Let's review a little bit. And then I want to move on to generalizations of what we've talked about so far.
I think we worked out the equations of an expanding universe.
They were Newton's equations ... let's talk about something else first.
Does Newton's equations really get it right?
Yeah. Newton's equations does get it right , for the most part.
Let me explain why.
Einstein's equations have to do with curved spacetime.
Now, the universe that we're ultimately going to study, has curved spacetime, alright?
And, in fact, some versions of it even have curved space.
That simply means that space itself, forget spacetime, just space itself, if you measure triangles on it, if you do various kinds of geometric exercises on it, you'll discover, perhaps, that space is curved.
At the moment it looks pretty flat, but it's possible that it will turn out on the average to be curved.
And, if it is curved, well, maybe it looks like a three-dimensional version, let's say, of a sphere.
Well we're going to study later, not tonight, maybe partly tonight, that's a portion of a sphere, we're over here, we look out, we can only see so much, we can't even really see that the sphere is curved.
But at a large enough difference we may be able to see that the sphere is curved
On the other hand supposing we just decide to look at very neighboring galaxies.
Now very neighboring galaxies can mean a billion light-years from us now.
Very neighboring galaxies much smaller than what we think the radius of curvature of this universe is.
Well then it looks flat, and if it looks flat it should mean
that at least for that portion, if we're not interested in the whole thing
but we're just interested in the local nearby behaviour, we should not have to worry about the fact that it's curved.
If that's correct, then it means that the way these galaxies move relative to each other
and how they move apart from each other, at least in the small here can be studied using Newton's equations.
That's what we've been doing.
We've been looking at the universe in the small
and studying how a small little fraction of it is expanding, or not expanding - whatever it's doing,
and it's perfectly legitimate and in fact entirely consistent with Einstein
with relativity, except for one thing
Except for one thing - we would run into trouble if the galaxies or whatever is present
galaxies, particles, whatever is present, if they were really moving past each other,
with a significant fraction of the speed of light.
One of the assumptions is that the neighboring things are moving relatively slowly with respect to each other.
Something very far away would be moving with a large velocity relative to you,
but as long as the things nearby are moving with non-relativistic velocities,
you can study - relative to you -
you could take a small patch of it,
now small could mean 10 billion light years, okay.
But you can take a small part of it and study it without using any relativity, really.
If we discover, that there are particles moving with close to the speed of light, past each other
then of course we would have to modify the equations.
But there are particles moving fast by comparison with the speed of light past us.
What are they?
[audience] Neutrinos are [inaudible]
Well, neutrinos for one. But, photons.
Now, I don't mean photons from the Sun,
I mean photons that would be there even if there was no Sun,
the Universe is filled in the same way that it's filled with galaxies,
it's also filled with radiation.
Homogeneous radiation.
And that homogeneous radiation does move with the speed of light.
That means that we have to modify our equations somehow to account for these very very fastly moving,
rapidly moving photons.
We're going to do that tonight, but I want to...
... uh ...
I want to just review what we did last time, quickly.
We first of all said:
suppose that space is homogeneous and filled with galaxies - I'm not going to try to draw all the galaxies,
they form a gas, if you like.
They kind of fill the blackboard,
with a certain number of particles per cubic metre.
In other words, a density.
A density called rho.
And that was the content of the Universe in kilograms per cubic metre if you like.
You could use some other units, but whatever units you like.
Physical units, kilograms per cubic metre, and we called it rho.
We laid down a grid on this Universe,
and laying down the grid, there was clear ambiguity
imagine that we laid down the grid at some specific time - like today.
We laid down the grid, and you could ask,
what is the spacing between the grid - a coordinate system
let's call it coordinates X
and the distance between X equals something, and X equals something plus one
in other words, one grid - uh - one grid separation here
one lot of separation - there's a certain distance associated with it
how big is that distance?
Well, we called it 'a', but how big is 'a'?
That depends on the grid that we laid down.
If we laid down a very coarse grid, it would be one thing.
If we laid down a fine grid, it would be another thing.
And so it would better be that our equations - at least at the moment -
do not prefer any specific value of 'a'.
We could lay down a different grid.
A different grid could be twice as dense,
so here's a black - forms one grid
and the black and green together form another grid.
If we looked at the more dense grid,
we would also invent an 'a', let's call it 'a prime'.
'a prime' is the distance between neighboring points on the dense grid,
that would be one half 'a'.
So if you ask me, what is the value of 'a'?
I'm gonna say I can't tell you until I know precisely what grid is laid down.
And so 'a' itself does not have a physical meaning,
At least at this stage. Later on we'll discuss more what a means.