Genady

... I'd toss a coin.

Do I understand correctly that they do not talk about measuring time dilation due to a passing gravitational wave but rather about changes in the background time dilation due to a motion of the sources of that time dilation?

Yes. Yes. It is a linearized GR approximation.

I've looked at the derivation again. In simple terms, it boils down to the fact that gravitational waves are transverse waves in spacetime. So, in coordinates where they move along t- and x-axes, they perturb the metric in the orthogonal y- and z-axes.

Here this analogy has been developed somewhat further: This comes from:

... should have been posted in the Speculations, by definition. Leave my stream alone.

The reply was: It does not actually answer the question, "how fast". To answer this question, one needs to take the derivative, \( (\frac 1 {1-t})' = \frac t {(1-t)^2} \). This grows infinitely when \(t \rightarrow 1 \). Thus, the answer to the question "how fast is 'fast enough'?" is, "infinitely fast".

I don't think so. I don't assume anything about how a depends on t . I only refer to how the distances depend on a .

Energy is just a time-like component of 4-momentum.

Yes, what is spatial and what is temporal depends on frame of reference. In the approximation that I refer to, the frame of reference is fixed in such a way that the gravitational waves are small perturbations in flat Minkowski spacetime which move along, say, x-axis. Then, they cause length contractions and expansions in the y- and z-axes.

The main feature of this property is that it is a scalar value that is conserved.

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.