There are plenty of similar topics about the physical or intuitive meaning of the equivalence principle or statement such as ''laws of physics are true in every frame of reference''. Usually answers rely on introducing tensors, covariant-derivatives and invariant formulations. While of course being perfectly correct, that technical style pretty much kills the intuition. So, I pose the challenge of explaining it more heuristically, what might mean that F=ma is true in an accelerated coordinate system.

In Newtonian case, we must add ''fictitious forces to make F=ma work in an accelerating frame, and such addition ''changes the laws of physics''. That is why Newton's physics do not fulfill Einsteins requirement ''the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion''. In GR Einstein accomplished this and now I ask what it means intuitively. How, in other words, does GR make these fictitious forces, the effect of which will surely still be there, real?

In case you are tempted to dodge the question and argue that F=ma is not a physical law because it is not formulated in an invariant way etc. I try to clarify.

  1. For Maxwell's equations there is a beautiful explanation how the electric and magnetic fields change into each other, when jumped from one inertial coordinates into another. And you can in principle explain how it happens simply by considering how the charge distributions with respect to the other system. In some simplistically symmetric cases (e.g. current carrying wire) if is very intuitively tractable. That is the style of explanation I am looking for, but here in the case of accelerated coordinates.

  2. F=ma is true at least approximately and in some sense. And as long as you are not near a black hole the spacetime is sufficiently flat so that you can consider the effects or gravity as a force (via linearized field equations, or, gravitoelectromagnetism). So it will not be necessary to introduce spacetime metrics here.

I suspect this question may still not be ''clear'' but be patient and do not vote it closed. I will give it a shot myself if I get no other candidates. Thanks.


closed as unclear what you're asking by DilithiumMatrix, Bill N, Gert, user36790, Hritik Narayan Dec 16 '15 at 4:58

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    $\begingroup$ I'm not sure what you're asking. $F=ma$ is only true in accelerated frames if you add fictitious forces designed to make it work, e.g. the centripetal force, but then it is tautologically true. You explicitly say inertial system when talking about Maxwell's equations, why are you comparing this to trying to explain $F=ma$ in non-inertial coordinate systems? $\endgroup$ – ACuriousMind Dec 15 '15 at 15:15
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    $\begingroup$ Comment to the post (v1): It is a bit unclear if the question essentially is 1. Is $F=ma$ true in GR? 2. Is $F=ma$ true in Newtonian mechanics in an accelerated ref. frame? 3. Is $F=ma$ true in post-Newtonian mechanics?, or 4. something fourth. $\endgroup$ – Qmechanic Dec 15 '15 at 15:34
  • $\begingroup$ Voting to close as unclear what you're asking. A good question usually has a question in it. A question mark is also a good start. $\endgroup$ – DilithiumMatrix Dec 15 '15 at 18:04
  • $\begingroup$ I think your question is: 1) Joe applies a force=ma to an object, but he is in a frame boosted wrt me. What do I see done to the object ? (similar to E then looks like a combination of E and B) 2) Same question, except Joe's frame is not boosted rather it is accelerating wrt me. 3) Once you answer 1+2 using tensors, explain the answers intuitively as can be done for E and B using how the charge current changes under boosts. Am I close to what you want? The transform to an accelerating frame is interesting, but I'm not sure how to do it. $\endgroup$ – Gary Godfrey Dec 15 '15 at 21:17
  • $\begingroup$ @Gary Yes you are close to what I mean. Thank you for the clarification. $\endgroup$ – AnssiM Dec 16 '15 at 10:29

As of what I know F =MA is the equation that predicts the change in velocity of any object based upon its mass and the forces applied in it. But it is known that this equation is not true in all reference frames. So this rule cannot be applied to all the reference frames. The RF(reference frame) in which this equation and other physical laws are valid is called an inertial RF.

Now if I assume that the space is Euclidean then it turns out that all the RFs which are accelerated with respect to an inertial RF are non inertial. But if the space is non Euclidean then this is not true.

  • $\begingroup$ "But it is known that this equation is not true in all reference frames" In Newtonian, yes. In GR it, or the law for which it should be an approximation, must be. I am asking if anyone can explain how. It should be doable with ''Gravitational fields'', if we restrict ourselves in near-Newtonian limit. $\endgroup$ – AnssiM Dec 17 '15 at 11:53

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