A man running on the treadmill

Question

Imagine a man is running on a treadmill. His inertia with respect to floor will be zero because he is not moving with respect to floor. If both he and the tread mill suddenly stops he will not fall, according to Newton's first law.

Imagine an ant on the conveyor belt. He doesn't worry about our reasoning. He just believes in 1st law, but can't explain it using the 1st law.

(Note: the ant will measure two momentum of the man. same but opposite in direction)

Now imagine we are a 2D being, living in a closed world like the surface of the Earth. Is Newton's first law not applicable here? Then what will be the definition of inertia in this world?

I think I should mention my current level of understanding. I read general relativity only in popular books.

Answer 1

You raise a very deep question that has not been satisfactorily answered. To rephrase, "without external objects to define a reference frame, how can we define an inertial frame?" This has typically been answered by invoking some concept of absolute space, but this answer is unsatisfactory. The position of the stars has been suggested to give space it's absolute-ness, but the stars themselves move. What are we then to do? Use the center of mass of all matter in the universe? How would we even define such a thing?

The issue becomes more difficult when dealing with angular momentum. When standing on the surface of a spinning sphere, we experience centrifugal force. Spinning tops precess, and hurricanes rotate due to Coriolis effect. If the earth weren't spinning, these phenomena would not occur. But how do we know the Earth is spinning? It clearly spins with respect to the sun, but how do we know the sun isn't spinning around the earth? At the end of the day, it is the presence of centrifugal force, the Coriolis effect, etc., that inform us (together with a dose of Occam's razor) that we are indeed in a rotating reference frame. But precisely why, and with respect exactly to what, our frame is rotating is not a resolved question.

Answer 2

I think Mach's principle is relevant here.

Answer 3

When the treadmill stops, the frame is no longer inertial (at constant velocity). The ant realizes this, since he feels a psuedoforce -- a jerk forwards which doesn't have any apparent origin.

Newton's laws are only valid in the "aether" frame or any inertial reference frame (a frame moving with constant velocity with respect to the aether)

Quoting Wikipedia:

Newton placed the first law of motion to establish frames of reference for which the other laws are applicable. The first law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to forces is a straight line at a constant speed

Basically, the first law establishes boundaries for Newton's laws. It is a way of saying "These laws are only valid in a case where an object has the property of inertia" (what we call "from an inertial reference frame")

If we look at Einstein, he was very particular about inertial frames as well. His special theory of relativity only talked about physics being the same in inertial frames.

His general theory of relativity has an analogue of this, which deals with non inertial frames. It has to do with Mach's principle, but it doesn't say quite the same thing as "physics is the same in all frames", as in this case the and needs to be aware of the ground(and/or frame dragging-like effects) to make an informed conclusion.