# How does Newtonian gravity violate law of inertia?

I came across a line on http://physics.ucr.edu/~wudka/Physics7/Notes_www/node49.html

" If the experiment is tried in vacuum any two objects when released from a given height, will reach the ground simultaneously (this was verified by the Apollo astronauts on the Moon using a feather and a wrench). This result is peculiar to gravity, other forces do not behave like this at all. For example, if you kick two objects (thus applying a force to them) the heavier one will move more slowly than the lighter one. In contrast, objects being affected by gravity (and starting with the same speed) will have the same speed at all times. This unique property of gravity was one of the motivations for Einstein's general theory of relativity"

This does not make sense to me. True, that different masses will accelerate at same rate due to the force of gravity, but that is because different magnitudes of forces are acting on them (a larger force on the heavier body and a small force on the lighter one, from the formula of gravity.)

So how does it violate inertia? It should have only violated inertia if the force acting on bodies of different masses was the same,and yet the acceleration was the same. What points am i missing? Can someone please explain how it is true and how it led to the General theory of relativity?

Newtonian gravity is not violating the law of inertia (and the article does not say that it violates) - the law of inertia just states that you need force to accelerate things.

Given a gravity field, and assuming inertial mass $=$ gravity mass, then mass of a falling object simply cancelled out in the equation of motion:

$$m_{inertial}\ddot x =G \frac{m_{gravity}M}{x^2}$$

$$\ddot x =G \frac{M}{x^2}$$

Hence, the inertial mass of a falling object doesn't play any roles in the time evolution of any falling objects.

However, such a phenomena is different to other forces, e.g. electric force:

$$m_{inertial}\ddot x =k \frac{qQ}{x^2}$$ $$\ddot x =k \frac{qQ}{m_{inertial}x^2}$$

There is no way to get rid of the inertial mass. The position of the charge always has something to do with its inertial mass in this case.

• "This unique property of gravity was one of the motivations for Einstein's general theory of relativity" what is so unique in this observation? – spatialdelusion Oct 21 '17 at 6:37
• @spatialdelusion What is so unique is that gravity is the only force for which this is true. – tfb Oct 21 '17 at 9:41
• The remarkable thing here is that the inertial mass always equals the gravitational mass. In Newtonian mechanics there's no fundamental reason that this should be true. – Harry Braviner Jun 12 '20 at 12:32

Nothing in that quote is saying that gravity "violates the law of inertia." It's just an interesting property of gravity, and led Einstein to consider some thought experiments involving elevators, which led to the development of general relativity. Namely, that if you had a physics lab in an elevator, that there was no experiment to could run that would determine if that elevator were sitting on the surface of the earth under the influence of gravity, or uniformly accelerating through space at an acceleration of $g$.

• ok, but in a gravitation field of earth objects will fall towards one point which would not be the case in an accelerating frame, so how is it indistinguishable – spatialdelusion Oct 21 '17 at 5:55
• @spatialdelusion Good question! The thought experiment assumes a uniform gravitational field. Either a gravitational field that is genuinely uniform, as would be produced by an infinite plane of uniform density, or else an "elevator" that is small enough that variations in acceleration are too small to measure. Effectively that "acceleration" and "gravity" are locally indistinguishable, not necessarily globally – Chris Oct 21 '17 at 5:58
• @cris thanks, but can uniform gravitation fields even exist practically? – spatialdelusion Oct 21 '17 at 6:13
• @spatialdelusion Perfectly uniform? No. So close to uniform you can't tell the difference over a small area? Sure. It's not really relevant anyway- the point was to postulate an equivalence between gravity and acceleration locally- meaning in an arbitrarily small volume. This has profound significance: for instance, it implies that light rays bend in gravitational fields, and that clocks run slower on earth than in space: because both of these things are true in an accelerated reference frame in the absence of gravity. – Chris Oct 21 '17 at 6:29