The invariant interval

Is there anything on which all observers agree?

Going back to Equation 15.3, we recall that x2 + y2 + z2 - c2t2 = 0 represented the special case of an interval between two events, which happened to be connected by the passage of a light signal.

Event 1 was the emission of a light signal from the origin of both frames as the two frames coincided. Event 2 was the same signal arriving at a certain point in space and time. The coordinates of that second event were x, y, z, t in frame S and x', y', z', t' in frame S'. The values of both space and time coordinates were different in the two frames, but the value of the expression remained always the same.

The invariant interval in space-time

It is not difficult to show that, by transforming the coordinates as before using the Lorentz transformation, we can write a more general expression for the interval between any two events not necessarily connected by the passage of a light signal. In general its value is not zero, but whatever the value, it is the same in all frames of reference. All observers agree on the value of the expression Ax2 + Ay2 + Az2 - c2t2, which is the interval in space and time (space-time) between any two events — which in more technical language is described as being invariant under a Lorentz transformation.

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