Authors' Mistakes #12 - Brian Greene

The Fabric of the Cosmos, by Brian Green, is a very enjoyable book about the Physics of the universe.

The author, Professor of Physics and Mathematics at Columbia University, knows a lot about Physics and also knows how to explain it. The first 75 pages have been a pleasure to read and I am looking forward to the remaining 450.

But – dare I say? – I believe I have spotted a mistake in what he wrote.

On page 73, Green says:

Things that are more massive and less distant exert a greater gravitational influence, but the gravitational field you feel represents the combined influence of the matter that’s out there.²⁰

This makes sense, because the gravitational force, which is proportional to the amount of matter, is inversely proportional to the square of the distance. Therefore, while large and close bodies exert a lot of pull, the influence only goes down to zero at infinity (i.e., nowhere).

But then, in endnote 20 of that chapter, he explains:

One qualification here is that objects which are so distant that there hasn’t been enough time since the beginning of the universe for their light – or gravitational influence – to yet reach us have no impact on the gravity field.

WAIT A MINUTE! Where should such matter come from? All the matter of the universe comes from the Big Bang and got to where it is now by moving away from a single point at speeds lower than the speed of light (obviously). Then, light and gravitational influence certainly have had enough time to reach us. Or not?

To put it in a different way, if the radius of the universe is given by c times T, where c is the speed of light and T is the age of the universe, there cannot be matter outside that radius.

I find it hard to believe that Green made such a mistake. And yet, I don’t see any fault in my reasoning. If you do, please explain it to me!

I am adding the following part on 2014-04-09.

I was wrong. But I believe that Green was still not right.

Imagine that we live in a mono-dimensional but limited universe. You can visualise it as an expanding circle that started with radius zero at the time of the Big Bang. All points on the circumference move away from each other. We are on a point of the circumference and no point on the circle is in any way different from any the others. Normally, to explain the expansion of the universe, the two-dimensional model is used, in which the universe is the surface of an expanding balloon. I prefer to use a circle because it is easier and we can understand what happens without unnecessary additional dimensions.

As we explore the universe with ever more powerful telescopes, we look at other points on the circumference that are further and further away from us. This means that, like in our real three-dimensional universe (if you believe that it is, indeed, real ;-), we see events that occurred deeper and deeper in the past. Note that the light moves on the circle, as that is the only dimension that universe has, exactly as in our real universe the light moves through three-dimensional space.

Now, as the universe expands, the radius of the circle grows. The light of a distant galaxy is red-shifted because of the expansion of the circumference, which moves that galaxy away from us. That is, the red-shift depends on how quickly the location of the galaxy and our location come apart, not directly on the fact that the radius grows.

If the radius has grown since the Big Bang with average speed ‘S’ (regardless of what units is used to measure the variation of distance over time), its length is now ST (by the definition of average speed!), where T is the age of the universe (or 13.8 Gy). The length of the circumference is therefore 2ΠST. Now, how long does it take to the light emitted from the farthest point (i.e., diametrically opposite to us or, more dramatically said, from the other end of the universe) to reach us? Obviously, ΠST/c.

Now the key question is: how does ΠST/c compares with T?

If ΠST/c > T or, more simply, S > c/Π, there are parts of the universe that we cannot possibly see regardless of how good our telescopes are, because the light from those parts takes longer than the age of the universe to reach us.

In fact, if there are such parts, we will never be able to see them, because the rate of expansion of the universe is increasing (for which Perlmutter, Schmidt, and Riess won the Nobel prize in Physics in 2011). Therefore, S (being the average rate of expansion) is also increasing, and will remain greater than c/Π.

So, in fact, when Green states that there exist objects which are so distant that there hasn’t been enough time since the beginning of the universe for their light – or gravitational influence – to yet reach us, he is correct (and I was wrong). But only if S > c/Π, because if the inequality is not verified, we can see the whole universe and I was right. That said, even if S > c/Π, Green is still partially wrong because he says yet. As the universe expands faster and faster, if the the light from certain objects requires now longer than T, it will always require longer than T.

Am I wrong in some other way? :-)

And what value has S, the average rate of expansion of the universe since the Big Bang? I have to think about that...