Why is Gravitational force proportional to the masses?.

Question

We know that two mass particles attract each other with a force

$$F~=~\frac{G M_1 M_2}{r^2}.$$

But what is the reason behind that? Why does this happen?

Answer 1

One could explain "well, gravity is the curvature of spacetime due to the mass-energy". But that would only lead to "well, why does mass-energy curve spacetime?" And, should someone produce a proposed answer to that, the follow-up question would have to be "but why is that so?" etc.

At some point though, one must accept that there are genuine fundamentals, genuine primaries that cannot be explained in terms of something "more" fundamental, "more" primary.

Gravity is considered one of those fundamentals. But the question "what is the reason for gravity" presumes that gravity isn't fundamental. So, the only proper "answer" to your question is "to the best of our knowledge, gravity is fundamental".

Answer 2

There is actually a really great reason behind this! When trying to find the "root" of a concept its really useful to go back to the origins of the concept when someone had to prove it to their peers.

This is along the lines of Newton's reasoning:

1. Suppose there is an attractive force between masses
2. By conservation of momentum the force must be equal
3. Suppose you have two masses $M_a$ and $M_b$, which are each composed of matter in the size of $a$ and $b$ units of mass. (first one 5kg second one 3kg)
4. Every Piece in $M_a$ must exert an attractive force in every Piece in $M_b$, so the proportionality of the attractive force is with $a \times b$ (Each kg in the 5kg must "pull" on each kg in the 3kg, so there is 15 total "pull"s happening)
5. Every Piece in $M_b$ must exert an attractive force in every Piece in $M_a$, so the proportionality of the attractive force is with $a \times b$ (Each kg in the 3kg must "pull" on each kg in the 5kg, so there is 15 total "pull"s happening)

So we can conclude; if there were to be a force between masses, then that force $\textbf{must}$ be equal on both masses and proportional to the product of their masses.

Newton then assumes the inverse square law, and demonstrates that his model of Gravity is able to describe all of Keppler's Laws and comet observations. (This is probably one of the coolest proofs in Physics and was done Geometrically, not with Calculus as its often thought; check out this playlist for an outline of the proof https://www.youtube.com/playlist?list=PLaTW6Ae6KGGWQ5c_7MzLwIK77zLGpv12C )

Answer 3

The masses in that equation play two different roles. Depending on which object you are interested in, one is a mass producing the field, and one is a mass experiencing the gravitational field.

The acceleration of an object due to a force is the force divided by the inertial mass of the object. Take that force and divide by M1. Then you get the acceleration of M1. You get the effect of the gravity of M2 on M1. In this case M2 is the active gravitational mass. It's responsible for the force acting on M1. Anlayzing the motion of M2 results in the same analysis, only then M1 is the active mass and M2 is the passive, or inertial mass.

Now gravity isn't a force, it's a pseudo force due to the curvature of space time. Like any pseudo-force, its proportional to the inertial mass of what it acts upon. The Equivalence Principle is helpful here. The curvature of space time is due to the presence of mass-energy, on small scales roughly proportion to the active gravitational mass.

So why do masses effect each other that way?

M1 produces a space-time curvature proportional to M1. M2 experiences that pseudo-force in proportion to its inertial mass. By the same analysis a symmetric effect produces the change in motion of M1.

As to why mass-energy causes space-time to curve, I'm not sure. I think we'd need a quantum theory of gravity for that.