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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.)

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    $\begingroup$ The notion of forces is absent in quantum mechanics. $\endgroup$
    – Archisman Panigrahi
    Commented Jun 5 at 0:53

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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 $g_{\mu\nu,1}+g_{\mu\nu,2}$.

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  • $\begingroup$ The second Newton's law does not imply the superposition principle of forces. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Jun 5 at 7:33
  • $\begingroup$ @GiorgioP-DoomsdayClockIsAt-90 Then what does? Or rather, why does the second law not imply the additivity of forces? Can you explain? $\endgroup$
    – Confuse-ray30
    Commented Jun 5 at 10:14
  • $\begingroup$ I had to write an answer. The space for comments was not enough. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Jun 5 at 16:47
  • $\begingroup$ I wouldn’t explain things in this way. Just because Newton’s second law says $\mathbf{F}=m\mathbf{a}$ a-priori is not a reason for why the $\mathbf{F}$ appearing here ought to be the vector sum $\sum_{i}\mathbf{F}_i$ of ‘all forces acting on the particle’. A-priori it might be conceivable that different forces contribute in a different manner to give the net force. That this is not the case is an extra assumption (sometimes explicitly mentioned, sometimes only implicitly), and is ultimately backed up and justified by experiments (and that is the true decider of course). $\endgroup$
    – peek-a-boo
    Commented Jun 5 at 17:02
  • $\begingroup$ See my answer to Why is Pythagorean theorem applicable to forces? for a somewhat related matter. $\endgroup$
    – peek-a-boo
    Commented Jun 5 at 17:04
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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.

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  • $\begingroup$ But that is because your fields actually change the source strength! The point is, that given the charge distribution (including the dipoles), the forces (electric fields) are additive. The fact that taking away some of the sources results in the other sources changing is not the point of the discussion. $\endgroup$
    – Confuse-ray30
    Commented Jun 5 at 16:57
  • $\begingroup$ I admit I should have worded my answer from a different perspective: Einstein gravity has a Levi Cevita connection which in turn is NON LINEAR in the metric. And the resulting geodesic equation thus does not depend linearly on two fields, and effectively means what I then stated at the end: One cannot add two fields together. This is in contrast to Maxwell equations, which are all linear, and so actually fields are additive. $\endgroup$
    – Confuse-ray30
    Commented Jun 5 at 16:59
  • $\begingroup$ @Confuse-ray30 If there is any meaning in stating the superposition principle as a different principle than the vector character of forces is precisely in the fact that some forces can fail to satisfy it. If the two principles were to coincide, there would be no room for different names. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Jun 5 at 21:34
  • $\begingroup$ I get your point. What you are saying is that including backreactions on the source makes the maxwell field equations non-linear. But I would define additivity not in terms of variant sources, but rather fixed. And then the additivity comes solely from the linearity of the equations in the force. This is not the case for GR, which even in the absence of backreactions are non-linear. $\endgroup$
    – Confuse-ray30
    Commented Jun 6 at 6:41
  • $\begingroup$ @Confuse-ray30 Yes, I agree. The superposition principle is just a different name for linearity. GR equations are structurally much more non-linear with respect to the sources than Newtonian equations of motion. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Jun 6 at 8:51

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