# Equivalence principle doubt

There is something about Einstein Equivalence Principle that I don't quite get. This is my reasoning:

1. Equivalence principle $$\rightarrow$$ locally, acceleration is equivalent to a gravitational field

2. Forces (which each observer, inertial or not inertial, agree about) causes particles to have a proper acceleration (they don't follow geodesics)

3. These particles which have a proper acceleration, from their point of view, feel a gravitational field (point one).

Now... is this gravitational field real or fictitious?

If it is real $$\rightarrow$$ it must depend on mass distribution around the object. So the force which caused the acceleration must be linked to the mass distribution somehow (since these two actions balance themselves so that the object is in equilibrium, from his point of view).

If it is fictitious $$\rightarrow$$ then it has nothing to do with “real” gravity, which depends on mass and is manifested as the curvature of spacetime. In this case, the Equivalence Principle seems to me just a coincidence which has nothing in common with the geometrized view of curved spacetime.

If my assumptions are correct... which of the two options is true?

You've somewhat misstated the equivalence principle. It says that the effects of a gravitational field cannot be distinguished from the effects of having an accelerating frame of references. That's different than saying they are equivalent, and it's enough of a difference to break your logic chain at your point #3.

Another route to seeing this is that equivalence principle says that the $$m$$ in $$F=ma$$ (inertial mass) is the same as the $$m$$ in Newton's gravitational law (gravitational mass). It didn't have to be that way in theory, but it is. That does not preclude, however, the existence of other forces.

A bit more, following the comments, by way of explanation. I prefer to think of the equivalence principle as the mathematical statement that $$m_{\mathrm{inertial}} = m_{\mathrm{grav}}$$, and the (probably more popularly stated) description about indistinguishability between gravitational force and an accelerating frame as a consequence of this equivalence between the two conceptual types of mass. One implies the other though, so I think there's no unique argument for starting with one or the other, other than historical convention.

Note the last statement, which I'll elaborate here: The equivalence between gravitational and inertial mass implies the indistinguishability between a gravitational force and an accelerating frame.

• If you imagine that you're in a box with no information about what's happening outside of your box, you now try to construct an experiment to determine if your box is in an inertial frame. No problem with this. Hold a pencil out, let it "drop" and see if it heads toward a wall of the box. If it does, then you're in a non-inertial frame.
• Assume now that it does move toward a wall (meaning that it accelerates since it started at rest in your hand) and try to figure out if you're in a "stationary" box subject to gravity or in an "accelerating" box. Now you're stuck. If the box is not subject to gravity and the frame (e.g. the box) is accelerating due to some other force, then the pencil will move toward the side with acceleration equal to that of the box. If the box is subject to gravity but "stationary" you have $$m_{\mathrm{inertial}} a = m_{\mathrm{grav}} g$$, where $$g$$ gives the local strength of the gravitational field (and can be signed to account for direction of the field). But since the two masses are equal, this just gives $$a=g$$, which is non-informative since you don't have an independent measure of $$g$$. In either case (or, by extension, any case that includes some elements of both), all you can tell is that the pencil accelerated according to $$a$$.

The rest of your question about whether the force is "real" or "fictitious" seems to be trying to apply Newtonian reasoning to a relativistic question, and also seems to be based on your subtle misstatement of the principle. A frame exists in a (possibly small) neighborhood of a point but not at a single point. To say that the frame is accelerating is to say that all of the points in the frame are (up to some order) moving rigidly "together" with a single acceleration. That's distinct from looking at individual particles (possibly described in a frame) where each particle has a different relative acceleration. Your question on this part seems to be either comparing different frames (if you build a frame around each particle separately) or confounding the motion of the frame with the motion of the various particles in the frame (if you build one frame around your collection of particles). That's different than what the principle describes.

• this makes me wonder why it is called "Equivalence" Principle at all... so in this case, the gravitational field they feel is fictitious... does that imply my last deduction ("if it is fictitious --> etc)? – Federico Toso Sep 11 '19 at 14:52
• The "equivalence" is between inertial and gravitational mass. – Brick Sep 11 '19 at 14:53
• Your use of the word "fictitious" here is probably not wrong, but it's not especially relativistic in my view. There's no invariant way to describe that - if anything it's an expression of coordinate-dependent concepts that relativity would like to supersede with coordinate-invariant, geometric concepts. – Brick Sep 11 '19 at 14:56
• Ok... but even in this light, I don't see an explanation of the "coincident similarity" between local acceleration effect (due to forces) and local gravitational effects (due to mass-energy). The geometrization of curved spacetime does not attempt to explain this coincidence. Do you agree? – Federico Toso Sep 11 '19 at 15:19
• Updated answer with more detail. Writing quickly, so hopefully I got a good balance of precision and detail. – Brick Sep 11 '19 at 16:15