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I have been told that "inertial movements, or distinction between proper and coordinate acceleration don't have meaning out of GR".

But now I'm confused, I always though of these concepts in terms of classical mechanics, because neither of them require us to think about the limit of the speed of light, and they seem to be pretty common concepts in the field of navigation systems.

My question is: can the equivalence principle, the notion of geodesics, and proper acceleration be understood solely in the context of classical mechanics, that is, Newtonian mechanics enhanced by the field force concept? Or we necessarily need the theoretical framework of general relativity in order to understand those?

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3 Answers 3

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can the equivalence principle, the notion of geodesics, and proper acceleration be understood solely in the context of classical mechanics

Yes. The formalism that demonstrates this is called Newton Cartan gravity. It is identical to Newtonian gravity in terms of experimental predictions, but it uses a geometrical mathematical structure like GR.

The equivalence principle holds for any theory of gravity where the inertial mass is equal to the passive gravitational mass. This includes Newtonian gravity. It is made clear with the Newton Cartan formalism.

Proper acceleration is simply the acceleration measured by an accelerometer. So any theory that describes acceleration will need to include the concept of proper acceleration, regardless of whether or not it uses that name. Otherwise it would be experimentally falsified by accelerometers.

So the only tricky one is the concept of geodesics. Geodesics still exist under the Newton Cartan formalism, and they still serve the purpose of describing free fall trajectories. The calculation is more cumbersome because instead of having one metric there are two, a time metric and a space metric. The connection has to be compatible with both.

In the end you get a valid geometric formulation of Newtonian gravity. I rarely use it directly. I prefer to use the usual formalisms form doing calculations. However, because it exists it makes it perfectly valid to speak of things like geodesics and the equivalence principle in Newtonian gravity. Those concepts are not unique to GR nor are they somehow “owned” by relativity. I find that the existence of the Newton Cartan formalism makes formulating Newton’s first law conceptually easier. So being aware of it is advantageous.

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  • $\begingroup$ 'Otherwise it would be experimentally falsified by accelerometers.' According to Wikipedia, 'an accelerometer is a damped mass, a proof mass, on a spring'. In the Newtonian framework, the spring deflection measures the weight of the mass, when at rest at the earth surface. And the acceleration is zero because the velocity doesn't change. I think the word 'falsified' is exagerated. $\endgroup$ Commented Feb 14, 2022 at 13:27
  • $\begingroup$ @ClaudioSaspinski the word falsified is not exaggerated. If a theory describing accelerations cannot correctly predict the reading of an accelerometer then it is falsified by said readings. That is the whole point of experimental measurements. Whatever quantity the theory predicts for the reading of an accelerometer, that is proper acceleration, even if the term is not commonly used in the theory. Hence, Newtonian gravity is not falsified by non-relativistic accelerometers because its predictions of accelerometer readings, aka proper acceleration, are correct $\endgroup$
    – Dale
    Commented Feb 14, 2022 at 13:47
  • $\begingroup$ @ClaudioSaspinski, both classical Newton and the N-C formalism admit proper (absolute) time and proper acceleration, but not proper position or velocity! You cannot measure them—they require choosing a reference frame, and the Newtonian equivalence of certain coordinate systems makes all coordinates and velocities relative, but the time and all accelerations absolute. In the classical Newton theory, you can read an accelerometer but only deduce that either your velocity is changing or you are in a gravity field, or both. N-C postulates these two cases equivalent, this simple. $\endgroup$ Commented Feb 14, 2022 at 20:56
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    $\begingroup$ Hi Dale, thanks for the very good answer, I was not acquainted with the Newton-Cartan theory, and its awesome (everytime time I cross the name Cartan I find something amazing about physics). $\endgroup$
    – Arc
    Commented Feb 14, 2022 at 22:46
  • $\begingroup$ @kkm, I believe newtonian mechanics in its original formulation cannot correctly describe the reading of accelerometers when both field and contact forces are present: it has a single acceleration $a$, and a single mass $m$ and thus cannot distinguish between acceleration promoted by gravity and acceleration promoted by contact force. That's why there's so much confusion when people analyze the case of a mass standing on the surface of Earth: gravity force downwards + normal force upwards result in zero net force, and thus it's an inertial referential, which is just wrong by AM measure. $\endgroup$
    – Arc
    Commented Feb 14, 2022 at 22:50
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I think they can all be understood in Newtonian mechanics.

Proper acceleration is what's measured by an accelerometer. A simple example is a mass suspended by many springs inside a hollow transparent sphere. The vector from the center of the sphere to the ball bearing's location points opposite to the direction of the acceleration, and its length is proportional to the magnitude of the acceleration (or at least a monotonic function of it). This kind of accelerometer doesn't detect gravitational acceleration. No self-contained device can detect the gross gravitational acceleration, since it accelerates everything equally. Only differences across the size of the device (tidal effects) are detectable.

The equivalence principle says that you can transform away the gravitational field to first order in some region. In Newtonian physics, you can always transform to a relatively accelerating reference frame whose fictitious acceleration cancels the gravitational field at a point. What's left is just the tidal force.

A geodesic is a path that is straight ($\mathbf x(t)=\mathbf x_0+\mathbf v t$) in coordinates where the gravitational field has been canceled along the path by a coordinate transformation. You can't necessarily cancel it along the whole path by transforming to a uniformly accelerating frame, but you can do it if you make the acceleration a function of $t$, which is no problem in Newtonian physics.

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  • $\begingroup$ Totally clear explanation, but why a ball bearing? A ball or even a bear would work equally fine. :) $\endgroup$ Commented Feb 14, 2022 at 21:02
  • $\begingroup$ Thanks, benrg, but - as I stated in a comment on the other answer - Newtonian mechanics has only one kind of acceleration $a$ and only one kind of mass $m$. Newton's laws do not make a distinction between acceleration promoted by contact forces (which are measurable by accelerometers) and field forces (which can't be measured by self-contained devices, as you correctly state). $\endgroup$
    – Arc
    Commented Feb 15, 2022 at 1:49
  • $\begingroup$ The worst misunderstanding is exemplified by a mass standing still on Earth's surface: gravity force downwards + normal force upwards result in zero net force, and thus it's an inertial referential, which is just wrong by accelerometers' measures. $\endgroup$
    – Arc
    Commented Feb 15, 2022 at 1:49
  • $\begingroup$ @kkm It just says "mass" now. $\endgroup$
    – benrg
    Commented Feb 15, 2022 at 6:36
  • $\begingroup$ @Arc I suppose Newton would have said that the gravitational field is absolute, but that's just philosophy. From an experimental perspective, Newtonian physics predicts that there's no way to distinguish a uniform gravitational field from no gravitational field. $\endgroup$
    – benrg
    Commented Feb 15, 2022 at 6:39
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can the equivalence principle, the notion of geodesics, and proper acceleration be understood solely in the context of classical mechanics,

Einstein's most important merit in the field of gravitational research is his understanding of free fall in a gravitational field, which is precisely NOT acceleration. Because there is no force acting on the falling body. In the free fall we don't feel any braking or accelerating force, just like we feel it in a vehicle when braking, starting or in curves.

This announcement was made still without the formula of the GR and could have come also from Newton. But he was already saturated with knowledge about the acceleration constant of the free fall and did not think about the difference between forced acceleration (circular motion, cannon shot or braking) and the free fall with the absence of any acceleration feeling.
BTW, heavy as well as light objects fall equally fast on a geodesic path, but need different amounts of force for an acceleration on the geodesic path. Newton could have already noticed this (and this is in no way to belittle Newton's genius).

Einstein has now tried to describe the gravitational field of an inhomogeneous space. Without experimental data. The General Theory of Relativity tries to describe the curvature of space by a formula. At the same time, Einstein corrected his Special Theory of Relativity.

General Relativity is highly speculative because all experiments always have a gravitational potential component and an acceleration component. A dial gauge must be accelerated to get to a point in space with a different gravitational potential. So, on its way to another measuring point, it is always subject to time dilation as well as to the changing influence of gravity.

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