Tidal affect on an object and the length contraction in Relativity Theory

According to the equivalence principle in general relativity theory; If an object are in free falling in a gravitational field,the object will not detect gravitational force on it.

From this principle, Einstein deduced that free-fall is actually inertial motion. Objects in free-fall do not really accelerate. In an inertial frame of reference bodies (and light) obey Newton's first law, moving at constant velocity in straight lines.

However, According to classic physical approach: The difference between the top force on the object and the down force will create a stretching affect on the box and we expect deformation on the box (tidal affect) because of Newton's gravitation law formula .As I understand according to general relativity theory, the change of box cannot be detected by the internal observer. (Assume that there is no friction in space during free falling).

I believe that the force difference on an object can be so huge near to a black hole. I wonder if the person in free falling box to huge mass can feel any affect in the box or not?Can outside a rested observer observe a deformation on the box?

Can The General Relativity formulas calculate that increasing length of the box in a gravitational field even if the internal observer does not detect any change in the box ?

If the answer is yes, Isn't it a contradiction with the length contraction of the special relativity because It states that the length will be getting less via the formula below.

$$L=L_0\sqrt{1-\frac{V^2}{c^2}}$$ where $L_0$ is rested lenght of object.

Thanks for answers

Answer

There are some qualifications on the equivalence principle that often aren't made clear in casual summaries. One is that the equivalence principle only works in the limit as the size of the spacetime region where measurements are performed approaches zero (or to use another term, in an "infinitesimal" region of spacetime)--in other words, you'd have to consider a limit where both the difference in height between top and bottom of the box approaches zero, and the time interval over which the experimenter in the box makes measurements approaches zero as well. For example, p. 67 of From Special Relativity to Feynman Diagrams: A Course of Theoretical Particle Physics for Beginners states it as:

In the presence of a gravitational field, locally, the physical laws observed in a free falling frame are those of special relativity in the absence of gravity. As explained above, locally means, mathematically, in an infinitesimal neighborhood or, more physically, in a sufficiently small neighborhood of a point such that, up to higher order terms, the approximation (3.5) holds.

The previous statement is referred to as the equivalence principle in its strong form or simply strong equivalence principle.

Another important qualification is the one alluded to in the comment "up to higher order terms" above--from what I've read the equivalence principle only holds for first-order effects in the Taylor series expansion of whatever physics equations you're looking at, and tidal effects are actually second-order, so the equivalence principle doesn't actually say they should disappear even in the infinitesimal limit. This page on the site of physicist John Baez mentions that the equivalence principle says a small patch of spacetime in general relativity behaves like SR Minkowski spacetime "to a first-order approximation", and that the truly "gravitational" effects are considered to be the second-order ones, which remain after you "subtract off the first-order effects by using a freely falling frame of reference" (like a local inertial frame used by an experimenter in an infinitesimal falling elevator):

Principle of Equivalence The metric of spacetime induces a Minkowski metric on the tangent spaces. In other words, to a first-order approximation, a small patch of spacetime looks like a small patch of Minkowski spacetime. Freely falling bodies follow geodesics.

Gravitation = Curvature A gravitational field due to matter exhibits itself as curvature in spacetime. In other words, once we subtract off the first-order effects by using a freely falling frame of reference, the remaining second-order effects betray the presence of a true gravitational field.

And p. 7 of Cosmological Physics likewise says the "important aspects of gravitation" are the second-order ones, and mentions that these are tidal forces:

It may seem that we have actually returned to something like the Newtonian viewpoint: gravitation is merely an artifact of looking at things from the 'wrong' point of view. This is not really so; rather, the important aspects of gravitation are not so much to do with first-order effects as second-order tidal forces; these cannot be transformed away and are the true signature of gravitating mass.

So if I'm understanding correctly, even in the limit as the volume of the box and time of measurements approach zero, tidal forces would still in principle remain measurable inside the box, and this doesn't contradict the technical formulation of the equivalence principle.