Genady

In the approximation of week gravitational waves, the spacetime metric is perturbed only in the two-dimensional plane perpendicular to the wave propagation.

No. Solar energy is a property of solar radiation.

(Energy - Wikipedia)

No, it does not mean that. As Markus will clarify that, I think the following quote from Misner, Thorn, and Wheeler will help.

Homework in combinatorics?

This is an example of how In this case, it is a concept of mass-energy contribution to a system from its parts.

As an illustration to the Markus' explanation above, consider this example from Gravitation by Misner, Thorn, and Wheeler:

Just to clarify, AFAIK, the concept of mass is not applicable to universe and thus it cannot be described as finite or infinite, regardless of the curvature.

This is not English. See English alphabet - Wikipedia. Plus, the topic of an alphabet is not Physics.

Reminder: this is Physics forum.

This question belongs to another forum 😉

When you take second derivatives of metric, you get rates of its deviation from the flat spacetime.

Curvature is encoded in second derivatives of metric. So, any tensor that depends on second derivatives of metric, describes curvature in some way. Einstein tensor describes curvature in a way that can be related to the spacetime's physical contents.

Let me try to explain it with a bit of algebra. In an expanding homogeneous isotropic universe, a distance between any two points - let's call them, galaxies - is proportional to a number, \(a(t)\), called scale factor, which increases with time, \(t\). So, for example, if a distance between some two galaxies at some moment is \(D\) then later, when \(a(t)\) is twice as large, the distance between these two galaxies is \(2D\). Thus, this distance increases with time as \(a(t)D\). If the universe is finite, then there is a largest distance in it, which, just like any other distance, is proportional to \(a(t)\). Let's call it, \(a(t)L\). The only way for the \(a(t)L\) to become infinitely large is that \(a(t)\) becomes infinitely large. But, if \(a(t)\) becomes infinitely large, then distance between any two galaxies, \(a(t)D\), becomes infinitely large. IOW, all galaxies become infinitely far from each other. We of course know that it isn't so. Thus, either the universe was finite and remains finite, or it was infinite to start with.

The "diagonal argument" can be expressed purely algebraically, without lists, rows, diagonals, and any other "visual aids". I think that such algebraic presentation would eliminate a lot of confusion.

A really large number is finite by definition.

No, in a continuous process it cannot. What is "number TREE3"?

Here is an example: a superposition of two Hamiltonian eigenstates, \(\psi_1\) and \(\psi_2\), with the energies \(E_1\) and \(E_2\): \(\frac 1 {\sqrt 2}(e^{-iE_1t}\psi_1+e^{-iE_2t}\psi_2)\). The probability is squared modulus of this function, which includes a time component, \((E_1-E_2)t\).

I just try to clarify the question because I don't know what they were referring to in the thing you've read. Let's consider an example, a particle in a Hamiltonian eigenstate \(\psi(x)\) with energy \(E\). It evolves in time as \(e^{-iEt}\psi(x)\). The probability density for it to be in position \(x\) is \(\bar {\psi}(x) \psi(x)\). This outcome does not depend on time and thus doesn't change if the sign of time is flipped.

You mean, like in Newtonian mechanics?

As QFT obeys SR, there is direction of time in the QFT as much as it is in the SR.

The domain of the function \(\frac 1 {1-t}\) excludes \(1\), as the domain of the function \(\frac 1 x\) excludes \(0\): (Domain of a function - Wikipedia)

Mathematically speaking, it cannot reach infinite size, but it rather increases unboundedly as \(t \rightarrow 1\) from below. At \(t=1\), the formula is undetermined: mathematically speaking, \(\frac 1 0\) is undefined.

I did not miss that point, but my counterpoint was that it does not work mathematically. It reaches the infinity and momentarily changes to \(- \infty \) and stays negative after that. Distance cannot do this. P.S. You want the size to reach infinity and to stay infinite. Try another formula.

The universe was much more homogeneous just after the Big Bang, as is evidenced by the isotropy of the CMB radiation. It became less homogeneous with time, presumably because of the gravitational clumping. OTOH, expansion works against clumping, and an accelerated expansion even more so. Does anybody know what will happen in the future to the "scale of homogeneity"? and becomes negative after that.