Genady

A friend has sent this question to me. I have it solved with linear algebra (and some hand waving). Can you find a shorter way to the answer? (It's not a homework, not mine anyway. My homework times long gone.) A gardener collected 17 apples. He finds that each time he removes an apple from his harvest, he can share the remaining fruit in two piles of equal weight, each containing 8 apples. Show that all apples are the same weight.

No, position and time are described by real numbers in QM.

It is true. For observer 1, a particle P is at rest and a particle Q accelerates toward it. For observer 2, Q is at rest and P accelerates toward it. Susskind answers exactly this question in the lecture that I've linked in the post above yours.

An easy derivation, with answers to the audience's questions relevant to the above discussion, can be found here, starting about 30 minutes into the lecture:

The center is just an origin of coordinates. Of course, the premise is homogeneity and isotropy of the entire space. For any coordinate system, a particle in its origin does not move anywhere, but all other particles accelerate toward it. The same as in the Hubble expansion.

Each particle is attracted equally from all directions, and these will cancel indeed. But, each particle attracts all other particles and they will all accelerate toward it. This is so for all particles and thus all particles accelerate toward each other. So, the entire thing homogenously and isotropically contracts, or slows its expansion. For a bit more precise treatment, take any particle as a center and consider particles on a sphere of radius R around it. Each particle on the sphere, according to the old Newton's theorem that holds in GR as well, is attracted to the center as if the mass of the entire ball of radius R is in the center, and effect of each larger sphere on it is 0. This holds for any point picked as a center. The fully precise result in GR follows from increasing mass density in Friedman equation.

Yes, it would cause a net gravitational attraction and that would cause slowing of the universe expansion.

Not proved, according to this: "In discussions of the cosmological constant, the Casimir effect is often invoked as decisive evidence that the zero-point energies of quantum fields are “real.” On the contrary, Casimir effects can be formulated and Casimir forces can be computed without reference to zero-point energies. They are relativistic, quantum forces between charges and currents. … I have presented an argument that the experimental confirmation of the Casimir effect does not establish the reality of zero-point fluctuations. Casimir forces can be calculated without reference to the vacuum ... . The vacuum-to-vacuum graphs (See Fig. 1) that define the zero-point energy do not enter the calculation of the Casimir force, which instead only involves graphs with external lines. So the concept of zero-point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity." Casimir effect and the quantum vacuum R. L. Jaffe Phys. Rev. D 72, 021301(R) – Published 12 July 2005

If there were much more of it around, it would affect the cosmological expansion. Perhaps it could be detected this way.

The distribution of the DM in space now is very much non-uniform. Since it doesn't clump, was its distribution as non-uniform soon after the BB?

Is it necessarily so? If it could interact electromagnetically by completely reflecting an EM radiation it wouldn't emit a thermal EM radiation, would it?

Many think that it interacts not only via gravity but also weakly, i.e. made of WIMPs.

I don't know why it's called dark, but not interacting electromagnetically makes it rather transparent than dark.

We also know that it was there by the time of "recombination", about 380 000 years after the Big Bang.

We know where it is and how much of it is there. We don't know what it is.

Nice. Except the very last statement. That is a kind of "wishful thinking" fallacy.