Required math: calculus
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 3; Problem P3.1.
We’ve already looked at the four-velocity in special relativity, but it’s worth a second look from a different angle. We can instead define the four-velocity in terms of two events separated by an infinitesimal spacetime interval . The four-velocity is defined as the derivative of with respect to the proper time , so that
Since the components transform using the Lorentz transformation, then so do the components of the four-velocity.
Since the spacetime interval is invariant (it has the same value in all inertial frames) the relation
holds in all inertial frames. In particular, it holds in the observer’s rest frame, in which , so we have
In this rest frame, then, we get
The square of is then
where is the metric used in special relativity:
Since this is true in the rest frame and the square of a four-vector is an invariant, it is true in all frames.
As an example, suppose we have an object that moves along a worldline given by (in some inertial frame)
( and is a constant). The component of the four-velocity is then
Using we can find the component:
using the identity . From this we can get the time in the inertial frame:
The velocity of the object as seen in the inertial frame is
Since tanh is bounded by , the velocity never exceeds 1, so never exceeds the speed of light.
We can invert the relation between proper time and inertial time to get
Using the relations (derived from )
Again, note that and also that as , so again the velocity remains less than .