How big is an inertial frame?
Consider a huge rod which is rotating about a fixed point in a plane, its length is 1 light year.
Thus light from its end closer to the fixed point to the end farther from the fixed point takes one year to reach.
Now the angular velocity of the point closer to the fixed point is much slower than the angular velocity of the farther point.
Thus the end closer to the fixed point has a relative velocity to the end farther away.
At some point in time, the clocks at the two ends are synchronized by sending a light signal to both ends from exactly from the centre of the rod (half a light year from any point).
My question is, are the two ends of the rod in the same inertial frame? Is it possible for 1 object to be in two inertial frames simultaneously?
A simpler question. A rod is 1 light year long and travelling at a a velocity c/2 along its length. There will be length contraction of the rod by gamma. Now imagine a light source at one end of the rod. The light from one end takes one year to reach the other end (as measured by either end), when the rod is stationary.
Now since the distance is contracted when the rod is moving, so must the time be different (as compared to an observer looking at this rod) for the two ends of the rod as light must be measured with the same velocity on both ends. The time must be same at both ends by symmetry, so they are indeed in the same inertial frame.
However, if there is length contraction, they will not measure one light year, they should each measure less than one light year.
An external observer observing this rod would see the distance as less than one year. But to measure the same velocity of light, must also measure the same time as the ends of the rod, thus implying the observer is in the same frame as the rod?
The second part of my question is this. In zero gravity, a person holds a ball in his hand. Thus the person and the ball are in the same inertial frame. Then the person "throws" the ball and as a result, both and ball and the person now acquire a relative velocity to each other (action and reaction). Are the two in the same inertial frame?
The third part of my question is this. In zero gravity, can a ship and an observer on the ship be in the same inertial frame? That is, do they share acceleration? Assuming the ship is a plane, when the plane accelerates, the person is floating around, hence the person and the ship cannot share acceleration unless the person is strapped to the ship.Thus in zero gravity, every object must have its own inertial frame?