# Inertial movement of a body on the surface of a planet being dragged away

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.

• How are you not subject to the same force that pulls the planet? Commented Feb 1, 2022 at 6:11
• @PM2Ring and John Alexiou, exactly! That's the point! My feeling is that if the planet is being pulled by sudden gravity then you fall with the planet, if it is a contact force then you would free-fall upwards. But I can't confirm these cases, how do I put them in mathematical formulation is the issue.
– Arc
Commented Feb 1, 2022 at 12:40
• BTW, a planet like the Earth is relatively fragile. I think it'd be rather difficult to achieve much acceleration with the rope method without disrupting the planet. But I Am Not A Geophysicist, so I might be wrong. ;) Commented Feb 1, 2022 at 12:52
• @PM2Ring, well, its a theoretical question, so not considering geophysical aspects of the planet, nor, of course, of the rope. Edited as you suggested, feel free to edit it if you think you have a good suggestion!
– Arc
Commented Feb 1, 2022 at 13:05

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.

• Because the gravitational field is moving away from you? Aren't you on a geodesic trajectory of the field?
– Arc
Commented Feb 1, 2022 at 16:11
• Considering relative motion. A person in a rocket acceleratng away from the earth still feels the force of gravity. Commented Feb 2, 2022 at 14:40
• Right, but a person in an accelerating rocket is subject to the normal force exerted by the rocket body, and thus his/her accelerometers will indicate proper acceleration, whereas the person on the surface of a planet being dragged away is not under any contact forces (although the planet itself is, by the action of the rope) thus his/her accelerometers will measure zero. That's why I believe it will be an inertial frame, but I really don't see how to write this down in the framework of classical mechanics. I suspect the issue is the composition of contact/field forces one on top of another.
– Arc
Commented Feb 2, 2022 at 17:31
• My point is, the fact that the planet is moving away from you does not eliminate its gravitational force. You will still be accelerated toward the surface, but with a smaller acceleration than the surface. Commented Feb 3, 2022 at 14:10
• So, if your acceleration is smaller than that of the surface, you loose contact with the surface, and thus there's contact force no more. So, accelerometers measure zero, and thus it's an inertial frame, right?
– Arc
Commented Feb 4, 2022 at 1:59

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.

• Hi, well, thanks for the answer, but I really don't see how it could help to answer the question in a practical way. There are no magnetic forces postulated in the question, nor there's an atmosphere. What I want is an answer based on, say, analytical mechanics, with sound mathematical developments based on Newton's laws and the law of gravitation. But thanks anyway.
– Arc
Commented Feb 13, 2022 at 1:50
• The fact that there were no actual attempts of a mathematically based answer - even with the bounty posed - points to me its really a difficult problem.
– Arc
Commented Feb 13, 2022 at 1:51
• I know there are no magnetic forces, but the gravitational and magnetic forces function in a similar manner topologically, since both are inverse square laws. The atmosphere would serve as a damping force if earth is pulled with it, just like friction. If earth is not pulled with it, you would not experience air resistance and the atmosphere would form a "bubble" involving you. And, yes, it is not a simple problem. Commented Feb 23, 2022 at 2:08