# Spacetime expanding at a rate equal to the speed of light

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1. Jul 5, 2013

### fet2105

Is there anyway to figure out the distance at which the universe is expanding (rate of stretching) at a rate that is equal to the speed of light?

2. Jul 5, 2013

### marcus

Yes.

Astronomers call that particular distance the Hubble distance, or "Hubble radius". It is not the current radius of the currently observable portion of the universe, that is much bigger than the Hubble distance, but it is a very useful benchmark distance.

It has changed quite a lot over time, and it will continue changing in future according to the standard cosmic model.

It is currently 14.4 billion ly.
That is equivalent to saying that at cosmological scale (not in our solar system or in our galaxy but over large reaches of intergalactic space) distances are increasing at a rate of 1/144 % per million years

The two statements are equivalent:
A the distance increasing at rate c is 14.4 billion ly (and other distances are growing proportionately)
B cosmic distances in general are growing 1/144 % per million years

You can check that they are equivalent: what is 1/144% of 14.4 billion ly?
Well, ONE percent of a billion is 10 million, so one percent of 14.4 billion is 144 million.
So 1/144 percent of 14.4 billion ly is exactly one million light years.

If a distance grows exactly one million lightyears per million years it is growing at the speed of light.

So if someone tells you that in the future the Hubble distance will increase to 17.3 billion ly,
then you can interpret that to mean that the percentage growth rate of distances will then be
1/173 of one percent per million years.

Here is a past and future history of the Hubble distance, showing how it has changed and is expected to change in future. It shows that in year 67.4 million (the far distant past) the Hubble distance "R" was 102 million ly. That is, about 0.1 billion ly. That is equivalent to distances growing about 1% per million years. One whole percent per million years, really fast growth.

$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&R (Gly) \\ \hline 0.0674&0.1021\\ \hline 0.0956&0.1445\\ \hline 0.1354&0.2044\\ \hline 0.1917&0.2890\\ \hline 0.2713&0.4086\\ \hline 0.3839&0.5775\\ \hline 0.5430&0.8160\\ \hline 0.7676&1.1522\\ \hline 1.0847&1.6251\\ \hline 1.5315&2.2869\\ \hline 2.1589&3.2044\\ \hline 3.0346&4.4534\\ \hline 4.2427&6.0957\\ \hline 5.8756&8.1265\\ \hline 8.0089&10.3976\\ \hline 10.6648&12.5991\\ \hline 13.7872&14.3999\\ \hline 17.2617&15.6499\\ \hline 20.0000&16.2548\\ \hline 22.8231&16.6519\\ \hline 25.7011&16.9035\\ \hline 28.6133&17.0597\\ \hline \end{array}}$$
And it shows that in year 13.8 billion (the present day) the Hubble distance R = 14.4 billion ly.
And that in the distant future R will approach 17.3 billion ly, as I said earlier.
The table is easily obtained with this calculator:
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
Basically you just have to press "set sample chart range" and press calculate.
that wlll give a table that is longer and has more columns and thre is a way to make the table shorter, just type in a different number of steps and press calculate again.
but to save you having to deal with an unfamiliar calculator I just made the table for you.
It shows the size of Hubble radius at various past and future times including the present (year 13.8 billion)

Last edited: Jul 5, 2013
3. Jul 5, 2013

### phinds

To add to Marcus's always excellent explanation of this subject, "stretching" is not really a helpful description in this contest. Space does not stretch, things just get farther apart.

Also, as is explicit in his discussion, but could be overlooked, all of this is relative to US as a frame of reference. That is, it is not meaningful to say "the distance at which the universe is expanding (rate of stretching) at a rate that is equal to the speed of light?" in any absolute sense but rather "the distance FROM US ... " (or from any other point you care to choose, in which case the answer will be 14.4 billion LY from that point)

4. Jul 6, 2013

### fet2105

Thanks for the explanations. As a followup question; will the Hubble radius ever decrease in size? I think of it as follows: the rate at which the universe is expanding is accelerating and that as a result objects will move away from us at a faster rate as time goes on. I always pictured this "ring" if you will which marks the distance at which objects are moving away from us at the speed of light (which I now understand to be the hubble radius) to close in on us. Am I thinking of something else?

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