Volume of fundamental unit of discrete space and expansion of space

Question

In this video, the following is said between 06:07 to 06:38:

Now one of the interesting questions is how do you get from the quantum of space and time to Einstein’s theory of general relativity? It’s because when you add mass and energy, you can distort the shape of the little volumes. Now that seems like it wouldn’t make sense because I said that there were quanta of lengths, areas, and volumes, but you need to remember that you are bending space and time and that has the property that you can distort the local definition of space in such a way that the volumes are unchanged. So these ideas of quantum spacetime are all good and all, but are they real?

Suppose that space is made up of small discrete cubes of volumes $10^{-105}(m^3)$, or 3D pixels each of volume $10^{-105}(m^3)$. When space is distorted due to the presence of mass or energy, the volume of cubes would get distorted. The volume would either remain the same, increase, or decrease from the standard Planck volume of $10^{-105}(m^3)$. But the volume of each space unit has to remain $10^{-105}(m^3)$ at all times because that's the fundamental unit of discrete space. Where am I wrong?

Also when the expansion of space takes place, do more discrete space volumes get created out of nothing, or the volume of a cube gets stretched and breaks down into two individual cube volumes, and so on?

Could you please help me with the queries above? Thank you!

Answer 1

You seem to think space-time needs to be built out of regular voxels of unit volume and length. But this is problematic: as Weyl pointed out, if you have a square grid of such pieces diagonals will not approach $\sqrt{2}$. Indeed, for any repeating pattern there will be preferred directions, something not observed in reality. On the other hand, for suitably disordered underlying graphs one can get spacetime isotropy.

Loop quantum gravity can get a normal-looking spacetime even on a regular grid by having positions be quantum superpositions; see the answer here. Treating lengths and volumes as quantum variables allows us to ignore the underlying network or voxel structure to a large degree, which is exactly what one would want for real physics. Graphs can also model curved spacetimes, which is what the loop quantum people are doing.

The exact dynamics of expansion of space in this setting appears to be complex: see the accepted answer to this question. But it does involve potentially adding new nodes and links to the spin network, it is just that they do not straightforwardly correspond to distance and volume.

Answer 2

Let me start by saying that in Quantum Loop Gravity (QLP) space is not literally built up out of enormously small units of space, like constructing a brick wall by gluing together the elementary units, the stones. It's quite hard to imagine how space then is built up out the Planck volumes of space (I can't).

It's true that for a cube that I made out of some flexible material, the volume is maximal when one (i.e. all) three angles (at the corners of the cube) is 90 degrees. If I distort the cube, letting the sides stay straight and of equal length (i.e. performing a transformation that preserves distances but not angles). Obviously, the volume will decrease. This is the case in the non-quantum space(time).

The unit volumes in QLG don't follow the rules of "normal" geometry. For example, there is a commutation relation between the volume and area of such a unit, similar to the one in QM. The more certain the unit's volume, the more uncertain the area of the unit becomes and the other way round. This gives already a little bit of sense of the theory. It's this uncertainty that can let the unit volume stay constant while changing its area. Basically. I don't know the exact details of the theory, but it is possible for this to happen.