B Is the Universe discrete or continuous?

[ProfuselyQuarky]

Apologies if this question has been asked already. I've been given resources to help me understand, but it's been hard for me to wrap my head around the answer and, for that matter, it is difficult to understand a text when you have to look up every other word (an exaggeration, but you know ... ). I've been taught that General Relativity says that the universe is continuous. How is that so if the universe is made up of matter, which is discrete?

 

[Dale]

In principle two discrete objects can be located a continuous range of distances apart from each other. So discreteness of matter does not imply discreteness of space.

 

[fresh_42]

In principle two discrete objects can be located a continuous range of distances apart from each other. So discreteness of matter does not imply discreteness of space.

But continuity of space can't be proven, doesn't it? So discreteness of space could be possible?

 

[ProfuselyQuarky]

But continuity of space can't be proven, doesn't it? So discreteness of space could be possible?

Which is one thing that makes me confused ...

 

[williamwaco]

I am ignorant too. That said I believe your premise that the universe is made of matter is incorrect. It contains matter.

 

[Ibix]

Discreteness of matter doesn't prove discreteness of spacetime, as @Dale says. General relativity models spacetime as a differentiable manifold - the differentiability implies continuity not discreteness.

However, it's only a model and it doesn't work right in some circumstances, for example where quantum effects are important in the source term (i.e., what's the gravitational field of an electron at 0.01nm from it?). So we expect it to be superceded by a quantum theory of gravity. That may imply a discretisation of spacetime (not just space) or it may not. I gather that at least some of our current candidates do discretise spacetime, but we don't really have any evidence either way yet.

 

[Dale]

But continuity of space can't be proven, doesn't it? So discreteness of space could be possible?

Which is one thing that makes me confused ...

But that is a different question from what was asked in the OP.

The question asked in the OP is how is the continuousness of spacetime in GR (spacetime is modeled as a smooth pseudo Riemannian manifold) compatible with the observed discreteness of matter (matter is modeled as quantized excitations of fields)? The answer is simply to point out that the assumption of incompatibility is wrong. It is perfectly possible to have discrete matter and continuous space, and in fact this is the mathematical framework used in all current physical theories.

Whether or not it is possible to formulate physics in a discrete spacetime is a different question which is not required nor implied by the discreteness of matter.

 

[Jeronimus]

How would it even be possible to have discrete space, hence "space pixels" in a sense, and yet measure the diagonal of a square to be √d²+d² , d being its edge length?

It seems impossible to model such a discrete space where if you were to walk an imagined square from bottom left to top right in two different ways,

1) bottom left -> bottom right -> top right
2) bottom left then diagonally to top right

you would measure about the same lengths we get within the "physical world" we live in. The amount of "space pixels" or discrete units you would move(jump) to, would not correspond or even be close to √d²+d² with d being the edge/side length of the square, if you were to traverse the square diagonally as in case 2).

Well, at least i cannot imagine how this would be possible. Maybe my imagination just fails me.

 

[Jeronimus]

To put the above differently. Imagine a big cube. Now form some other 3 dimensional objects you consider to be your indivisible unit of space or "space pixel" .

How could you possibly arrange your space units inside this cube, such that you would end up traversing roughly d√3 units of space when moving diagonally from one corner to the other, whereas if you moved to the same endpoint by walking the edges of the cube, you would end up with roughly 3*d units as you do in the real world?

edit: Also keep in mind that any of those units of space you filled the cube with, would have to be capable of acting as the corner of another imagined cube subject to the same rules above.

 

[PAllen]

To put the above differently. Imagine a big cube. Now form some other 3 dimensional objects you consider to be your indivisible unit of space or "space pixel" .

How could you possibly arrange your space units inside this cube, such that you would end up traversing roughly d√3 units of space when moving diagonally from one corner to the other, whereas if you moved to the same endpoint by walking the edges of the cube, you would end up with roughly 3*d units as you do in the real world?

edit: Also keep in mind that any of those units of space you filled the cube with, would have to be capable of acting as the corner of another imagined cube subject to the same rules above.

You are assuming space is discrete, but rulers are continuous mathematical objects. If the ruler is also discrete at the same fundamental level as spactime (or larger, since it is matter), then it cannot probe your hypothetical microsquare boundaries. Instead it would effectively count the number of micro-squares diagaonally versus horizontally, coming up with the same answer as the continuous model to any achievable precision (assuming the discretiztion is very small).

Note, I am not arguing that spactime is discrete, just that it far from trivial to experimentally determine. (I've heard nasty rumours from spacetime that make me think it is not discrete ; [stolen from a science fiction story that had one scientist stating space is discrete and time is particular; the listener responded "how very nice of them"]).

 

[Jeronimus]

You are assuming space is discrete, but rulers are continuous mathematical objects. If the ruler is also discrete at the same fundamental level as spactime (or larger, since it is matter), then it cannot probe your hypothetical microsquare boundaries. Instead it would effectively count the number of micro-squares diagaonally versus horizontally, coming up with the same answer as the continuous model to any achievable precision (assuming the discretiztion is very small).

Forgive my ignorance but i really cannot wrap my mind around that. I would love to assume that both space and the ruler are discrete but i cannot see how i would program a 3D game for example in which both the space is discrete AND the ruler is discrete, and then when i move the ruler to measure a discrete square inside my discrete world i would end up with the diagonal being roughly d√2 (given a sufficiently large square made up of a lot of space units).
How would i model such a world inside my computer?

edit: For example, i could not simply program a 3D world made up of tiny cubes in a standard grid, my ruler is also made of and then place my ruler diagonally inside a larger cube or square.
To measure the length, i would have to measure the cubes or squares the ruler occupies until it reaches the opposing corner.
Never mind the problem of it being impossible to define what walking diagonally in discrete steps within this micro-cube/square world would mean, the ruler could never possibly occupy d√2 squares or d√3 micro-cubes no matter how i would define walking diagonally.
If for example i defined it as moving one square to the right, then up, then right and up again and so on... i would end up with the ruler measuring 2d instead of d√2. If i defined it as moving to the next microcube which is diagonal, i would end up with just d.

So you are telling me that there is actually a mathematical model i cannot think of, which solves those issues.

 

[PAllen]

So you are telling me that there is actually a mathematical model i cannot think of, which solves those issues.

Yes, and it is really very simple. First, there are no 1d or 2d objects in the real world only 3d. And 3d objects and 3d space both consist of volume elements. For the simplicity, assume all fundamental volume elements are spherical. Then rectangle 3X4 units has 3 spheres by 4 spheres with diagonal of 5 spheres. The way you measure it is with a ruler consisting of a chain fundamental spheres. 5 ruler spheres ruler matches 5 sphere diagonal.

There are full blown sophisticated models of this that are much more realistic. Some overly simplistic models of this type have been ruled out by experiment, but most have not. Neither have they been supported by experiment. But your objections are certainly not the issue.

 

[Dale]

This is now getting into "beyond the standard model" territory. With that excellent reply, this thread is closed.