The "equivalence principle" may be the most remarkable principle in all of science. It's the product of Einstein's early attempt to understand gravitation, and to assimilate it to a general relativity that would include all accelerations, just as the special theory included all uniform motion. But the principle should have been abandoned once E. realized that gravitation is a geometric deformation of spacetime. As such, gravitation isn't relative: there is always at least one center of mass determining a body's path, or resisting its path. And for every inertial acceleration there is absolutely a force being applied. They are not two sides of the same phenomena, as the "equivalence principle" more or less explicitly claims.

The only reason the elevator on or suspended above the earth's surface and the elevator being towed in space appear similar to the occupants (but not identical: drop two balls and they'll converge in a grav field but fall parallel if being towed) is that there is an inertial acceleration in both cases: due to the resistance to gravitation in one case and the mechanical force of the rocket in the other.

The only way an "equivalence" can be claimed is if we are prevented from making the distinction by limiting the comparison to a small space or a brief period of time. In other words, it's actually a principle of NON-equivalence, like saying red and blue are equivalent if the laboratory is made dark enough.

The equivalence principle would make acceleration relative like velocity is in an inertial system. Unfortunately, acceleration is only relative with respect to absolute space. The fact that the gravitation field is zero at infinity is part of absolute space, and cannot be used by gr to constitute the limit at large distances and weak fields. In other words, velocity can be transformed away in an inertial system, but there is no way to transform away rotation. This is made evident by the uniformly rotating disc.

The equivalence principle is local. The effects of rotation are disregarded in this approximation since local means flat and Euclidean. Non local effects due to gravitational masses/rotation in summary arbitrary accelerations can be calculated by means of covariant transport on the space time manyfold.

In other words, equivalence in non-equivalent, except for practical purposes.

However, GR is not flat and Euclidean! And what does local have to do with absolute as opposed to relative motion?

>However, GR is not flat and Euclidean!

In GR the effects of gravity and inertial acceleration are indistinguishable (equivalent) only locally. The effects of rotation can't be made equivalent to inertia on extended scales of course. Any reference frame should be equally valid to describe Nature, from this and the transformation rules between arbitrary reference frames (covariant transport, tensor analysis) provide the laws of Einstein gravitation, space metric = matter energy contents and vice versa.

Gravitation is indistinguishable from a force-free state of rest, not inertial acceleration, no matter how local. Falling in a gravitational field, orbiting a planet, etc, are force-free experiences. It's only when gravitation is interrupted that inertial acceleration occurs. So the reason standing in a suspended elevator and being accelerated through space are indistinguishable if the reference frame is local enough is because both involve inertial acceleration. Inertial acceleration = inertial acceleration. What could be more trivial?

Then what is the transformation, equivalent to the Lorentz transformation for velocities, that would make accelerations relative?

Inertial accelerations aren't relative; in principle, an impelling force can always be identified. And a test particle inside a box will always be found to be pressing against one side, indicating that the particle, the box, or both are being accelerated by a force.

quoteI

nertial accelerations aren't relative; in principle, an impelling force can always be identified

end of quote

???????????????????????

What does an impelling force have to do with Bernard's question ?

Nothing.

So what are you saying ?

I was answering your question. Why were you asking?

Einstein's equation R_ii=0 is his law of gravitation for empty space. Only the gravitational field is present in the absence of matter(=mass). But this is Newton's law of absolute space saying that rotation is not relative to another frame, but to absolute space. Einstein constructed his general theory on Mach's principle, and found later that it violated it. Mach's principle would be relative space where the distribution of matter determines the centrifugal force acting on a body. According to Einstein, "there can be no inertia relative to space", and found later (post 1917) that there could. But general relativity is pre-1917!

This can lead to an idia that the space time is indeed dynamic space time not static as assumed previously.

James, I am sorry to say that I do not understand your answer

Disagreements about the meaning of the importance of the principle of equivalence is, in a sense, extensions of the old debates on the principle of least action. For me, some scientists have regarded the principle of least (or stationary) action with almost mystical reverence, as a kind of expression of the divine will, while others have seen as a merely descriptive device, applicable in many, but not all, without deep meaning subjacente.A "sum over all paths", goes some way towards the "explain" why action is stationary so common, but only by quantum superposition, which then leads to the measurement problem in quantum mechanics . It is interesting that the non-local aspects and even teleological general relativity were already present in the debates older less action, and the same concerns can be seen in "correction" of Einstein.A assertion that "every continuous path is infinitely straight "and therefore can not distinguish right ways of curved paths in a base infinitesimal, is essentially equivalent to the statement that" stationarity infinitesimal "can not be the cause of anything. In other words, if we say that over an interval sufficiently small, the difference between the motion of a particle that moves in a curved path can not physically be differentiated from a particle moving in a straight line then can not explain the fact that the particle moves in a straight path (even on the space-time) - unless you invoke the principle of least action over a finite interval.

Thus, we are led to a non-local application of the principle that if we insist can not have any causal effect on infinitesimal intervals. In fact, the sum of all paths are explicitly non-local, which of course is what leads to the problem of measurement. An alternative approach is simply to reject any causal implications and consider the geodesic motion (stationary action) as purely descriptive, or suggest that it is tautological to object free to move along geodesics, because ultimately, define and geodetic rest of our model space-time paths of free objects. The question then becomes coherent, ie, why there is such consistency in the movements of objects, so that you can define a single coordinate system of space and time do not necessarily flat, in terms of the movement of such great number of entities that are at least approximately geodesic.

>

that's correct

The Modification of Newton's Gravitational Law and its Application in the Study of Dark Matter and Black Hole: https://www.researchsquare.com/article/rs-373969/v1

The Physical Cause of Planetary Perihelion: Precession:https://www.researchsquare.com/article/rs-536456/v1