We have previously seen that the velocity does not transform from one reference frame to another using the standard Lorent transformation to 4-vectors, simply because the velocity is not a 4-vector. Similarly, the force is not one either (nor is it part of one, such as the 3-momentum). We therefore have to develop the transformation rule.
Let us develop the transformation of force from a system S
where a particle is moving at a speed v
to any system S'
which is moving relative to system S
at a speed V
Calculating the transformation is very similar to the transformation of velocities. We begin with the momentum transformation (and not the coordinate transformation as we did for the velocities). We have:
with β and γ defined using the coordinate system velocity V
Using the chain rule, we can write:
However, we have from the Lorentz transformation for the time, that
For the x component, we have (again, using the Lorentz transformation) that:
We have seen, however, from the definition of force, that dE/dt
, and thus
To summarize, the complete transformation is:
If the velocity at S
vanishes at a given moment (namely, that S
is the rest frame of the particle), then v
=0, and we have
is the system moving relative to the particle and S
is the rest-frame).