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 followup to The Lightness of Fine Tuning Part 1 and Part 2.
This article is a followup to The Lightness of Fine Tuning Part 1 and Part 2.

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 lefthand corner also expands, along with a
hypothetical 10 lightyear 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 massenergy 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
twentyfive 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.

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 stuck 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
– v^{2} / ก^{2})
and
t_{E} = t_{G} . √ (1
– v^{2} / ก^{2})
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 FineTuning – 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.
Okay. I see I have had some visits from some scientific types over at reddit (http://redd.it/11ejwu).
ReplyDeleteI just want to respond quickly to the bollocks statement.
Firstly, I did say up front that this is not a standard way of looking at relativity, so it perhaps shouldn't even be there on reddit (since this is far from being a peer reviewed journal  it's just my musings).
Secondly, there are some misrepresentations, or perhaps misunderstandings. It is probably my fault for not explaining fully, but ... well, I wasn't quite expecting the audience and it was not my intent to defend the thesis quite so formally in blog format.
There are two elements to the above, the first being the explanation about how I think of the expansion of the universe. supracedent is pretty much right, the universe **is** expanding uniformly but he is wrong, I did not say it wasn't. What I at least tried to say, is that where there are concentrations of massenergy, there is an effect where the massenergy does not expand at the same rate as empty space  a phenomenon that we know as gravity. I had not thought specifically about chemical bonds to be honest, but since I was thinking about expansion at the granular level, I don't think they would apply. It seems that he took the ruler on the desk too literally. The ruler is sort of like one of Einstein's rods, it is used to measure things, rather than being a literal wooden ruler with writing on it. What I was trying to say there was that (using supracedent's words a little): if "gravity is so strong locally that our solar system won't grow with the expansion of space at any perceptible level", then a local (conceptual) ruler won't expand as much as an intergalactic (conceptual) ruler.
The second part is about how I came to terms with the idea of how two spaceships approaching each other, while both are travelling at greater than half speed of light (implicitly in reference to a third observer) don't approach at greater than the speed of light. When I thought of this, my universe was completely empty with the exception of the two spaceships (and the third observer, I guess). supradendent's problem there is with the idea of the origin of the universe being at the big bang (which is Lemaitre's idea and seems to be accepted by everyone else) and with the idea of using Euclidean geometry. I can only use Euclidean geometry because the universe is empty, I cut away unnecessary dimensions and (although I never clearly stated it) I assumed that the spaceships were not sufficiently massive to bend spacetime to any appreciable extent. I was not trying to imply that Euclid was universally applicable. transformatronic was right about Feynman's derivation being Euclidean, even Einstein's derivation was Euclidean. The complaint is, therefore, rather unfair.
Again, however, this is not a standard derivation of the equations for time dilation and length contraction. I haven't even gone all the way to explain how the time equation relates to time dilation in this article, I do that in http://neophilosophical.blogspot.com/2012/08/specialrelativityfromgalileoto.html.