Why are forces superimposable in Classical Mechanics? Does this also apply in higher theories like General Relativity and Quantum Mechanics?

Question

In classical mechanics, forces are treated as vectors and are added linearly. Is this principle to be treated as an axiom or is there some underlying principle from which this is derived? And given that general relativity deals with the curvature of spacetime, do these gravitational effects simply superpose on each other, or do they add up in a non-linear fashion?

Add up question: What if these dark matter interactions (all I know about dark matter is from documentaries and YouTube (particularly Veritasium) videos) we speak about are because of some uncanny additive nature of gravity?

(I am just a high school student; any references or suggestions would be very helpful.)

Answer 1

Since Newton's laws are treated as axioms in classical mechanics and the forces act linearly in the second Newton's law, from that perspective, it is axiomatic that forces are additive. But of course, it's physics, not maths.

In general relativity, the field equations for gravity turn out to be non-linear. So if some matter (read energy) distribution $T^{\mu\nu}_1$ sources a gravitation field $g_{\mu\nu,1}$ in isolation and simlarly for "2", then the total field will NOT be $g_{\mu\nu,1}+g_{\mu\nu,2}$.

Answer 2

The vector nature of the forces implies the possibility of summing them according to the parallelogram rule.

The principle of superposition of forces states that a single resultant force has the same effect as the sum of the individual forces acting on a body.

Therefore, the superposition principle assumes but is not equivalent to, the possibility of a vector sum of forces. The superposition principle states that if we partition the set of bodies interacting with a given object into disjoint subsets and we measure/evaluate the partial force of each subset on the body in the absence of the other subsets, the vector sum of these partial forces is equal to the net force measured/evaluated when all the subsets are present. Such a property is different from the property of forces of being vectors. It concerns how the vector force on a body depends on the other body's dynamic state and physical properties.

A couple of examples could clarify such a point.

A typical situation of validity of the superposition principle is that case of the forces between point-like charges. Each pair interacts with Coulomb's force, and the net force on a charge is the vector sum of the pair-wise Coulomb contributions.

However, if we add the possibility of an induced electric point-like dipole to the point-like charges, the situation is quite different. In general, the induced dipole moments when $N$ of such bodies are present are not the same as in the presence of only a pair at the time. Then, even though the net force on a body remains the vector sum of all the monopole plus dipole force vectors, it is not true that each contribution is the same as without the other bodies.

The cases of General Relativity and Quantum Mechanics may show some analogy, but in both cases, the concept of force is abandoned. Therefore, the superposition principle needs to be reformulated for other physical quantities.