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I read here (Feynmann Lectures, Lecture 42) that "Just as time scales change from place to place in a gravitational field, so do also the length scales. Rulers change lengths as you move around." (Rulers also change as you re-orient them; see footnote 2, in the link.) That reads to me like chemical bonds and other internal forces holding the physical ruler together do not stop it from changing length due to changes in space-time induced by the mass.

However, in cosmology, where all of space is expanding, galaxies become further apart from each other, but an individual galaxy, itself, does not expand (due to the forces that hold it together), people do not expand (also due to the internal forces that hold us together), nor do physical rulers. That is, lengths of physical rulers do not change because of their internal forces. I believe distance, in cosmology, as measured by the physical ruler is called "proper distance", vs "co-moving distance", which does expand as the universe does.

In the first paragraph, the change to space affects the length of the ruler, regardless of the ruler's internal forces but, in the second paragraph, the change to space does not affect the ruler because of the ruler's internal forces. I am confused regarding why the physical ruler's internal forces do not prevent length change in the first paragraph, but do in the second. After all, in both cases, space is changing in a way that affects length or distance. Maybe the reason is that the type of change to space is different since in one case it is caused by matter and in the other case it is caused by dark energy?

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  • $\begingroup$ Related, possible duplicate: physics.stackexchange.com/q/2110/123208 $\endgroup$
    – PM 2Ring
    Sep 30, 2019 at 6:47
  • $\begingroup$ BTW, dark energy is a bit of a red herring here. Space doesn't need dark energy in order to expand (see FLRW metric), but dark energy causes the expansion of space to accelerate. $\endgroup$
    – PM 2Ring
    Sep 30, 2019 at 6:54
  • $\begingroup$ My question is different than the possible duplicate in that it is about the intersection of general relativity theory and big bang theory, which is not in the duplicate. This question also includes the answer to the possible duplicate, stated as part of the background to the question. $\endgroup$
    – David
    Oct 2, 2019 at 23:13

2 Answers 2

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It's a good question and you are right, there seems to be an unresolved contradiction at the heart of cosmology.

It can be solved by presuming that, as the universe expands, all objects - people, atoms, galaxies etc... expand too.

This leads to an alternative interpretation of redshift. If the size of atoms and Plancks constant were lower in the past, then from $E=hf$, the energy of photons arriving from a distant star would be lower, hence the redshift.

Here, https://vixra.org/abs/2006.0209 in figure 3 is a cosmology that doesn't have the problem you highlighted with your question.

The alternative approach naturally predicts that the matter density will be measured as 0.25 or 1/3 depending on how it's measured. This seems to be the case.

So perhaps the alternative theory answers your question.

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When Feynman talks about the lengths of rulers changing he is talking about the coordinate length of the rulers. That is, suppose I have a ruler on Earth and try to measure its length in terms of degrees longitude $\lambda$ and latitude $\phi$ such that

$$length = \sqrt{\lambda^2+\phi^2} $$

then obviously the length of the ruler depends on where on Earth it is and what its orientation is. However, the length of the ruler measured by a local observer using, say, a laser as reference always remains the same.

Note that this isn't even properly related to gravity, but is simply a consequence of the coordinates used. In a similar vein the size of galaxies measured in co-moving coordinates changes as space expands. The proper size of the galaxy on the other hand stays the same.

So, to answer you question, there is no difference in the changes in length of a physical rulers due to gravity or due to the cosmological expansion of space.

PS. For sake of not over complicating things I have the ignored the potential effects due to tidal forces in either case.

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  • $\begingroup$ Thank you. I think Feynmann may have meant, for instance, that if a mass has a flat surface, in the $z=0$ plane, and that surface has a huge, but finite, area, and the mass is uniformly distributed between $z_0<z \leq 0$, below that surface, then the length of a ruler near $z=0$ and near the center of that surface, depending on its orientation, will be different than the length of the same ruler at $z \gg 0$. $\endgroup$
    – David
    Oct 1, 2019 at 17:05
  • $\begingroup$ However, in the cosmological expanding universe case, if you measure the length of a physical object, with a physical ruler, (both held together by their internal forces) at t0 and again at t1 (where the universe expanded between those times), you will get the same numbers. The internal forces of the ruler and object kept the measurement of that "proper distance" the same. Hence, in both cases, although we have changes to space that affect length, in one case the internal forces of physical things are a factor and in the other case they are not. $\endgroup$
    – David
    Oct 1, 2019 at 17:07
  • $\begingroup$ Please see abyss.uoregon.edu/~js/cosmo/lectures/lec06.html : "space contracts near mass and dilates away from it". "Time dilates near mass and contracts away from it". In the extreme, black holes "spaghettify" us. So, the length of a ruler, very near a massive object, will be different that it will be if it is far away from it, despite the internal forces that hold it together. However, the cosmological expansion of space does not affect the "proper distance", between one end of the ruler and the other, due to the internal forces that hold it together. Why is that? $\endgroup$
    – David
    Oct 2, 2019 at 6:00
  • $\begingroup$ The expansion of spacetime is a time dependent scaling of a metric of flat spacetime, so spacetime stays flat and things move with, not through, space. And time is not expanding/changing, anywhere. Hence, this is nothing like gravity which could do work against actual forces, such as electromagnetic. This is the reason given for why expanding space time is not changing atoms, or sizes of rulers, etc. However, if the magnitude of a force is a function of distance, then, as distance scales out, why wouldn't the configuration of things, where that configuration depends on that force, also expand? $\endgroup$
    – David
    Oct 7, 2019 at 23:41
  • $\begingroup$ @David so are black holes bigger on the inside? $\endgroup$ Mar 10, 2021 at 19:20

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