# Hubble's law (again)

1. Nov 7, 2006

### Thrice

Since the galaxies are receeding with velocity propotional to distance, I'm curious what happens at or beyond the point where the velocities approach the speed of light & how that's possible. One source I looked at said the galaxies are invisible. Another source said the spacetime is what's stretching & there's no speed of light limit to that. Yet another source said the concept is incoherent because you can't compare vectors at different points in spacetime (in different tangent spaces).

I don't know much about differential geometry yet, but I do know you physicists need to get your junk together. What is going on? Some math please.

2. Nov 8, 2006

### pervect

Staff Emeritus
Try Ned Wright's cosmology FAQ, http://www.astro.ucla.edu/~wright/cosmology_faq.html#FTL

tutorial http://www.astro.ucla.edu/~wright/cosmo_01.htm

and Lineweaver & Davis "Expanding confusion" paper.

http://arxiv.org/abs/astro-ph/0310808

The short answer is that it is not possible to compare vectors in a coordinate independent manner at different points in space-time. However, there is a coordinate dependent defintion of distance that most cosmologists use, comoving distance. The rate of change of this distance with respect to cosmological time defines a sort of velocity. This is what goes into Hubble's law.

So there is a way of defining the velocity, but one should be aware that this defintion is a) coordinate dependent and b) not compatible with special relativity.

What's particularly enlightening is that the "comvoing distance" does NOT reduce to the familiar notion of distance in SR in the limit of an expanding universe with a very low mass density.

So cosmologists do have something in mind when they talk about recession velocities and the "distances" of distant objects, but it's based on a particular coordinate system that's convenient and common, and it's also not SR-friendly.

3. Nov 8, 2006

### Thrice

Thanks, pervect. I'll work through that.

Incidentally I don't know if this is the right place to ask this, but would you guys be interested in starting a relativity wiki? Something along a more pedagogical approach, almost textbook-like. I can probably figure out how to do it in a few days.

4. Nov 8, 2006

### yogi

Ed Harrison in his book "Cosmology, the Science of the universe" takes the point of view that expansion is not limited by c as would be the case in SR - the Nebula are being wafted outwardly by expansion - it is space itself that is stretching rather than galaxies moving wrt to space - so in this sense there is no limit to the velocity since we are not dealing with relationships between relatively moving inertial frames. I think this is consistent with what pervect said

5. Nov 9, 2006

### pervect

Staff Emeritus
I wouldn't be terribly surprised if you have some more specific questions after doing some reading.

As far as the wiki goes - you are talking about a wiki in the generic sense, not Wikipedia, right? I rather doubt I'll find the time, I havaen't even been keeping up with what I really want to do for the Wikipedia wiki.

6. Jan 7, 2007

### Thrice

Hmm GR locally reduces to SR, right? So roughly would it be accurate to say the reason is because the metric in GR can't always be split into a "space" and "time" with definite meaning?

That other post on the front page reminded me that I hadn't posted a reply here. Incidentally, I think http://en.wikipedia.org/wiki/Distance_measures_(cosmology)" [Broken] page is quite useful for illustrating the different distance measurements.

Last edited by a moderator: May 2, 2017
7. Jan 7, 2007

### pervect

Staff Emeritus
GR reduces to SR locally. This means that if you have two bodies that are at the same point in space-time, or very close to each other, that you can determine their velocities using the techniques of SR.

Velocities are mathematically represented by vectors. If the two bodies are distant, you have to define a mechanism for transporting a vector at one point of the manifold to another. As I remarked in another recent post, the tangent spaces are different at distant points of the manifold, this is what causes the difficulty.

So an extended discussion is needed to define exactly how a velocity "here" is transported to a velocity "there" in GR. This involves issues like chosing to use parallel transport vs Fermi-Walker transport vs other possibilities, and what curve connecting the two points to use to perform the tranpsort.

8. Jan 8, 2007

### Chris Hillman

Relativity wiki?

Some other disaffected former Wikipedians and myself discussed something like that, but our concern was with rectifying the absence of effective quality control at Wikipedia by controlling write access to qualified editors, allowing signed essays, and introducing other innovations. So, I wouldn't be interested unless your wiki had security features and limited write access.

Your wiki should probably use MediaWiki since this seems to currently feature the most convenient implementation of latex-like pseudocode for mathematical markup.

9. Jan 8, 2007

### Thrice

Well I've started the wiki since I made this post. I have a thread on it https://www.physicsforums.com/showthread.php?p=1207342", see what you think.

Last edited by a moderator: Apr 22, 2017
10. Jan 8, 2007

### Thrice

I've ran a wiki before & I've become convinced that's really not worth it. It simply kills your wiki unless you already have a very large audience. And quality control is quite easy to do with something this size. Of course i'd be willing to implement it if most everyone disagrees.

The first is already (sort of) up & any suggestions on the second are appreciated.