Could we be on the inside of a concave hollow universe? Recently I was discussing this theory again (a little drunk, I admit) and then tried to find answers, but couldn't find anything satisfying.
There is a theory (or several theories) that we could be living on the inside of a hollow sphere. The universe we know would then be inside this sphere.
This means:


*

*All lengths are shortened when going nearer to the center

*Only the light rays exactly moving towards the center, are straight, all the others are circular and go through the center

*The speed of light decreases the closer the light is moving the center.


(This is from the German Wikipedia the English Wikipedia also has something on this, but not as much information. )
At the English Wikipedia is this quote from Martin Gardner:

Gardner notes that "most mathematicians believe that an inside-out universe, with properly adjusted physical laws, is empirically irrefutable". Gardner rejects the concave hollow Earth hypothesis on the basis of Occam's Razor.

This is basically what I am interested in:
Is it possible to create a sound physical theory that states, that we live inside a hollow universe, and that is equal to our common theory of the universe.
This would mean, that there is no way to empirically decide if this theory is true or the "normal" theory about the universe.
On the German Wikipedia it states, that this theory contradicts Noether's theorem. It also states, that the general theory of relativity doesn't allow to have a coordinate system that spans the whole universe and so the transformation to a hollow earth is not possible. But it also states, that there would have to be additional axioms which make it possible?
As stated, I am not interested in any specific hollow earth theory, but if it is, in general, possible to decide if we live on the outside of a planet somewhere in the universe or on the inside of a hollow universe without resorting to faith and believes.
 A: For any transformed shape of the Universe, one may always easily define the theory in such a way that its results will be absolutely indistinguishable from the original theory. For example, one may describe the Earth and its vicinity by the polar coordinates
$$ R, \theta, \phi $$
so that $R\gt R_E$, the Earth's radius, corresponds to the space outside the Earth. It's possible to translate all of ours laws to a new coordinate system where $R$ is replaced by $\rho = R_E^2 / R$. In this system, $\rho \gt R_E$ will be the solid Earth while $\rho \lt R_E$ will be the air and the outer Universe. The most distant galaxies will appear near $\rho \sim 0$.
When done correctly, the two theories will be absolutely physically indistinguishable. They will be so indistinguishable that professional theorists wouldn't even talk about two theories. They would talk about one theory expressed in two different ways or two different coordinate systems.
The coordinate system with $\rho$ "looks" like it has the Universe inside the sphere. However, there are still ways to check that in some "operational sense", the Universe is still infinite and nearly flat – and equivalent to the usual Universe which is "mostly outside the Earth". The usual Universe and its corresponding coordinate system is arguably "more natural" because things like the metric tensor, which we need to define the motion of particles and evolution of fields, is pretty much constant in these coordinates.
We say that the laws of physics have a simpler form in the coordinates in which most of the stars are outside the sphere. We favor descriptions of the same theory in which the form of the laws is simpler. But the more contrived transformations of the laws are in no way "ruled out". We just don't seem to gain anything by those coordinates. And even if we started with those "hollow sphere" laws, we would be able to mathematically realize that there exists an equivalent description in which the metric is nearly constant and the stars live in a big space. We could still define distances using the usual definition of one meter (using the second and/or a number of wavelengths of some radiation) and this method would still imply that "what is inside the hollow sphere" are actually huge distances, especially near $\rho\sim 0$.
There are many analogous questions. For example, one may say that geocentrism – the Earth is the center of the Solar System or Universe – is true because one may pick coordinates (and we often do!) in which the Earth sits at the origin etc. All the cosmic effects of the Sun, planets, other bodies, and the cosmic expansion and evolution may be translated to some effects in these geocentric coordinates.
But there's a sense in which the heliocentric coordinates are "more natural" because in the geocentric coordinates, we need extra terms for huge forces that direct the motion of the other planets and all those unnatural forces are there just to convert the motion from the more natural, nearly heliocentric system to the geocentric one. They disappear in the inertial frame! And this simplicity matters.
Even though general relativity allows us to use any coordinate system, including geocentric ones, I think that its users would mostly agree that heliocentrism is at least "morally" right and the heliocentric system is "more scientifically correct". The laws of physics simply have a simpler form in this system and don't contain any terms whose true origin seems to be unclear or deliberately masked.
At the end, the two theories you mentioned are examples of descriptions that differ by "appearances". In the 20th century (relativity, quanta) and especially in recent 20 years (since the "Duality Revolution" in string theory), theoretical physics became very careful about "not depending on appearances or emotions". Good professional physicists really talk about theories in such a way that the dependence of their conclusions on "unobservable differences in the philosophy or appearances" disappears. 
Coordinate transformations (and field redefinitions) are actually viewed as trivial examples to produce "equivalent theories" or alternative descriptions of the same theory. The Duality Revolution has revealed lots of pairs of theories of physics that superficially look qualitatively different, contain completely different kinds of objects, different shape (or number) of the dimensions of the spacetime, but they're still fully and exactly indistinguishable when all of their features are taken into account. Such "hard to prove" equivalences are known as the dualities. The AdS/CFT "holographic" correspondence may be considered an example of a duality, too. In the case of dualities, there is no sharp answer to the question which of the equivalent description is "more right" or "more real". They are equally right and equally real. But when we use the alternative descriptions at all, both sides of the equivalence are much more natural than the hollow universe or the geocentric system.
A: 
Could we be on the inside of a concave hollow universe?

Nope. Nor are we on the inside of a hollow Earth. 

Recently I was discussing this theory again (a little drunk, I admit) and then tried to find answers, but couldn't find anything satisfying.

That's because it's bunk.

There is a theory (or several theories) that we could be living on the inside of a hollow sphere. The universe we know would then be inside this sphere.

Only there's no evidence for any such theory. Compare and contrast with General relativity, which is one of the best-tested theories we've got. See Clifford M Will's paper at http://arxiv.org/abs/1403.7377 

All lengths are shortened when going nearer to the center. Only the light rays exactly moving towards the center, are straight, all the others are circular and go through the center. The speed of light decreases the closer the light is moving the center.

And we have no evidential support for any of that. 

Gardner notes that "most mathematicians believe that an inside-out universe, with properly adjusted physical laws, is empirically irrefutable". 

So are fairies. Gardner's statement is meaningless, and mathematicians aren't physicists. They don't follow the scientific method. If you mention that little thing called scientific evidence they just shrug.

Is it possible to create a sound physical theory that states, that we live inside a hollow universe, and that is equal to our common theory of the universe.

No it isn't. Like Luboš was saying, you can express a theory in more than one way, but when it delivers a totally different message, like we live in a universe shaped like a holographic chocolate teapot, then it just isn't the same theory. 

This would mean, that there is no way to empirically decide if this theory is true or the "normal" theory about the universe.

The evidence distinguishes one hypothesis from another. 

On the German Wikipedia it states, that this theory contradicts Noether's theorem. It also states, that the general theory of relativity doesn't allow to have a coordinate system that spans the whole universe and so the transformation to a hollow earth is not possible. 

We don't live in a hollow Earth, end of story. 

But it also states, that there would have to be additional axioms which make it possible?

I'm sorry, Josef, but it's bunk. 

As stated, I am not interested in any specific hollow earth theory, but if it is, in general, possible to decide if we live on the outside of a planet somewhere in the universe or on the inside of a hollow universe without resorting to faith and believes.

Just look out the window.  
