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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?

David
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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.

TimRias
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There is confusion on this subject. Perhaps it would be better to think of it this way: the ruler doesn't physically change length - rather it goes out of calibration. According to our ruler, in this relativistic frame light travels further than one foot in one nanosecond. We know that the speed of light is a constant in all reference frames, and from general relativity we know that our nanosecond is longer in this reference frame, so accordingly our ruler must be made longer and its units re-calibrated to the new length. The units of length would change in direct proportion to units of time. Our units of measure change size but not the things we're measuring.

Jim Kelley
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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.

John Hunter
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