Required math: calculus
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 3; Problem P3.2.
Another example of four-velocity in special relativity. We start with an object whose velocity (3-d velocity, that is) in an inertial frame is
where is a constant with relativistic units of , and is the time measured in the inertial frame (so it’s not the proper time of the object). The suffix indicates the motion is along the axis, as usual.
The relation between a proper time interval and the time interval measured in a frame moving at speed along the axis with respect to the object is given by the time dilation formula
This gives us the time component of the four-velocity:
We can integrate this to get in terms of :
where is the constant of integration. If we require when , then and we get
From this we get
From the definition of four-velocity, we have
We can now find and as functions of :
using software to do the integral.