# Hubble solution

There are some very interesting answers to the Hubble puzzle I posed below (see the comments section of the original post). Let’s review the question and then I’ll try a solution in simple language:

Hubble discovered that distant galaxies are moving away from us (or any other point) with a velocity that is proportional to their distance. The law arises from experimental observation and is usually written as

v = Hd

where v is the recessional velocity of a galaxy, d is the displacement of the galaxy from us and H is the Hubble ‘constant’, or the slope of the graph.

Every physicist reads this law as v1/d1 = v2/d2 =v3/d3 = H and it works fine. However, if we consider what Hubble’s Law says about any one particular galaxy, the equation clearly implies that the velocity v of a galaxy A (relative to some point) is proportional to its displacement d (relative to that point). But for non-zero velocity, the displacement d must be changing in time – therefore Hubble implies that the galaxy’s velocity is also changing in time – which is another way of saying that galaxy A is accelerating!

So there’s the puzzle: Doesn’t Hubble’s Law predict that distant galaxies are accelerating away from us? If so why all the fuss/surprise about the recently observed acceleration of the universe?

My solution (simple version): Yes, Hubble’s Law implies that distant galaxies are accelerating away from one another. However, this has nothing to do with the so-called acceleration of the universe. The latter term refers to the recent observation that the universe expansion seems to have speeded up (an acceleration of the acceleration if you like.)

My solution (more sophisticated version): if the solution above sounds a bit cumbersome, it’s because it should really be framed in terms of general relativity. Of course relativity affects the puzzle, contrary to what I said in the original post (teachers!). Relativity tells us that that the expansion of the universe is an expansion of space-time (or space expanding as time unfolds). Hence, the common ‘explosion-picture’ of galaxies rushing away from one fixed point is simply wrong. Instead, space itself is expanding and this expansion has a scale factor. The recent evidence of ‘acceleration’ simply suggests that the scale factor has increased in the last few million years. (This is a surprise because most cosmologists expected the expansion to slow down, if anything, due to gravitational effects). In this context, the idea of a non-constant scale factor is not so way out – for example, the theory of cosmic inflation posits that an exponential expansion of space occurred in the very early universe, before the expansion settled down to the scale factor we see today.

That’s my best explanation in simple language. For a slightly more technical explanation, see the comments by Chris Oakley and SomeRandomGuy in the original Hubble post. Chris, you have won the princely prize of a guest post – let me know when you read this!

Filed under Cosmology (general)

### 7 responses to “Hubble solution”

1. “My solution (simple version): Yes, Hubble’s Law implies that distant galaxies are accelerating away from one another. However, this has nothing to do with the so-called acceleration of the universe. The latter term refers to the observation that the universe expansion has recently speeded up (an acceleration of the universe acceleration above if you like.)” – Dr Cormac O’Raifeartaigh

Thanks for putting the mainstream official case so eloquently. I think that this is wrong for two reasons: first, the universe isn’t “recently” speeding up. The acceleration is observed at the greatest distances, i.e. the earliest times after the big bang. Second, there is no evidence that the “dark energy” causing the acceleration of the universe is evolving with time. Whatever is causing the cosmological acceleration, it is only a very small acceleration, 6*10^{-10} m/s^2 over immense distances, and you need to look to immense distances to detect it’s effect on recession rates.

Maybe you have Smolin’s book, “The Trouble with Physics”, where Smolin finds that the acceleration of the universe is quantitatively equal to approximately 6*10^{-10} m/s^2. Smolin found it a coincidence that a = cH or RH^2. Presumably you do too, despite my derivation in 1996 of this acceleration from the Hubble expansion law, v = HR. a = dR/dt = d(HR)/dt = RH^2.

If you look at Hubble’s law, H = 1/t where t is time since big bang in the observer’s frame of reference in flat spacetime with cosmological acceleration cancelling out gravitational attraction over large distances, and R = cT, where T is time past.

So v = HR = [1/t]*[cT] = cT/t

The whole point of Hubble’s law is that when you look to greater distances R in spacetime, the increase in v is matched by the proportionate increase in R, so v/R = constant = H. If H = 1/t, when looking out in spacetime, the fact that H is constant makes also t constant.

So t is not a variable in spacetime! Two variables in the equation v = cT/t are v and T. Hence, in spacetime, a = dv/dT = d(cT/t)/dT = c/t = cH = 6*10^{-10} ms^{-2}.

This is physically and mathematically legitimate, and makes an accurate prediction. The only objections I’ve ever received have been based on errors, misunderstandings, or the idea that physics is mainstream orthodoxy and the obnoxious but prevalent idea any new developments based on deeper understanding of the basics must be dismissed as wrong automatically.

This is a very tiny acceleration and is therefore only observable over immense distances. Perlmutter’s group back in 1999 came up with a computer program to detect the supernova signatures automatically from CCD equipped telescopes, which was an innovation.

The acceleration is only observable over vast distances, corresponding to relatively short times after the big bang. Therefore I don’t think that you can claim that this acceleration is “recent”.

In spacetime you are looking back to earlier times with bigger distances. In the time taken for light to travel from a distant star to you, the star will presumably have receded a further distance. One way to get around the two distance scales is through spacetime, using the travel time of light to measure how far away things are. If you stop thinking about distances and think about times past instead, then the velocity-distance relationship of Hubble becomes a velocity-time relationship. The funny thing is that the maths predicts the correctly observed cosmological acceleration.

“My solution (more sophisticated version): … Relativity tells us that that the expansion of the universe is an expansion of space-time (or space expanding as time unfolds). Hence, the common ‘explosion-picture’ of galaxies rushing away from one fixed point is simply wrong. Instead, space itself is expanding and this expansion has a scale factor. The recent evidence of ’acceleration’ simply suggests that the scale factor has increased in the last few million years. (This is a surprise because most cosmologists expected the expansion to slow down, if anything, due to gravitational effects)….” – Dr Cormac O’Raifeartaigh

“The recent evidence of ’acceleration’ simply suggests that the scale factor has increased in the last few million years.”

I’m worried that your “few million years” timescale is not consistent with Perlmutter’s 1998 discovery of cosmological acceleration, using specifically supernovae at half the age of the universe, i.e. 7,000 million years.

Also, there has been quite a lot of criticism of the concept you mention of “expanding space”:

“Popular accounts, and even astronomers, talk about expanding space. But how is it possible for space, which is utterly empty, to expand? How can ‘nothing’ expand?

” ‘Good question,’ says Weinberg. ‘The answer is: space does not expand. Cosmologists sometimes talk about expanding space – but they should know better.’

“Rees agrees wholeheartedly. ‘Expanding space is a very unhelpful concept,’ he says. ‘Think of the Universe in a Newtonian way – that is simply, in terms of galaxies exploding away from each other.’ ” – http://www.newscientist.com/article/mg13818693.600-all-you-ever-wanted-to-know-about-the-big-bang—-everyweek-questions-about-the-big-bang-flood-into-the-new-scientist-officesowe-thought-it-was-about-time-to-let-some-experts-loose-on-the-subject-.html

I don’t think that Dr Chris Oakley, SomeRandomGuy, or yourself really grasp this problem. I think I do after twelve years of battling against ignorance everywhere, although there is always room to improve the communication of facts (although the more forcefully the facts are presented, the more angry the opposition to progress!).

2. cormac

Hi Nigel, I think the solution you suggest is intrigiuing, and would certainly not dismiss it. My only caveat is that it is a little above the level intended for this blog, an introduction to the ideas of cosmology.

The purpose of the puzzle posed was
(i) was to illustrate what is meant by talk of the acceleration of the universe expansion
(ii)to illustrate the importance of discussing Hubble’s law in the context of general relativity.

I feel that teachers/communicators have a duty to present the mainstream view first (just as you and I would have encountered it) – after a while, one learns enough to question the accepted wisdom!

That said, I’m sure many of readers will have thoroughly enjoyed and learnt from the discussion on the Hubble post

3. Pingback: Advice « Gauge theory mechanisms

4. Hi Cormac,

OK – I will prepare a guest post, which all will be relieved to hear will have nothing to do with cosmology. In fact, I have already started writing. It concerns the irreducible representations of the Poincare group and my (brief) interaction with your dad around 1983-1985.

5. tyler

its not hubble who discovered it!