# Paradox of non inertial forces in relativity

Ok, so I've been thinking of a certain paradox in relativity that I can't seem to resolve:

We have two observers, one in a "stationary" frame outside the Earth. This observer is looking down on Earth at another observer sitting at rest with respect to the surface of the Earth, but obviously, rotating with the Earth and with respect to the outside observer. The outside observer can calculate the mass of the Earth and the velocity of the earth observer due to the Earth's rotation and calculate what the Earth observer would determine to be the net force acting on them: gravitational force minus centrifugal force.

Then, the Earth observer begins to move, at constant velocity, equal and opposite the rotation of the Earth, such that both observers are now at rest with respect to each other. The outside observer determines that the Earth observer would have noticed an increase in force toward the center of the Earth, due to the fact that there is no more centrifugal force, but it seems as though the Earth observer should feel themselves getting lighter, due to a centrifugal force from his new rotation around the spherical Earth.

Who is right? It seems like the outside observer should be right, and that the net force on the Earth observer should increase, leading to the Earth observer to conclude there is another "fictitious" force at work pulling him in to the center of the Earth as he starts to rotate opposite the spin of the Earth with respect to the Earth's surface. The strange thing is that this force would also be unexplainable to a third observer at rest in the Earth's rotating frame, watching the Earth observer racing around the planet.

I think I kinda understand this situation, but can someone maybe help me make heads and tails of it? Thanks!

ghwellsjr
Gold Member
This sounds like the old joke:

Can you loan me $20? You say, I only have$10. OK, I say, then loan me that and you'll owe me the other $10 and since I'll owe you$10, we'll be even.

If the Earth observer has a negative centrifugal force while he is traveling around the Earth in the first case, then everyone will agree that that centrifugal force goes away when he stops traveling around the Earth in the second case. The rotating Earth under his feet is the same in both cases and has nothing to do with his centrifugal force. There is no "fictitious" force in the second case that needs explaining by any observer.

Ok, so I've been thinking of a certain paradox in relativity that I can't seem to resolve:

We have two observers, one in a "stationary" frame outside the Earth. This observer is looking down on Earth at another observer sitting at rest with respect to the surface of the Earth, but obviously, rotating with the Earth and with respect to the outside observer. The outside observer can calculate the mass of the Earth and the velocity of the earth observer due to the Earth's rotation and calculate what the Earth observer would determine to be the net force acting on them: gravitational force minus centrifugal force.

Then, the Earth observer begins to move, at constant velocity, equal and opposite the rotation of the Earth, such that both observers are now at rest with respect to each other. The outside observer determines that the Earth observer would have noticed an increase in force toward the center of the Earth, due to the fact that there is no more centrifugal force, but it seems as though the Earth observer should feel themselves getting lighter, due to a centrifugal force from his new rotation around the spherical Earth.

Who is right? It seems like the outside observer should be right, and that the net force on the Earth observer should increase, leading to the Earth observer to conclude there is another "fictitious" force at work pulling him in to the center of the Earth as he starts to rotate opposite the spin of the Earth with respect to the Earth's surface. The strange thing is that this force would also be unexplainable to a third observer at rest in the Earth's rotating frame, watching the Earth observer racing around the planet.

I think I kinda understand this situation, but can someone maybe help me make heads and tails of it? Thanks!
This has more or less been answered, but perhaps you will see this clearer if you leave out gravity.
So, take a rotating space station instead (or even a merry-go-round). Using the rest frame of the space station, the astronaut feels a force which he/she ascribes to the spinning around of the rest of the universe. That force is not felt when the astronaut walks along with the rest of the universe.

Dale
Mentor
2021 Award
it seems as though the Earth observer should feel themselves getting lighter, due to a centrifugal force from his new rotation around the spherical Earth.
This is not correct. If the earth observer uses a ring interferometer or a gyroscope or even a Foucalt pendulum then he can determine that he is in a rotating reference frame. From that information he will be able to make the same prediction that the space observer will make.

Ok, that's what I thought. Thanks, guys!

ghwellsjr