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Invariant Interval

Consider two events P₁ and P₂ occurring in space-time characterized by co-ordinates (x₁, y₁, z₁, t₁) and (x₂, y₂, z₂, t₂) in a Lorentz frame S. The space-time interval s₁₂ between these two events is defined as,

s₁₂ = {c² (t₂ – t₁)² – (x₂ – x₁)² – (y₂ – y₁)²– (z₂ – z₁)²}^1/2

If the same two events are characterized by two co-ordinates (x₁’, y₁’, z₁’, t₁’) and (x₂’, y₂’, z₂’, t₂’) in another Lorentz frame S’, then corresponding space-time interval is

s₁₂’ = {c² (t₂’ – t₁’)² – (x₂’ – x₁’)² – (y₂’ – y₁’)² – (z₂’ – z₁’)}^1/2

Using Lorentz transformations, it is simple to check that

s₁₂’ = s₁₂

That is, space-time interval between two events is an invariant under Lorentz transformations.

If the two events are infinitesimally close together both in space and time, then the interval ds between them is given by

ds² = c² dt² – dx² – dy² – dz²

= invariant

Denoting spatial distance between two events by l₁₂ and time interval by t₁₂, we rewrite,

If > 0, or ct₁₂ > l₁₂, the interval is called time-like interval; if < 0, or l₁₂ < ct₁₂, it is called space-like interval; lastly, if s₁₂ = 0, or l₁₂ = ct₁₂, it is called light-like. The invariance implies that two events (or points in space-time) separated by a time-like, or space-like, or light like interval in one inertial frame, are separated by same kind of interval in all other Lorentz frames (i.e. for example s₁₂ < 0 implies s₁₂’ < 0).

Time-like interval: Two events are called time-like if the interval between them is real, i.e. > 0. Hence, it is possible to choose a Lorentz frame in which l₁₂’ = 0, so that

S₁₂’² = c² t₁₂’² > 0

That is, two events separated by a time-like interval in one frame may appear to occur at the same space-point when observed from another (a particular) frame, though separated in time. There exists no Lorentz frame where these events can occur simultaneously (i.e. t₁₂’ = 0, because that would mean s₁₂’² = – l₁₂’² < 0, which contradicts interval invariance since > 0).

Obviously, two events can be causally related only if they have time-like interval, so that they are never seen to be occurring simultaneously. The concept of earlier and later in time has an absolute meaning for time-like events.