Actually it isn't the equivalence principle-it's the mathematical statement that it's possible to choose Minkowski coordinates at any one spacetime point of the manifold.  The equivalence principle then implies that the curvature effects between different spacetime points are equivalent to the effects of acceleration, since the spacetime is assumed to be described only by the metric. 

So the exact size is zero, if the manifold is curved. 

Actually it isn't the equivalence principle-it's the mathematical statement that it's possible to choose Minkowski coordinates at any one spacetime point of the manifold.  The equivalence principle then implies that the curvature effects between different spacetime points are equivalent to the effects of acceleration, since the spacetime is assumed to be described only by the metric. 

So the exact size is zero, if the manifold is curved. 

I believe a sharp formulation would be that the errors involved in approximating a finite region by flat space go to zero as the square of the region's size. That shows you do not need a specific size, and is enough to obtain the results usually derived via the equivalence principle.

@ Low: correct: I should have said positions. My point is, that formulation is elementary and does not involve the tensor concept. It is not such a strong restriction, since every description of a physical system winds up predicting specific values for position.

In flat spacetime the observables are Lorentz invariant, in curved spacetime they're invariant under general coordinate transformations. In neither case is position or velocity a meaningful quantity, so it's not useful to perpetuate such misunderstandings. It is in the non-relativistic approximation, that these make sense as observables.   In this approximation the equivalence principle reduces to the equality of  the mass parameter that enters in Newton's second law with the mass parameter that enters in Newton's law of gravitation. 

Relativity consists of two theories : one about the geometry of the universe (where we live), and one about gravitation. The latter has a meaning only in a non "flat" representation, that is without the frames, Minkoswski space and Poincaré group. The principle of equivalence is about Kinematics, that is the relation between the motion of material bodies and their inertial forces, and states that the gravitational charge (which rules the behavior of a material body in a gravitational field) is the same as the inertial charge (which rules the relation between inertial forces and motion). In the common understanding : gravitational mass = inertial mass. And it can be extended to rotational motion and inertial tensor in the spinor framework. Anyway all this requires the representation of the gravitational field, that is the General Relativity framework.

Rommel @

You asked originally about the mathematics related to the elevator. The following remarks may be of some help.

Inside a lab on Earth it is a very good approximation to consider the gravitational field to be homogeneous. Then the spacetime metric is given by the so-called Rindler metric (discussed in many places). This has a vanishing Riemann tensor and can be transformed explicitly to the standard Minkowski form that holds inside the freely falling elevator. In particular, the light rays are there straight.

What about the light rays for observers standing on the floor of the lab? As for every static metric these are spatial geodesics of the associated Fermat metric. For the Rindler spacetime the associated Fermat metric is the metric of hyperbolic3-space in the upper half chart. (Leaving out one horizontal dimension this is the metric of Poincaré's upper half plane.) As is well known from Riemannian (surface) geometry its geodesics are half-circles.

(By the way, this light bending is so tiny that it will probably never become observable.)

Anatoly A. Logunov (April 2000, monograph relativistic theory of gravity RTG):

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Rommel Nana Dutchou : Contrary to assumptions today, see a scientific report in Nature [1], arbitrary motion and acceleration are in fact described by Minkowski SR [2], without any equivalence to gravitation, as described in GR. GR will have to be modified. It is a misconception to think that SR only applies to inertial motion [3].

[1] Article Testing quantum mechanics in non-Minkowski space-time with h...

[2] https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html

[3] Preprint Special Relativity Applied to Arbitrary Motion