How can the Equivalence Principle hold when we consider tidal forces?

[Bernard]

Tidal Forces: "It arises because the gravitational force exerted by one body on another is not constant across it". which implicitly implies that the acceleration is not constant on that body.

Equivalence Principle: "weightlessness sensation occurs when one free falls in gravity" - which implies that in an upright position, you don't feel your feet accelerating more than your head in a large gravitational field such as that of a black hole, which is a contradiction in itself.

 

[Ibix]

The equivalence principle only holds over regions of spacetime small enough that tidal effects are negligible. So your second paragraph describes a situation where the equivalence principle is not expected to apply.

 

[Dale]

Equivalence Principle: "weightlessness sensation occurs when one free falls in gravity" - which implies that in an upright position, you don't feel your feet accelerating more than your head in a large gravitational field such as that of a black hole, which is a contradiction in itself.

If you use a standard textbook definition then you will see that this is not how the equivalence principle is described. You will either see the explicit limitation to uniform gravitational fields, the absence of tidal forces, or the term "local".

 

[Bernard]

If you use a standard textbook definition then you will see that this is not how the equivalence principle is described. You will either see the explicit limitation to uniform gravitational fields, the absence of tidal forces, or the term "local".

So this only holds in uniform gravitational fields. Now Earth does not have a uniform gravitational field (since field lines are not parallel and tidal forces can be seen on earth). Then why do we need to correct our time on the GPS satellites due to general relativity if the slow running of clocks is implied by the strong EP which does not hold since our gravitational field is radial?

 

[Dale]

GR is more than just the equivalence principle. GR holds over non-local regions of space time, even when the equivalence principle does not.

The equivalence is just a tool to help people figure out what the laws of physics should look like in GR based on what we know on a local scale.

 

[PeterDonis]

So this only holds in uniform gravitational fields.

No, it holds over a region of spacetime small enough that non-uniformities in the field cannot be detected.

the slow running of clocks is implied by the strong EP

The fact that the strong EP implies the slow running of clocks does not mean slow running of clocks is not present in circumstances when the strong EP cannot be applied. Slow running of clocks--or more precisely different proper times along different worldlines in spacetime--is a much more general phenomenon.

 

[vanhees71]

If it comes to the question, what the equivalence principle really means, my feeling is that the only precise meaning of the weak principle is that the spacetime manifold is a 4D pseudo-Riemannian space, and at any point of space time the tangent space is a Minkowski space, i.e., the pseudometric is of signature (1,3) (or equivalently (3,1) depending on the sign conventions chosen). This implies that at any point there's a local inertial frame, where any local (!) law of physics takes the form of that law in special relativity. Only at the very point where you use such a "free-falling rotation free reference frame" you don't have gravity. Practically it's approximately true with some precision only for regions small compared to any curvature measure at this point.

 

[atyy]

Tidal Forces: "It arises because the gravitational force exerted by one body on another is not constant across it". which implicitly implies that the acceleration is not constant on that body.

Equivalence Principle: "weightlessness sensation occurs when one free falls in gravity" - which implies that in an upright position, you don't feel your feet accelerating more than your head in a large gravitational field such as that of a black hole, which is a contradiction in itself.

There are (at least) two forms of the equivalence principle.

1. The equivalence principle applies only locally, and fails when curvature and tidal forces are considered.
http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/ (see section 25.7)
http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html (se section 9.5 and the discussion about local flatness)

2. The equivalence principle applies exactly to all of general relativity when it is taken to be the principle of minimal coupling (in the action).
http://www.blau.itp.unibe.ch/newlecturesGR.pdf (sections 5.1-5.4)

 

[vanhees71]

I'd say 2. is even the strong equivalence principle.

 

[martinbn]

2. The equivalence principle applies exactly to all of general relativity when it is taken to be the principle of minimal coupling (in the action).
http://www.blau.itp.unibe.ch/newlecturesGR.pdf (sections 5.1-5.4)

I think section 1.1 Motivation. Einstein's equivalence principle is more relevant to this discussion.