4-force

Classes in relativity usually avoid discussing force even though every physics student started in non-relativistic physics with . It is not immediately obvious if this is still true under relativity and if not what is the correct formula. Of course, discussion of regular 3-vector in relativity is not very convenience. It is more natural to construct relativistic (i.e. covariant) relations using 4-force , 4-momentum , 4-acceleration , 4-velocity , and with proper time . There is a problem: should we define 4-force by acceleration or momentum? .

Momentum is more fundamental than acceleration, so:

We still want Newton's law to hold in some sense. Consider the collision of two particles. Newton's third law would read like this:

But this is strange because it means both particles have the same proper time even though they might be travelling with very different velocities.

Using above definition, we have:

with relativistic 3-force

Using the identities for 4-velocity and :

gives the proper rate at which a force is changing the rest mass of the particle. If a force preserves rest mass , the term in the parenthesis above:

This is just the work-kinetic theorem in Newtonian case: energy increment is purely kinetic. Force is necessarily mass preserving in Newtonian physics. There is no such restriction in relativity.

Moreover, mass is no longer so sacrosanct in relativity as in Newton. For rest-mass preserving force, we have,

So in general, while (3-)force, velocity, and acceleration are coplanar, but force and acceleration are not parallel to each other. If we break up force components parallel and perpendicular to velocity, we have:

So we now have a dated but still useful concept of "longitudinal mass" and "transverse mass" . So force has different effects when it is exerted parallel or perpendicular to motion.