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When we stand still on the surface of Earth, this is clearly a non-inertial frame.

Inertial frames of reference are characterized by accelerometers measuring zero, so you fell weightless when you: 1) free fall, 2) are pushed into a parabolic trajectory, or 3) orbits around a celestial body. These are all inertial movements because they all follow the geodesics of the gravitational field.

But standing on the surface of Earth means that you are: a) being attracted downwards to the Earth's center by gravity (a field force), and b) is being hold still by the normal force of the Earth's crust upwards (a contact force). So not an inertial frame of reference.

But now suppose that the planet you are in is being fastly pulled by an enourmous force away from your feet, a force far greater than the gravitational one that pulls you into the center of the planet.

Does that amounts to free falling upwards? Would you be in an inertial frame of reference then? Does it make a difference if the planet is being pulled by gravity (field force), compared to being pulled by a rope (contact force)?

Would like to see mathematical demonstrations within the framework of classical mechanics for both cases.

Arc
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If the planet is being pulled by a rope, gravity from the planet is accelerating you, and a coordinate system moving with you is not an inertial system.

R.W. Bird
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Just answering the comment you made about the different force types because I think it could help you put your line of thought into mathematical terms. I'm not going to prove anything, I'm just going to try helping you proposing some thought experiments.

Imagine we are on earth and it is an "orphan" planet (i.e. a planet not orbiting a star) on our Galaxy and let's assume we can observe large time quantities. When we are traveling aimlessly through the universe we are not under gravitational influence of considerably massive bodies except the Earth (it is negligible). Somehow, at some point, we enter the sphere of gravitational influence of a star and start to orbit this star.

How would we perceive this event? (Gravitationally wise)

Now, if you have a fridge and a fridge magnet, you can take off the fridge magnet from the fridge easily, just by pulling it off. Pulling the fridge would be much harder, but it should be mathematically possible. (Someone correct me if I'm wrong)

So, in analogy, it should be mathematically possible to push the earth out of you. Then there are three different phenomena: Earth has its atmosphere? If not, how this event would unfold? If yes, then the earth is pulled away with or without the atmosphere? And so on.

This is all hypothetical, just reminding. Hope I have helped.

Fractal Admirer
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