New Transformation Equations and the Electric Field Four-vector

1. Time Dilation from the Spacetime Interval

We can use the invariance of the spacetime interval to compare time measurements between two events in two reference frames in uniform relative motion. Let a primed frame of reference be in uniform motion with four-velocity relative to an unprimed frame. The spacetime interval between any two events will be the same in both frames, therefore

where is the Kronecker delta. If, in the primed frame, we have two events which are separated only by a time interval, in other words they happen at the same place, we get

The interval , in this case, is the proper time, since the space interval between the events in the primed frame is zero, so , and therefore

Dividing both sides of Eq. (4) by we have

Remembering , now, that the components of the four-velocity of the primed frame are , we can write

and, since , we have

We see from Eq. (9) that the time interval between the two events, as measured by an observer at rest in the unprimed frame, is less than the time interval between the two events as measured by an observer at rest in the primed frame. This is the same result we obtained in Eq. (13) of "Additions: August 17, 1999" on this website, with the situation reversed. That is, the proper time, here, is the time measured in the primed frame, rather than in the unprimed frame, and the components of the spacetime vector are replaced by the coordinate differentials. We have also included, here, all spatial components of the four-velocity in Eq. (9). Therefore, we conclude that, as in special relativity, time in a moving frame appears to run slower. This effect is, of course, reciprocal.

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