# Why do objects with mass have gravitational force that is proportional to their mass?

Why do objects with mass have gravitational force that is proportional to their mass, i.e the larger the object the more gravitational force it has?

• – AccidentalFourierTransform Mar 31 '16 at 16:40
• We don't know, but it is most likely one of the most fundamental equivalences in nature and probably linked to the microscopic structure of spacetime. – CuriousOne Mar 31 '16 at 18:30

Why is the force of gravity proportional to the mass of the body exerting it?

Just because.

This answer may disappoint you, but it's the truth. In Newtonian physics, the equation $$F = G \frac{m_1 m_2}{r}$$ is axiomatic, just like Newton's laws of motion. None of these is derived from more fundamental propositions, either.

So your question, while perfectly reasonable, is like asking, "why does a body in motion stay in motion unless a force acts on it?" Or, "why is the force that it takes to accelerate the body proportional to its mass?". Or, "why does every action have an equal and opposite reaction?" The answer is the same for all these axioms: That's just the way it is in Newton's theory. There is no "why". The buck of asking "why" stops the moment you hit an axiom of a theory.

You tagged your question "Newtonian Physics", but I want to add that a relativistic approach to it would not make the answer any more interesting or insightful. In Einstein' General Relativity Theory, one body creates a curvature in space-time that's proportional to its mass. The other body then 'feels' a force that's proportional to the curvature of space-time, and to its own mass. Why? Just because. These, too, are axioms, axioms of General Relativity theory in this case. so the buck stops, once again, before it ever gets going.

• There are no axiom police who decide what's an axiom. One person's axiom can be another person's theorem. – user4552 Jun 26 '19 at 2:12
• True, Ben, but Newton gets to decide what's an axiom in his theory of gravity, and Einstein gets to decide what's an axiom in his. So if you have a problem with the law of gravity being an axiom, I suggest you invent your own theory of physics with its own set of axioms. It would help, of course, if your theory's predictions about the real world be at least as accurate as Einsteiin's and Newton's. – Thomas Blankenhorn Jun 26 '19 at 6:38
• Thank you for your feedback, though! After thinking about it, I reworded my post to make clearer that I'm talking about the way particular physical theories work. – Thomas Blankenhorn Jun 26 '19 at 7:34

The gravitational force an object experiences is proportional to its mass because of intimate ties between gravity and acceleration. The mass of an object influences the gravitational field but it does not determine how the object moves through the gravitational field. Suppose we fix a mass $M$ at the origin. This generates a gravitational field $$g=GM/r^2$$ where $r$ is the distance from the origin and the field is directed towards the origin. If we place a test mass $m$ in this field the force is $$F=mg=GMm/r^2$$ and it looks like the mass $m$ is important for determining the motion of the test mass. However, notice that when we want to determine the motion of the test mass we need the acceleration and $m$ drops out $$a=F/m=GM/r^2.$$ In a deeper sense we can think about general relativity where the mass of an object determines the curvature of spacetime. However, the mass does not affect how the object moves through spacetime. Objects of different mass will just move in a straight line through curved spacetime without any regard for their mass differences (unless there is some other force, e.g., electromagnetic).

• You basically seem to be saying that passive gravitational mass is proportional to inertial mass because Newton says so, which is fine I guess, but then you seem uncertain about how to make the connection to active gravitational mass. One way of making such a connection is that if we didn't have active gravitational mass proportional to passive gravitational mass, we wouldn't have conservation of momentum. – user4552 Jun 26 '19 at 2:30
• @BenCrowell And a number of tests of the equivalence principle (both the weak and strong versions) have been done. The weak (Newtonian) version is good for 1 part in $10^{15}$ or better. – hdhondt Jun 26 '19 at 3:03

It's an empirical relation. Empirical relations are those relations which can't be derived but verified through experiments and observation. F=ma is also empirical.

If we ignore general relativistic corrections, then gravity itself does not contribute to gravity. This means that the gravity from each part of the object adds up to the gravity from the other parts. The gravity of an object is therefore proportional to the mass of the object.

• The OP clarified in the second part fo the sentence with " i.e the larger the object the more gravitational force it has?" So, it seems to me about the proportionality, rather than the dependence being on mass rather than something else. You could take that dependence as defining gravity. Obviously, if gravity were proportional to charge, then it would be an electromagnetic force :) . – Count Iblis Mar 31 '16 at 17:07
• This means that the gravity from each part of the object adds up to the gravity from the other parts. What does this mean? – user4552 Jun 26 '19 at 2:10
• @BenCrowell The superposition principle. – Count Iblis Jun 26 '19 at 15:29