A Little Expansion on the Lightness of Fine-Tuning

A warning for people studying relativity in a formal setting, please use the derivation methodologies recommended by your teacher or lecturer.  I don’t believe that what follows is wrong, but it’s not entirely standard.

This article is a follow-up to The Lightness of Fine Tuning Part 1 and Part 2.

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In the 1920s it became clear that the universe was expanding – due largely to work by Edwin Hubble.  What Hubble noticed was that the further away a galaxy is from us, the faster it is receding (moving away from us) and that there is a linear correlation between distance and speed of recession (as expressed by Hubble’s Law).

Something that is blatantly obvious, and which is probably rarely mentioned for that very reason, is the fact that not everything is expanding at same rate.

If everything was expanding at the same rate, we could not notice it.  What we do notice is that the space between galaxies seems to expand while the space between the two ends of a ruler on our desk doesn’t.

Hopefully Alice can help explain.  When in Wonderland, Alice has a bad habit of changing size without notice.  She first experiences this phenomenon after seeing a beautiful garden that she wants to visit, but she’s too big to get through the door.  Shortly after drinking a potion (helpfully labelled “Drink me”), she finds that she is appropriately sized.

Now, the standard explanation is that Alice shrank until she was small enough to enter the garden.  This makes sense because Alice ingested the potion and we expect the potion to affect her in some way.  However, Wonderland is clearly a fantasy world, and our normal cause and effect logic need not necessarily apply.  All we can say is that Alice was no longer too big for the door, which could mean that when she drank the potion, the door (and the rest of Wonderland) expanded to match her size.

From one perspective, this makes more sense.  Alice was a standard human, and standard humans don’t change size when they drink potions.  Wonderland, on the other hand, is a fantasy world, and the normal rules of causation don’t seem to apply.  Cat’s don’t talk, or disappear in stages, babies don’t turn into pigs irrespective of how much pepper they breathe in, rabbits don’t serve as advisors to royalty, dormice don’t drink tea, and so on.  There’s no reason to assume that drinking a potion won’t cause Wonderland to change.  If Alice did change size, there are a range of issues that would arise – she’d suffocate unless the air changed size as well, her head would overheat unless the laws of physics changed and then there’s a conservation of mass issue which C.S. Lewis clearly didn’t think through properly.

Alice’s Problem	Assumed Solution	Another Possible Solution

All Alice can tell is that she and the door changed in relative size.  This could have been because she shrank or because the door got bigger.

If everything in the universe was expanding at the same rate – if there was no change in relative size – we’d never know about it.  What we do notice is that the universe is expanding unevenly.  In other words, some parts of the universe are expanding more than other parts.

To explain how this expansion works, imagine we have a piece of rubber sheeting on which we have drawn a very simple version of the universe:

Then we stretch the whole thing to double the size:

Note that the little Earth in the bottom left-hand corner also expands, along with a hypothetical 10 light-year ruler – so to anyone in the universe, it would look the same size as it was before the expansion.  This, which is equivalent to Alice getting bigger at the same rate as the door, isn’t quite what happens.  What happens is that the space between concentrations of mass-energy expands at a greater rate to produce this effect:

Now the universe does look bigger!  (The little Earth has changed size, but not as much.)

Next we have to imagine that this representation of the universe is the surface on a spherical balloon.  This allows the entirety of the “universe” to expand without there being a centre, which is precisely what happens with the real universe.

Once have we managed this little feat of imagination, we have a two dimensional representation of the universe mapped onto a three dimensional shape.  In reality, the universe as we comprehend it is three dimensional, mapped onto a four dimensional manifold.  The fourth dimension involved is time.  The universe expands with time or, as we could also say, the expansion is time.

To take the next step, it will be helpful to eliminate another dimension.  We can do that by imagining that we take our three dimensional representation of the universe (the spherical balloon) and we neatly slice it in half.

Now we have a circle:

At the centre of this circle we have t=0 – which is the centre of the universe (in other words, the centre of the universe is not the answer to a “where” question; it’s the answer to a “when” question).  On the circle itself, we see that we have cut through two spaceships which are clearly not to scale.  Imagine that these two spaceships are moving towards each other at close to the speed of light:

At this point, we are (conceptually) where I was one cold winter evening about twenty-five years ago as I was walked home thinking about spaceships approaching each other at significant fractions of the speed of light.  If you look at the drawing, you can see that each arrow is about the same size, representing about the same speed.  Note also they are at an angle to each other.  Also remember that we’ve cut the dimensions back to two, one spatial dimension and time.  Furthermore, because we are showing velocities here, the arrow out from the middle of the circle represents the rate at which the universe expands (which I’ve labelled as ก).

The speed at which the spaceships are approaching each other is not the simple sum of the speed represented by the red arrow plus the speed represented by the green arrow.  Using vectors we can see that the combined value is actually less than this:

The narrower arrows represent the apparent speed of approach of the other space ship (according to each space ship, which assumes itself to be stationary).  The white double arrow represents the closing speed of both space ships according to another observer.  Note that this is all representational and not intended to be to scale.  What these illustrations show is that when two space ships approach each other obliquely, rather than moving directly towards each other as we might expect if the universe was flat, then their speed of approach is going to be less than the sum of their individual speeds – no matter who observes it.

This is in fact what happens.  And how oblique they are to each other is directly proportional to their speed of approach.

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For a while, I didn’t go much further than this.  I didn’t really need to since, after all, I’d worked out (to my own satisfaction) that it makes sense that space ships approaching each other at more than half the speed of light each won’t approach each other at more than the speed of light combined.  I was worried though, at first, that this little theory of mine meant that the universe had to be expanding and that as a consequence things that were further away would be moving away at greater speeds than things that are close.  Fortunately, I discovered a couple of years later, Edwin Hubble had already dealt with that problem back in the 1920s.

Then it struck me that if the universe was expanding the way I had postulated that it was then the centre of the universe was in the past, not in a location.  Again, someone else had sorted that out a long long time ago - Georges Lemaître was the one who first formulated the idea that Fred Hoyle later described as the “Big Bang” and proof of it arrived via the discovery of cosmic microwave background radiation in 1964.  (I knew about the Big Bang, of course, what had not occurred to me was that there was no specific spatial location for it, the Big Bang happened everywhere a long time ago.)

Then I was struck by the idea that if space ships approach each other obliquely, then there would be an interesting phenomenon.  Imagine that you are on a lighthouse, looking at a tanker that is aimlessly sailing in circles nearby.  When you look at it side on, you get to see the full length, but when you see it at an angle, you see less of the length.

It also occurred to me that unless acted upon by a force, things travel in a straight line – not along curves.  And on the surface of a circle, a straight line is called a tangent (mathematical straight lines only touch a mathematical circle at one point unless they pass through the circle).  This means if you were on the surface of a sphere and you moved in a straight line, you would leave the sphere.   (We do this all the time while walking on the Earth, but we don’t get very far from the surface and gravity pulls us straight back down again anyway – walking, and running even more so, is the fine art of constantly falling down without falling over.)

After some pondering, I arrived at something like this as a representation of movement in an expanding universe (harking back to your evil twin, you are notionally stationary, your evil twin is notionally in motion – relative to you):

When your evil twin leaves the “surface of the universe” in this conception, this results in a little journey into the future – your evil twin arrives at the future more quickly than you do as a stationary observer.  Being oblique to the “stationary” universe, your evil twin will also appear foreshortened.

When we look closer, we can see that there are some triangles involved:

The angle θ created by the sine relationship between v and ก (as opposite and hypotenuse) is the same as the angle created by the cosine relationship between x_E and x_G (as adjacent and hypotenuse) and the angle created by the cosine relationship between t_E and t_G (as adjacent and hypotenuse).

It seems a bit complicated at first, but a final year High School maths student should be able to show that

x_E = x_G . √ (1 – v² / ก²)

and

t_E = t_G . √ (1 – v² / ก²)

Here are the workings, look at them briefly, if that is all you can bear, or in depth if you want to:

The final two are precisely the same equations that we arrived at in Part 2 of The Lightness of Fine-Tuning – except that we now have ก where we previously had c.  This is because, if my conception of things has any validity, the rate of the expansion of the universe is equivalent to c.  In other words, the granularity of the universe is related to the rate of expansion of the universe, both being equivalent to what we know as the speed of light.

I have shown how I arrived at this in a very rough way here, but there is a more thorough explanation available, if anyone is interested. Just leave a comment or email me for details.