Questions about the expansion of space on galactic scales

[KurtLudwig]

TL;DR Summary

Is the Hubble constant the same at scales of our galaxy as it is for the whole universe? Is it an average, but locally there are variations, where some regions expand faster and others slower?

How would the expansion on a scale of 10 kpc be measured, by a red or blue shift?
How can expansion of space be differentiated from the velocity of stars?
It seems that the expansion of space weakens the effects of gravity?

 

[Bandersnatch]

There's no expansion in bound systems, including galaxies.

 

[mathman]

The "variations" result from the fact that galaxies are moving.

 

[KurtLudwig]

Now I am at a loss. Isn't our universe expanding?
Then why wouldn't galactic space expand?

 

[Ibix]

Isn't our universe expanding?

Stuff is getting further apart, yes, but only on large scales. On smaller scales, stuff may be getting closer together or staying the same distance apart.

Then why wouldn't galactic space expand?

"Space expanding" isn't a particularly accurate description of what's going on, although it's fairly common because there isn't a "soundbite" explanation that is genuinely accurate. Basically, stuff continues to fly further and further apart because nothing is stopping it. It can't do that in a flat spacetime, but a more-or-less uniform density of stuff produces a curved spacetime where it's possible for everything to be flying apart from everything else.

However, on small scales, gravity is sometimes enough to stop stuff flying apart. That includes things like galaxies and the local group. These aren't expanding because they were sufficiently over-dense to stop stuff in a "small" region from escaping.

 

[Bandersnatch]

Now I am at a loss. Isn't our universe expanding?
Then why wouldn't galactic space expand?

Liddle gives a brief answer in chapter 3.2., albeit without any mathematical support. I know you have the book.
You can get a feel for why that is by calculating the escape velocity from some mass (a galaxy or a cluster of galaxies) and using Hubble's law to find out at what distance does the recession velocity exceed that escape velocity. This will invariably be larger than the size of the structure under consideration, until you get to the sizes of superclusters or thereabouts.
This should suggest to you that a system (e.g. two galaxies, some distance apart, or stars within a galaxy, or planets in a stellar system) that is bound by gravity strong enough to overcome the recession velocity, will not expand but stay bound. Such systems are decoupled from the Hubble flow, and with time form isolated islands in the increasingly empty expanding universe.

 

[KurtLudwig]

Thank you Bandersnatch for the above detailed explanation. I have re-read 3.2 and will re-read Modern Cosmology.

… but a more-or-less uniform density of stuff produces a curved spacetime where it's possible for everything to be flying apart from everything else.
Please explain

 

[phinds]

Thank you Bandersnatch for the above detailed explanation. I have re-read 3.2 and will re-read Modern Cosmology.

… but a more-or-less uniform density of stuff produces a curved spacetime where it's possible for everything to be flying apart from everything else.
Please explain

Explain what? Your question has already been accurately answered. Bound systems don't move apart within themselves (because gravity holds them together) but do move apart from each other.

 

[KurtLudwig]

You are right.

 

[PAllen]

...
Basically, stuff continues to fly further and further apart because nothing is stopping it.
It can't do that in a flat spacetime, but a more-or-less uniform density of stuff produces a curved spacetime where it's possible for everything to be flying apart from everything else.

...

I'm sure you know this, but this can happen in flat spacetime. It is the Milne cosmology. You have to assume gravity doesn't exist or more absurd, that nothing has mass, else spacetime would be curved. But you certainly can have isotropic, homogeneous expansion with arbitrarily large superluminal recession rates and arbitrarily large 'cosmological redshift' in flat Minkowski spacetime. It is unfortunate that a majority of cosmology sources falsely claim these things require curved spacetime.

 

[Ibix]

I'm sure you know this, but this can happen in flat spacetime. It is the Milne cosmology. You have to assume gravity doesn't exist or more absurd, that nothing has mass, else spacetime would be curved. But you certainly can have isotropic, homogeneous expansion with arbitrarily large superluminal recession rates and arbitrarily large 'cosmological redshift' in flat Minkowski spacetime. It is unfortunate that a majority of cosmology sources falsely claim these things require curved spacetime.

I do know that, and you are correct that I overstated the case. However, you are also correct that the Milne cosmology is weird and you pretty much have to require GR to be wrong for it to make sense as a physical cosmology (as opposed to an interesting mathematical trick).

 

[PAllen]

I do know that, and you are correct that I overstated the case. However, you are also correct that the Milne cosmology is weird and you pretty much have to require GR to be wrong for it to make sense as a physical cosmology (as opposed to an interesting mathematical trick).

Well, it isn't quite a mathematical trick, IMO. What it shows is that the origin of key features of cosmology is not curvature, per se, but the ability to have an everywhere isotropically expanding congruence (isotropy everywhere implies homogeneity). Only very special GR solutions allow such a congruence, but SR (flat spacetime) is one of the solutions that does, so these properties are not deviations from SR behavior, as often misleadingly stated by some authors. It also emphasizes that expansion is best (IMO) viewed as a property of the congruence not of space. It is the global geometry of the universe that allows the existence and detailed properties of the congruence.

 

[Ibix]

It also emphasizes that expansion is best (IMO) viewed as a property of the congruence not of space.

To paraphrase Apple, "there's a tensor for that". Agreed.

However the Milne cosmology remains a very special case. In particular, the expanding congruence in general FLRW spacetimes covers all of the spacetime, while Milne's covers only part of Minkowski spacetime. So I think (?) it's correct to say that you can't have an expanding congruence in Minkowski spacetime that covers all of it.

 

[PAllen]

To paraphrase Apple, "there's a tensor for that". Agreed.

However the Milne cosmology remains a very special case. In particular, the expanding congruence in general FLRW spacetimes covers all of the spacetime, while Milne's covers only part of Minkowski spacetime. So I think (?) it's correct to say that you can't have an expanding congruence in Minkowski spacetime that covers all of it.

True, but a Milne patch is a well defined, unbounded manifold (each spatial slice has infinite area). However, it can be analytically continued to the whole Minkowski space - eliminating geodesic incompleteness. Other FLRW cosmologies cannot be continued, so the geodesic incompleteness is irremovable. Just another way of saying there is a true singularity.