manifold of spacetime points, Minkowski space, with a spatial-temporal geometry specified in eq. (4.1). This geometry makes a comparison of spacetime distances possible. Furthermore, it allows to distinguish between curved and straight lines and the computation of distances along arbitrary curves in four-dimensional spacetime.

The distance function (Δs) 2 defined in eq. (4.1), is a measure of separation between two events that are time and/or space separated in Minkowski space. It assigns real numbers to event pairs, but, unlike the distance function in Euclidean space, spacetime distances are not positive definite. One, therefore, classifies spacetime intervals according to their sign

4.2

(Δs) 2 >0,    (Δs) 2 <0,    (Δs) 2 =0

This places intervals between two events in spacetime into three exclusive categories, on which all observers agree because spacetime distance is an invariant:

  1. Timelike (positive squared length): two events are in principle connectable by a signal moving from one event to the other at less than light speed. There could be no reference frame in which the two events occur at the same time.
  2. Spacelike (negative squared length): there is no reference frame in which the two events occur at the same place, so they must occur at different places and be some spatial distance apart.
  3. Null (zero squared length): two events are connectable by a signal moving exactly at light speed.

The distinction between these categories has physical significance. Ordinary massive particles can never attain the speed of light. So, all events during their existence must lie on the path of a timelike curve. Null curves are reserved for photons and other massless particles travelling with the speed of light.

  • Curves traced by objects in Minkowski space are called worldlines, a naming introduced by Minkowski.
  • Unlike temporal durations and spatial distances, spacetime intervals are objective in the sense that the spacetime interval is not relative to any particular reference frame.