Why do we have to revise the definitions of momentum and force in special relativity?

Why do we have to revise the definitions of momentum and force in special relativity?

I am going to downplay the theoretical reasons, because they aren’t what’s important.

What’s important is that the Newtonian definition of momentum is useless because experimentally in relativistic particle collisions it isn’t conserved. When you change the definition to $\mathbf{p}=\gamma m\mathbf{v}$, this relativistic momentum is conserved. So it is a much more useful definition!

It is also then the spatial part of a Lorentz four-vector, which makes it easy to write covariant equations that look the same in all Lorentz frames, but that’s less important. Theories are constructed to explain experiments, and have no a priori justification.

Conservation of momentum is more fundamental than what happens when you change reference frames. It holds in all reference frames under both Galilean transformations and Lorentz transformations, as long as momentum is appropriately defined. Conservation of momentum arises from spatial-translation symmetry, and both Newtonian space and Minkowskian spacetime have this symmetry.

Forces (and here I mean three-forces) don’t get redefined in relativity. For example, the Lorentz force is the same whether you are considering non-relativistic or relativistic motion. Newton’s Second Law — which doesn’t define force but rather tells you the effect of a force — changes so that a three-force now tells you how fast the relativistic three- momentum (with the $\gamma$-factor) changes. Why? Most importantly, because this agrees with experiments!

Let us propose a 3-vector quantity of the form $$ {\bf p} = m f(v) {\bf v} $$ where $\bf v$ is the velocity of a particle, $m$ is an invariant scalar (a property of the particle) and $f(v)$ is some function of speed, to be discovered. Let us then propose that this quantity $\bf p$ is conserved in a two-body collision: $$ {\bf p}_{A,i} + {\bf p}_{B,i} = {\bf p}_{A,f} + {\bf p}_{B,f} $$ where $A,B$ refer to two bodies, and $i,f$ stand for "initial, final". It can be shown that, in order that this equation be respected in all inertial frames, the only possible functional form for $f(v)$ is $$ f(v) = \frac{1}{\sqrt{1 - v^2 / c^2}} $$

This is not the only way to arrive at relativistic momentum, but it is a conceptually simple way. In the proof one makes use of the Lorentz transformation as it affects velocity.

You can see from this that the quantity defined this way is the only vector quantity depending only on mass and velocity, and having the same direction of velocity, that can possibly satisfy a conservation law without running contrary to the two basic postulates of relativity (principle of relativity and light speed postulate). Those two postulates do not themselves say that momentum is conserved, but they do say that if something momentum-like is conserved, then this is the only possible formula for it.

One can then go on to consider inelastic collisions, and deduce the conservation of $\gamma m$. One then defines energy as $E = \gamma m c^2$. For zero speed this reads $E = m c^2$ and this is one way to derive that famous formula (one defines energy as that scalar quantity related to motion which is conserved in collisions, and one sets the fixed proportionality factor by having the result agree with Newtonian kinetic energy plus a constant in the low-velocity limit).

For starters, your question can be reduced to "why do we need to revise momentum" only, because force can be defined in terms of momentum via

$$\mathbf{F} = \frac{d\mathbf{p}}{dt}$$

The reason we are interested in force as a measure of the changes in momentum is because, seen from this perspective, it is interaction that is the more basic concept, and one kind of interaction between objects is for them to cause a change in their momenta. Such an interaction can then be quantified by measuring how much the momentum changes, which is what force (and for momentary interactions, impulse) refers to.

So why do we need to re-define momentum? The answer to this is that, for the idea of an altered spacetime geometry to be useful, we must have some sort of "play" that occurs between the spacetime geometry and the dynamics defined on it, so that there is a logical relation between the two and the form of one cannot be changed without impacting that of the other. This means that the different spacetime geometry must cause a different dynamics to prevail, and so momentum, as a dynamical quantity, should be expected, on that dint alone, to have to change. Explaining the reason it changes to this particular form takes a bit more work, however.