What is the Principle of Equivalence and how was it determined?

[loislane]

Well, if you know what happens in every local inertial frame, then doesn't that imply what happens globally? Under the assumptions that:

1. We're talking about test particles and weak fields whose effect on gravity is negligible, and
2. There are no direct couplings of the equations of motion to curvature or higher-order derivatives of the metric.

Nonminimal coupling can never be ruled out except experimentally, but I would say that if there are nonminimal couplings, that to me means that the EP does not hold for situations in which nonminimal coupling is relevant. Or to put it another way, to me, the impact of the EP is the claim that there is minimal coupling of matter and fields to gravity.

If you assume minimal couplig from the start you are assuming the EP in exact(not just approximate) form. The problem with that assumption is that as commented by PeterDonis it leads you to a completely coordinate dependent formulation of the geodesic equation.

 

[stevendaryl]

If you assume minimal couplig from the start you are assuming the EP in exact(not just approximate) form.

Exactly. To me, the EP is the claim that gravity effects particles via minimal coupling.

 

[loislane]

I think that there is a very close analogy with Newtonian gravity. In Newtonian gravity, you have a gravitational potential $\Phi$, and the path of a test particle is given by:

$m \dfrac{d^2 x^j}{dt^2} = -m \partial_j \Phi$

The motion of the particle only depends on the first derivative of $\Phi$

Newtonian gravity certainly has tidal effects, which involve the second derivatives of $\Phi$. But the existence of tidal effects follows from the above equation of motion, except in the special case in which $\nabla \Phi$ is a constant vector.

I think you are relying too much on Newtonian gravity, you do know is not exactly correct, don't you?

 

[PeterDonis]

To me, the EP is the claim that gravity effects particles via minimal coupling.

I agree with this formulation (the only potential quibble I would have would be to say "spacetime geometry" instead of "gravity"--"how spacetime tells matter how to move", so to speak). My previous comments weren't really about the physics but about ordinary language terminology. I agree with everything you have said about the physics.

The problem with that assumption is that as commented by PeterDonis it leads you to a completely coordinate dependent formulation of the geodesic equation.

Christoffel symbols are coordinate dependent, but covariant derivatives are not; they are proper tensorial objects. So the geodesic equation expressed in terms of covariant derivatives is properly covariant.

As I said above, I was not really commenting about the physics; I was commenting about terminology. I thought the phrase "curvature does not affect the motion of particles" might be misleading. But the "minimal coupling" formulation is saying the same thing, just in different words. The physics is the same either way.

 

[stevendaryl]

I think you are relying too much on Newtonian gravity, you do know is not exactly correct, don't you?

The particular points being discussed are true of both Newtonian gravity and General Relativity.

 

[PeterDonis]

I think you are relying too much on Newtonian gravity

No, he's not. He is saying that, if you want more than what has already been said in this thread, you are basically asking us to provide you a textbook on GR in the limited space of a PF thread. That's not going to happen. If you want more details, please consult a textbook.

 

[PeterDonis]

At this point the OP's question has been more than thoroughly answered. Thread closed.