If I had a measuring tape between earth and a distant galaxy ...?

First, I assume that only one end of the tape is fixed -- say, held by a person on earth. If both ends are fixed (held by people in two distant galaxies), the expansion would stretch and tear the tape (just as would happen on earth if you connected a measuring tape to two cars and drove them apart).

But what happens if the tape is anchored only to earth?

I think the best answer to your question is different today than it would have been in 1990 (before the acceleration of the cosmic expansion had been found).

In 1990, the answer would have been this: The expansion started in the early universe, but is now just following the rules of gravity. Distant galaxies are still moving away from us simply because they started moving away long ago. In this case, a distant galaxy would become farther and farther from the end of the measuring tape, but the tape would not be affected by the expansion of the universe.

I think the answer today, however, is this: There is an acceleration to the expansion. Although we're not sure of the reason for this, it acts as though there is a repulsive acceleration that is proportional to distance. In the case of the earth and sun, this acceleration has the effect of causing the earth to be slightly (extremely slightly!) farther from the sun than it would be otherwise; but because this repulsive acceleration is far less than the attractive acceleration due to earth-sun gravity, the earth-sun system is gravitationally bound.

This same acceleration applies to your tape measure. It exists between any two points on the tape measure; and the greater the distance, the greater the acceleration. As Newton showed, acceleration translates into force.

Suppose you apply a stretching force between two points on the tape measure. If the force is very small, it will have essentially no effect. As the force increases, the tape may stretch, and it will eventually break.

So my answer is that the tape will not stretch in the same way as the expansion of the universe, but it will stretch (and break) in the way that the metal (and the forces that hold the metal together) responds to a force that increases with distance. The mysterious cosmic acceleration will apply a stretching force to the tape, and the molecular bonds within the tape will act to hold it together; eventually, something will give.

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You often hear that the solar system is gravitationally bound, and does not participate in the expansion of the universe. (I mentioned the reason for this above.) This is really very similar to my answer about the tape measure. The solar system is bound by gravity, and the tape measure is bound by the internal forces that keep it together as a solid. The difference is that the size of the solar system is small on an astronomical scale, so the amount of cosmic acceleration is very small. If a tape measure were billions of light years long, however, the cosmic acceleration would be large. (I'm trying to imagine going to my local Ace Hardware and asking for a billion-light-year tape measure.)

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Regarding Planck units: As far as we know, the fundamental constants are truly constant in our universe; they do not change over time and do not depend on location. Therefore, the Planck units of space and time are of fixed size, and do not change as the universe ages and expands. What is expanding is space. It's not new particles that are being added, but simply new space. I'm not aware of any energy cost in increasing the volume of space relative to the Planck length.

I am no expert, and I may have interpreted your question wrong, but here is my take on it.

I think space would expand on both ends of the tape and in between, while the nuclear force holding the atoms together would keep the tape at the same consistent size. Instead of space being added, I think it is more along the lines of the same amount of space being stretched (perhaps made less "dense" too?), much like how muscles are built through expansion, not addition, of muscle cells.

Basically, it would appear that both Earth *and* the other galaxy would move away from the tape at the same rate (ignoring proper motion) unless we were holding onto the tape somehow.

As a tired old man who has spent much of his years pondering what Alberts equation was missing, I believe that is EXACTLY what happens! I have approached many astrophysicists with the idea that what they fail to see is that not only is the Universe expanding, but all matter in the Universe is expanding along with it. That means your measuring device is too! They all look at me as if I am a fool but it explains one seemingly unexplainable event also. The speed of light. This would mean the speed of light is NOT a constant, we simply perceive it as a constant because we dont realize that WE are also expanding. Say you had 2 balloons both being inflated at the same proportion while being moved apart from each other in direct proportion too. You had 2 people on those baloons that were also expanding at the same rate. Although they are moving away from each other, from their perspectives, they arent!

Of course they laughed at Columbus and many others so maybe some day they will look back long after I am dead and realize I was right all along. From my point of view, there never was a big bang, the bang was only our perception of a Universe that has ALWAYS been expanding. Just because we look back at it as a small, dense ball doesnt mean it was!

As a matter of fact, I dare and defy ANYONE to give me one iota of proof that this is NOT a possibility!!!

It is also perfectly clear from this point of view that time is a medium that we cannot see. Time is a constant, we simply travel through it.

I'm not an expert on the subject, but I think of the expanding universe as an oil slick, expanding in all directions, and each point in the slick getting farther away from each other point. Add in a third dimension, and the picture is more complete. And as the tape is part of the slick, the units would simply expand along with the slick, and so even if the galaxy did move away, how could we tell? I've read the answers before me; it's a fascinating topic. Thanks for asking.

There are many things (parameters) of Space that one can measure. But in general, to measure Space means its dimensions. The contemporary notion is of 4 dimensional Space. We were (or rather I was) told that the implication is that we are all on the surface of this 4D universe (at the same instance that is 1dimension, t=const.) and radius = B = cT & T is time from the big-bang, upon which the whole theory rests. A diametrically opposite point, to where I stand is 2B (=D) away. I am not very sure about my 4D geometry (of 4D sphere), but its surface hyper-area (by induction) can be [(πD)³ / π] = 8π²B³ where B = cT. Of course the switch between time and spatial dimension is rather tricky that one can falter and trip over.

"...the universe has no center nor edge; how can something be added and from where?..."

Imagine that you are a creature whose "universe" is the thin film of a soap bubble. Imagine a child (of whom you can never be aware because (s)he does not exist within your universe), this child is inflating your soap bubble. Your universe is expanding, but what is being added, and where?

From your point of view, everything in your soap bubble universe seems to be moving away from everything else. Just like what astronomers seem to see when they look at OUR universe.

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I don't know what to say about a physical measuring stick that was billions of light years long. When you're looking at the far end, you'd looking at where it was billions of years ago. No doubt, Herr Einstein would have something to say about that. I'm no Einstein.

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