Equivalence principle, geodesics, and proper acceleration are exclusive to general relativity, or they can be understood in classical mechanics terms?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.