What happens to the Planck length due to the expansion? Does it grow with the scale factor?
planck length is defined using fundamental constants G hbar c, which are not changed by expansion.
most things we are familiar with are not changed by expansion
here are some things that are not changed
the size of a molecule
the distance to the sun
the length of a ruler
the size of the north american tectonic plates in the earths crust
the distance to Alpha Centauri
the Compton wavelength of an electron
fundamental constants like G, hbar, and c.
none of these things are affected by the expansion of largescale intergalactic distances
(I'm not 100% sure of the following, so correct me if I'm wrong.)
Just some extra info, because I think the above might be a little confusing by itself. You have to be careful when speaking about "distance" when dealing with the expansion of space, because there are several ways of defining the concept of distance. Two of them which are relevant in this case are:
1. Comoving distance. This one is defined as the distance between two points on the comoving coordinate grid, and if there is nothing at work besides the expansion of space, two objects in the universe will retain the same comoving distance as space expands. It is unmeasurable with a normal apparatus, but very useful as a theoretical tool.
2. Physical distance. This is the distance you would measure with an ordinary apparatus. This one increases with the scale factor of space's expansion. However, due to the gravitational attraction between bodies that are close together, the distance between objects that are close to each other (Earth and Sun, etc.) does not increase noticeably with the expansion of space.
If anyone else has more clarifying remarks, I'll be glad to hear them.
what I mean by the distance NOW to some object is the physical distance now, which you could measure with ordinary apparatus, if you had taken the trouble in the past to set up for it.
the idea of simultaneity and time I'm invoking is the fairly common one of the CMB restframe. sometime people call it universal time.
comoving distance can be normalized to agree with what I mean---physical distance---at the present moment.
So in certain cases physical distance can AGREE with comoving.
The physical distance at the present moment (CMB rest time) is the distance that works in the Hubble Law. So it is basic. The recession speed at the present moment is equal to the physical distance at the present, multiplied by the current value of the Hubble (around 71 km/second per Megaparsec).
Ned Wright used to have a nice discussion somewhere on his website of how you could actually measure the physical distance to some galaxy at present, or equivalently the comoving, if you had prepared the setup. probably it is still there.
You distribute a lot of observers in a long line between here and the galaxy, and they all have radar distance measure devices. And all are AT REST WITH RESPECT TO THE CMB, so all their clocks can be approximately synchronized. And all at the same moment, which we call now, they measure the distance from each one to the next. And add it up.
The reason Ned Wright has a lot of observers in a long chain is that he wants the distance to be measured quickly, like in a few months or years, so that there isn't time for the galaxies to move much while the distance is being measured.
I couldn't find that discussion recently, it is or used to be in a section explaining comoving distance (because the comoving was the same as the physical at some present moment which you could have measured today if you had set up for it.)
In reality, physical and comoving distance have to be inferred using the LCDM model and the astronomical distance-scale ladder, which is itself a chain of interence, as I am sure you know. Alas the long chain of observers is not available.
the key thing though is to picture them all stationary with respect to the CMB (so none of them see doppler hotspots in the CMB sky). then asking them to each measure the radar distance to their neighbor all at the same time makes sense. otherwise it may not be defined
you asked about the expansion of small distances like earth-to-sun. Does it happen? I think not. We are talking about physical distance, just so there is no confusion.
The physical distance between nearby things, that are part of a gravitationally bound system, does not increase in accordance with the regular percentage Hubble law rule that large distances increase.
And, I suppose that also the radius of an event horizon of a big hole (black or white) is approximately constant (measurured in Plancklenght) as long as one can neglect Hawking evaporation, even if at some time in its existance there occurs a bounce?
If the Planck length expanded at the same rate as the expansion of the universe we would not be able to tell the universe is expanding. Light travels one Planck length in one Planck time interval. If the Planck length was half the current length when the universe scale was half what it is now then the scale of the universe back then would appear to be the same as the scale of the universe now. In a way the scale of the universe (or the scale of anything) is relative to the Planck length.
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