**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.

### Like this:

Like Loading...

*Related*

Pingback: Four-momentum conservation | Physics tutorials

Pingback: Particle orbits – conserved quantities | Physics tutorials

Pingback: Schwarzschild metric with negative mass | Physics pages