There are two standard forms of the equivalence principle. The Weak form states the equivalence of inertial and passive gravitational mass, which implies that all massive particles will fall along the same space-time trajectories in the presence of only gravitational effects. The Weak Principle is what is tested by the experiments of, e.g. Eötvos. Note that it has no consequences for the behavior...

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There are two standard forms of the equivalence principle. The Weak form states the equivalence of inertial and passive gravitational mass, which implies that all massive particles will fall along the same space-time trajectories in the presence of only gravitational effects. The Weak Principle is what is tested by the experiments of, e.g. Eötvos. Note that it has no consequences for the behavior of light. The Strong principle states the empirical equivalence of experiments done "at rest" in a constant gravitational field and in a constantly linearly accelerating lab with no gravity. That principle does have implication for light. But the Strong equivalence principle is restricted to comparing constant linear acceleration to a constant field. It does not even properly hold for labs on Earth, where the gravitational field is not constant, although the differences are of second order. (Consider stretching your hands apart and dropping two masses, measuring the distance between the hands and the distance between the place where they hit the floor. In a linearly accelerated system in flat space-time, those distances will be identical. In a lab on the Earth, they will not, since the gravitational field is not constant (they will hit slightly closer together, as they are both falling toward the center of the Earth, as it were. Similarly, a water droplet in a space station orbiting the Earth will be slightly elongated by tidal effects, and one in inertial motion in flat space-time will not. So not all "free-fall" is the same.)

The Strong Principle does not have any application at all for rotating systems. If you check your own reference (Misner, Thorne and Wheeler) you will verify this. Check Wald (for example) as well.

In the first part of this paper, two coordinate systems are laid down on flat space-time. Obviously, these coordinate systems do not change the space-time geometry at all: it is flat in both. The line element, of course, takes a different algebraic form relative to the different coordinates, as it must. This is just the same as using different coordinate systems on Euclidean space, and has no connection to the Equivalence Principle.

To clarify the situation, one cannot just talk about "accelerating" systems: the Strong Principle compares linearly accelerating systems in flat space-time to systems with a constant field. But rotation is not a linear acceleration. There is no gravitational field that will will mimic, as it were, the apparent effects of rotation. It you think any gravitational field can produce a "centrifugal" force, try to specify how. No stress-energy tensor will produce the same apparent physics is a non-rotating lab as there is in a rotating lab.

I see what Kündig states, but the claim is not accurate.

Regards,

Tim

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Dear Tim,

I am happy to see that you did not reply to my rebuttal on the motion in accelerating frames as you should have understood to be wrong on that issue. I regret to inform you that you are wrong also on your interpretation of the Equivalence Principle. First of all, and contrary to your claims, standard forms of the Equivalence Principle are 3 instead of 2. You missed the most famous...

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Dear Tim,

I am happy to see that you did not reply to my rebuttal on the motion in accelerating frames as you should have understood to be wrong on that issue. I regret to inform you that you are wrong also on your interpretation of the Equivalence Principle. First of all, and contrary to your claims, standard forms of the Equivalence Principle are 3 instead of 2. You missed the most famous one, i.e. the Einstein Equivalence Principle, which is the one that I use in my Essay. It states that the weak Equivalence Principle holds, and that, in addition, the outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime. This has consequences for the behavior of light. In fact, it is well known that among the most important tests of the Einstein Equivalence Principle are gravitational redshift experiments. Thus, your claim that only the strong Equivalence Principle has consequences for the behavior of light is wrong. You are wrong also when cite the Strong Equivalence Principle which has nothing to do with the issue here. The Strong Equivalence Principle states that "The gravitational motion of a small test body depends only on its initial position in spacetime and velocity, and not on its constitution" and "The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime." In other words, differently from the Weak Equivalence Principle which considers point masses, the strong Equivalence Principle considers small test bodies (with respect to the source of the gravitational field) i.e. objects that exert a gravitational force on themselves, for example the Earth in the gravitational field of the Sun. This is not the current case, because here we are considering photons, which do not exert a gravitational force on themselves. Differently from the Einstein Equivalence Principle the Strong form considers also gravitational experiments in a freely falling laboratory.

But the issue where you are really wrong is in claiming that Equivalence Principle does not have any application at all for rotating systems. As I explained in the technical endnotes of my Essay, the Equivalence Principle is also involved in the observation by Einstein that the gravitational “force” that an observer experiences locally when standing in the gravitational field generated by a massive body (the Earth) is equivalent to the pseudo-force experienced in an accelerated (non-inertial) local reference frame. Although you seem to do not understand this issue, a rotating frame is an accelerated (non-inertial) reference frame. You miss the point "locally". Rotation generates locally a linear acceleration. Locally, the apparent effects of rotation can perfectly mimic the gravitational field of the earth. You must merely choice an angular velocity and a distance from the center of the rotor for which the square of the angular velocity multiplied for the distance from the center of the rotor equals 9.80665 m/s^2, i.e. the gravitational acceleration of the earth. In the rotating frame of my Essay, i.e. the frame in which the observer sees the detector at rest , that local acceleration will have radial direction. Thus, the analogy with a local gravitational field is absolutely complete. You cited MTW and Wald. The fact that in those books the possibility of using the Equivalence Principle in rotating frames is not considered does not mean that that possibility is wrong. Those textbooks does NOT claim that it is wrong, merely they do not consider it. The book of Pauli considers it instead. I add that, differently from you, I am not particular interested in what is written in textbooks. My own knowledge of a theory does not come from blindly accepting wisdom from textbooks; textbooks do contain mistakes, so I only accept scientific facts when I fully understand the arguments on which they are based. My above argumentations are completely sufficient in the current case.

Cheers, Ch.

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Dear Christian,

A rotating laboratory is not "freely falling", i.e. subject to only gravitational effects. Put otherwise, a body "at rest" in a rotating frame (i.e. with constant spatial coordinates) is not following a geodesic. That is why it isn't an inertial frame.You are not separating linear acceleration from rotation, which are quite different. The Strong Equivalence Principle is...

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Dear Christian,

A rotating laboratory is not "freely falling", i.e. subject to only gravitational effects. Put otherwise, a body "at rest" in a rotating frame (i.e. with constant spatial coordinates) is not following a geodesic. That is why it isn't an inertial frame.You are not separating linear acceleration from rotation, which are quite different. The Strong Equivalence Principle is confined comparisons between effects in various circumstances, none of which include rotating labs or rotating frames. (The physical effects of rotation were, of course, whole point of Newton's bucket experiment, which gets the same explanation in relativity as it does for Newton: rotation is an objective feature of some motions.)

One application of the principle is this: experiments done in a lab "at rest in a constant gravitational field" (to a good approximation, a *non rotating* lab on the surface of the Earth, but that is only approximate) will display the same phenomena as a *linearly accelerating* lab in flat space-time. That gives the "bending of light". Another is the (approximate) equivalence of a non-accelerating lab in flat space-time to a *non rotating* lab in "free fall" on Earth (again this is only to first order since the field on Earth is not constant). There is no principle equating a rotating lab with a non-rotating lab in any gravitational field. Phenomena in a rotating lab will not even be spatially isotropic (referring to the rotating coordinates).

It is kind of odd that you say I cited MTW, I only brought it up because you cited it as one source for the application of the Equivalence Principle to rotational situations. Since you seem to concede that they nowhere make such a claim, perhaps you should remove that citation. Now you say you do not trust textbooks. If so, then don't cite them, especially when they do not make the claim you are trying to establish.

There is a reason that the SLAC was built as a linear accelerator, rather than a closed circuit like the LHC, and that reason has to do with the difference between linear acceleration and non-linear acceleration. Your claim that rotation is the same as linear acceleration is not accurate. And no Equivalence Principle, including the one you cite above, equates rotating labs or rotating frames to non-rotating labs or non-rotating frames.

Regards,

Tim

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The equivalence principle simply states that freely falling frames are equivalent to a purely inertial frame independent of gravity. Similarly a frame that is accelerated and one on the surface of a gravitating body are equivalent:

we ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. — Einstein, Albert (2003)....

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The equivalence principle simply states that freely falling frames are equivalent to a purely inertial frame independent of gravity. Similarly a frame that is accelerated and one on the surface of a gravitating body are equivalent:

we ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. — Einstein, Albert (2003). The Meaning of Relativity.

Departures come about for two reasons. The weak EP is a form of Galileo's principle, and it says the motion of a particle in a gravity field is the same as on an accelerated frame. This insures Galileo's observation on the independence of mass of a body with respect to its motion. With the weak equivalence principle the main departure is due to tidal forces and the radial direction of gravity. so the WEP requires that the size of the frame in a gravity field, say the dimensions of a lab sitting on the surface of a gravitating body, be very small relative to the dimensions of the gravitating body. The Einstein EP (EEP) says that any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime. This is the inertial idea of being in a freely falling frame, such as the infamous elevator. Again if this frame falls through a region of a gravity field so that tidal forces are apparent there are departures. The strong EP (SEP) says that the velocity of the frame relative to any outside frame, such as a distant coordinate system, is not a determinant of the measured physics on that local frame. There is again locality of measurements required to eliminate tidal forces. This means that gravitational physics is purely geometric. This is “strong” because it makes reference to regions of spacetime that are removed from any local frame.

When it comes to the rotating frame and the EP, we have certain stipulations that are required. Certainly for the WEP and EEP we require that the dimensions of any local frame be small. This holds for the SEP as well, but we have another stipulation that physics in the lab frame be independent of motion relative to the outside world. This does not happen with the rotating frame. One clear departure is the Coriolis acceleration 2ωxv, which in the rotating frame is rather apparent if there is some motion of a particle relative to the rotating frame. An observer on a frame which observes motion of a freely moving particle as cycles or circles, with no central gravitating body present, suspects then that they are on a rotating frame. As a result the additional caveat for the SEP with rotating frames is that the motion of a particle not under any local force in that frame must have a small velocity v

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