Question about the equivalence principle

[nwall]

Let us suppose the same domain referred to a second body of reference K', which is rotating uniformly with respect to K. In order to fix our ideas, we shall imagine K' to be in the form of a plane circular disc, which rotates uniformly in its own plane about its centre. An observer who is sitting eccentrically on the disc K' is sensible of a force which acts outwards in a radial direction, and which would be interpreted as an effect of inertia (centrifugal force) by an observer who was at rest with respect to the original reference-body K. But the observer on the disc may regard his disc as a reference-body which is "at rest"; on the basis of the general principle of relativity he is justified in doing this. The force acting on himself, and in fact on all other bodies which are at rest relative to the disc, he regards as the effect of a gravitational field. Nevertheless, the space-distribution of this gravitational field is of a kind that would not be possible on Newton's theory of gravitation. (footnote: The field disappears at the centre of the disc and increases proportionally to the distance from the centre as we proceed outwards.) But since the observer believes in the general theory of relativity, this does not disturb him; he is quite in the right when he believes that a general law of gravitation can be formulated--a law which not only explains the motion of the stars correctly, but also the field of force experienced by himself.

- Albert Einstein, Relativity: The Special and General Theory, Section 23

I'm confused about how gravity could account for all of the observations of the observer on the rotating disc. If the observer dropped a ball, he would see it fall away from the disc initially, as expected if there was a gravitational force pulling in all directions around a disc at rest, however, wouldn't the ball also accelerate in the direction opposite of the disc's spin (as judged from the observer who calls the force centrifugal) and begin circling the disc from the perspective of the observer who calls the disc "at rest"? If the gravitational field just "increases proportionally to the distance from the centre as we proceed outwards" why would the ball begin to circle around the disc?

Also, is it possible to describe the motion of the sun, planets, and fixed stars as due to a gravitational field that "increases proportionally to the distance from the centre" of the Earth?

 

[selfAdjoint]

Sure, there would be coriolis and also tidal forces. But Einstein here is speaking only about local effects (he doesn't make that clear, but he is - his applications of th EP are all local). In the limiting case of small neighborhoods, both coriolis and tidal forces go to zero.

 

[R0man]

The equivalence principle, as described by Einstein, states that there is no way to distinguish between a gravitational force and an inertial force. In other words, the observer on the rotating disc experiences a force that is equivalent to a gravitational force, even though there is no actual gravitational force present. This is because the observer is in a non-inertial frame of reference (the rotating disc) and is therefore subject to inertial forces, such as the centrifugal force.

To address your first question, the observer on the disc would not see the ball accelerate in the direction opposite of the disc's spin. This is because the observer is also rotating with the disc, so the ball would appear to fall straight down from their perspective. This is consistent with the idea that the observer is in a non-inertial frame of reference, and therefore experiences inertial forces that are not present in an inertial frame.

As for your second question, it is not possible to describe the motion of the sun, planets, and fixed stars as solely due to a gravitational field that increases proportionally to the distance from the centre of the Earth. This is because the motion of these objects is also affected by other factors, such as the rotation of the Earth, the presence of other celestial bodies, and relativistic effects. The equivalence principle applies to small, local regions of space, but cannot fully explain the motion of all celestial bodies in the universe.

In summary, the equivalence principle allows us to understand the effects of gravity and inertial forces in a unified way, but it does not fully explain the complex motions of all objects in the universe. It is an important concept in the field of relativity, but it is not a complete theory of gravitation.