*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...

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