# An example of a theory that respects the Weak Equivalence Principle but violates the Einstein Equivalence Principle

The Weak Equivalence Principle has any one of the following forms:

1. the inertial mass is equal to the gravitational mass
2. there exists a preferred class of trajectories through spacetime, known as inertial or freely-falling trajectories, on which unaccelerated particles travel - where unaccelerated means "subject only to gravity"
3. the motion of freely-falling particles are the same in a gravitational field and a uniformly accelerated frame, in small enough regions of spacetime

and the Einstein Equivalence Principle has the following form:

In small enough regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field by means of local experiments.

The difference between the WEP and the EEP lies in the fact that WEP considers only the motion of freely-falling particles whereas EEP considers any local experiments.

The following extract describes a theory that respects the WEP but violates the EEP:

we could imagine a theory of gravity in which freely falling particles begin to rotate as they moved through a gravitational field. Then they could fall along the same paths as they would in an accelerated frame (thereby satisfying the WEP), but you could nevertheless detect the existence of the gravitational field (in violation of the EEP). Such theories seem contrived, but there is no law of nature that forbids them.

In the theory mentioned in the extract, what does it mean for the particles to fall along the same paths as they would in an accelerated frame and how does that satisfy the WEP? Similarly for the EEP?

P.S.: The extract is copied verbatim from Sean Carroll's textbook.

• IIRC, it is conjectured that WEP=EEP. I don't think there is a known counterexample, because this would disprove the conjecture, which I believe is still unresolved... Apr 29, 2016 at 15:35
• What about a charged particle accelerating in a gravitational field? Apr 29, 2016 at 15:53
• Aren't we only considering the effect of the gravitational field on the motion of the particle? Apr 29, 2016 at 15:56
• @PeterR I think the conjecture WEP=EEP was proved for charged particles, at least for rotationally symmetric grav. fields (e.g., see the $TH\epsilon\mu$ formalism and Schiff’s conjecture) Apr 29, 2016 at 16:03
• Isn't that trivially the case for theories with torsion? The question, technically, seems to be whether a theory with torsion can be distinguished from a theory without torsion that has an additional torsion field that can not be linked to gravity and the structure of spacetime. I have seen statements to that respect on this site before, which means that mathematics may make this a matter of perception or preference, if it is true that some torsion theories are indistinguishable from gravity plus classical force fields. Apr 29, 2016 at 17:52

In the quote cited you could imagine that point particles move in a straight line at a steady velocity and don't rotate when far from massive objects.

So to a small accelerating frame, they look like a non rotating point particle moving at a constant velocity would look to an accelerating frame.

But maybe point particles near a massive object have similar looking worldlines. But they rotate.

These classical point particles would need some kind of point particle structure to be able to do this. For instance they could have a magnetic moment. And far from massive objects and in zero external electromagnetic field the particle moves in a straight line at a constant velocity and the magnetic moment does not experience any torque.

And so then you could ask what that looks like to an accelerating frame.

But what if that same charged point particle has its magnetic moment experience a nonzero torque when near a massive object even when there is no external electromagnetic field.

So the world line for the elevator and the gravitational field might look the same.

But the dynamics of the magnetic moment could be different. Which means you could distinguish between being in the frame of an elevator and being near a massive object.

You just have to do it by looking at more than the worldline of an object only experiencing gravitational forces. You'd also have to look at electromagnetic interactions.