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...

More than one way of reading

I have been reading a little book written by the French psychoanalyst and professor of literature Pierre Bayard.

It is full of interesting ideas.

Bayard convincingly makes the point that “reading” has not a single, universally applicable meaning, and challenges us to admit that sometimes (or often, or always) we talk about books that we haven’t read from cover to cover.

Furthermore, he points out that, inevitably, we only retain a fraction of the information contained in a book. Therefore, even if we have indeed carefully read a book in its entirety, we can only talk about our impressions and interpretations that its content has elicited in us.

And as our memories fade with time, can we knowledgeably talk about books that we read months and years ago?

This last point stroke a chord with me. In a couple of occasions, I even bought a book I had already read, thinking that I hadn’t. It was somewhat bewildering to discover another copy of the same book in my bookcase. If I can completely forget some of the books I read, I have certainly forgotten the content of many of the books I still remember to have read.

What is the length of the reading-forgetting cycle? For me, scaringly short. I used to have a very good memory. I could read a poem a few times and be able to recite it. But, with the passing of years, my short- and medium-term memories have been deteriorating. Sometimes, immediately after reading a novel, I wonder how it begun. And my memory for names, whether they are characters in a story of real people in real life, is simply appalling.

Hopefully, something of what I read is still spread across a number of neurons and influences my thought.

On the basis of these considerations, why should I be more entitled to talk about a book I read long ago than somebody who skimmed it a couple of days ago or recently read an article about it?

It really seems that “reading” is in the eye of the beholder, to paraphrase a well known cliché.

Bayard suggests that we qualify each book with an acronym that indicates its “reading status”:

UB	book unknown to me
SB	book I have skimmed
HB	book I have heard about
FB	book I have forgotten

It seems reasonable, but I contend that UB is totally useless because, as soon as somebody mentions to me a book I didn’t know it existed, that book immediately acquires for me the status HB. Then, what’s the purpose of defining the status UB? Perhaps Bayard reserves UB for books of which he only knows the title, or part of it, but that doesn’t make sense because the title already tells something about the content. UB could only be used for all books of which I don’t even know the titles, but I certainly don’t care about them.

I also have a small issue with FB, because you can only forget what you once knew. So, does FB apply to books that were once SB or HB? And what about the books we actually read (in the strictest sense of reading all words in them)?

Notice that Bayard doesn’t define any acronym for books actually read. This is because he considers “read” too ambiguous to be used (or perhaps because he never really reads any book). But I like to think that if I have actually read more than 50% of the pages of a book, and have not forgotten of its existence, I should be able to use RB. Obviously, with time, many RBs will silently morph into FBs.

I religiously (interesting spontaneous choice of adverb...) maintain a list of all books I own and/or have read. There are books I own and have read, books I own and haven’t read [yet], books I read but don’t own (either because I borrowed them or because I gave them away), and books I don’t own and have never read. The last category only includes a handful of books I want to remember for whatever reasons. Consistently with what I said in the previous paragraph, I flag a book as “read” if I actually read more than 50% of its pages.

All in all, the classification I find most appropriate (if there has to be one) is as follows:

HB	book I have heard of read about but never held in my hands
SB	book I have held in my hands and skimmed
RB	book of which I once read more than 50% of the pages

Whether I have forgotten what I once knew about the book or not, refers to a totally different dimension and applies to all books I came in contact with, regardless of the way in which that happened.