How does the universe preserve distance? Assume I have some physical system, no matter of it's nature. Suppose also that this system can be descripted as some scalar field. So my physical object is just a function q = f(x, y, z). Imagine that we stretched out the space proportionally along all axes by some factor. You can say that my object has changed. It's bigger now. But actually, if you look closely, it is the same. It works the same way. Every part interacts with each other the same way. It's just resized. But you can't detect it (if you have resized too).
So what stops the universe to have hydrogen atom of 10m radius. If every part (every particle, wave, scalar field, vector field, whatever..) is resized then it should work the same way (may be until we get interation with it). So how does universe preserve distance?
P.S. A lot of commenters point out to the consept of measure. Let me clear one thing.
Suppose we take a line segment of 1 m and a line segment of 2 m. How do they differ? The most popular answer will be that they have different length. But how do they differ internally, I mean as sets of points. Both of segments are continuous sets of points. We can map one of them to another without loss of any point. So this segments have the same internal structure as sets. They are equivalent. This is what I'm asking for. How does the universe differ segments, that are internally the same.
 A: If you define your metre to be one tenth of the radius of a hydrogen atom then a hydrogen atom has a radius of $10$ metres.
But in the SI system the metre is defined to be the distance travelled by light in a vacuum in $\frac 1 {299,792,458}$ seconds, and the second is defined to be the time duration of $9,192,631,770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental unperturbed ground-state of the caesium-133 atom.
So in the SI system the radius of a hydrogen atom is definitely not $10$ metres.
If everything in the universe (including fundamental constants like the speed of light) were somehow "rescaled" then the SI metre would be rescaled too.
A: The universe doesn’t preserve distance in general. Distant objects can collide and nearby objects can separate. So there is not some generic aspect of the universe which treats distance as any sort of a conserved quantity.
Now, having said that, we recognize that systems with some sort of stable characteristic length scale are special and we can examine them to see what is special about them.
Atoms are a good example because they have a stable length scale. Their length scale is given by a minimum in the potential energy. At distances larger than the minimum the Coulombic force is attractive, while at distances smaller than the minimum quantum mechanical effects like the exclusion principle and the uncertainty principle produce a repulsive effect. The balance between these two produces a stable structure with a size determined by the minimum in the potential.
Another good example is a star. Hydrogen gas far away is attracted by gravity, and as it gathers closely the gas is pushed away by pressure. This balance between the attractive force of gravity and the repulsive force of pressure again leads to a minimum in the potential energy. To make the star bigger would require working against gravity and to make it smaller would require working against pressure.
So, at least in many cases, systems with a characteristic length scale have opposing forces which create a length at which some overall potential energy has a minimum. This minimum energy configuration is then the stable length scale since it requires work to make the length either longer or shorter.
A: Basically, when we talk about structured objects such as the Riemannian manifolds that we use to model spacetime, we find that there is not a single and unique such manifold. But that there are others that are isomorphic to it. It's one aspect of the problem of points in relativity, where in a sense, they're shown not to exist - but in the right sense, do. This is emphasised in Category Theory where isomorphism is taken seriously and is in fact part of the very structure of the theory. Mathematically, this is how Einstein's notion of General Covariance is understood abstractly.
