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In high school, it was taught that formula for describing circular orbital velocity around a central body is derived by equating Newton's law of gravity with the centripetal force formula (under the logic that the inwards centeipetal force required is provided by the gravitational "force").

It was only recently that I discovered that gravity isn't actually a force but is actually a distortion of space time. (I came across this while wondering why light bends around large masses). Does the fact that gravity is not a force make the above derivation of orbital velocity any less valid? Because the above derivation assumes that gravity is a force.

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You cannot longer use Newton's formalism $F = m a = -GMm/r^2$ if you introduce the fact that the geometry of space time is changed by the presence of the central body $M$.

It is true that the test mass $m$ still moves around $M$ because of gravity, but you should think of gravity not as a force any more, but as an emergent property of the curvature of space-time. Fortunately there's a whole body of mathematical tools that allow you solve this problem in particular.

Actually, it is one the most well known problems you can analytically solved using general relativity: the two body problem

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  • $\begingroup$ Thank you for the link to the two body problem, this has confirmed to me that the reasoning (equating centeipetal and gravitational force formula) as I described in my question is flawed. I'll upvote and tick this answer $\endgroup$ – user166520 Sep 14 '17 at 14:38
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TL:DR

Newton's law of gravity (used to describe velocity, and not force) is not wrong, it is just imprecise and has a limited scope for which it is accurate (I would not equate less widely applicable with less valid).


No, the fact that gravity is not a force, as Newton described it, does not mean that his calculations for describing circular orbital velocity are incorrect. Given that his formulae were derived based upon measurements of the same world as Einstein's formulae were, they are both designed to (and do a good job) describe the motion of large masses under the influence of gravity.

Newton's laws are, as you would expect given the time difference between Newton and Einstein, less precise than Einstein's, and they fail under particular circumstances, but overall, they do a pretty good job at describing a large chunk of gravitational effects at the precision that is necessary for a vast majority of application.

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gravity isn't actually a force but is actually a distortion of space time.

Some basics at the beginning. If you are sitting in a rotating carousel did you feel a centrifugal force? And could you imagine that an astronaut does not feel such a force despite his revolution around the earth?

So what is the difference between these two circular motions? Under the influence of masses any body (as well as light) follows a geodesic paths.


Some excursus: The curvature of this paths depends - for small masses in relation to the earth or other massiv bodies - only from the velocity with which the body moves. Since light moves in vacuum exclusively with the velocity c the local curvature of the space can be described unambiguitly on the basis of the speed of light. Than less the velocity of a body than more it's path is curved near a massive body. The straightest possible path one achieve with light.


Independent from the velocity with which you moves in space and even in the presence of massive bodies which makes your path a curve you don't feel any force. That is one reason why gravitation is not a force.

Does the fact that gravity is not a force make the above derivation of orbital velocity (Newton's law of gravity with the centripetal force formula) any less valid?

Considering the above said the equation $F = m a = -GMm/r^2$ is still the right equation

  • for bodies with slow motion and
  • of tiny mass in relation to the central body.
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It is true that (true) gravity arises out of the curvature of spacetime but that doesn't mean that the gravitational effect is not a force. All the aspects that are expected of a force are certainly possessed by gravitational effects. Sure, it is special in the sense that charge that causes gravitation is the same as the inertial mass, but that doesn't make the effects of the gravitation on a particle disappear. It just makes us realize what is the cause of this effect called the gravitational effect. Understanding the mechanism or reason behind the existence of the gravitational effect doesn't make it less of a force.

Understanding the mechanism of how the gravitational effect arises makes us realize that the gravitational effect is not generically described by Newtonian laws of gravity but Newtonian laws are true only in some limit (low speeds, low mass-energy-momentum density) of the generically true theory of gravity - Einstein's Theory of General Relativity. Thus, the calculations done using Newtonian gravity to make predictions about satellites going around the Earth are pretty much safe as long as the velocities of these satellites are small enough. But for some involved calculations (e.g. those of the synchronization of the GPS satellites), one inevitably needs to use Einstein's Theory of General Relativity.

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  • $\begingroup$ You invest your time and effort to give a better answer, but it doesn't get picked, because "correct" answers are picked by those who ask questions and therefore don't know what the answer is, much less what the correct answer is. So you don't get any up-votes and if you dare to get creative, you are down-voted without a comment. Gotta love this forum! $\endgroup$ – safesphere Sep 15 '17 at 8:04

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