Let us now address the question of the expanding universe and what does it
really mean. This is usually something that baffles
many people when they first hear about expanding universe.
What is it expanding into and how? The important notion here is that there
really are two different kind of coordinates in general relativity
cosmology. There are, there are the comoving
coordinates which expand as the universe expands.
Examples of those would be unbound systems like any two distant galaxies
that are not gravitationally bound together or wavelengths of massless
quanta like photons. But they're expanding relative to what?
There has to be a fixed set of coordinates and those are called proper
coordinates, those was what you usually think of as
coordinates, which are fixed and space expands relative to them.
If it wasn't the case, we couldn't tell there was an expansion, because
everything would be expanding. Well, things that do not expand with
expanding space or atoms or molecules or solid bodies like planets or in fact any
gravitationally bound systems like the solar system, stars and galaxies.
So, galaxies don't puff up with the expanding universe, they stay the same
size, but they do go apart.
Okay, so what's the universe expanding into?
Well its, in expanding into itself. There is nothing outside to it,
except for purely hypothetical multiverse, but same would apply in each
one of its constituent bubble universes. Consider a positive curvature universe,
that's like the surface of a sphere except in one extra dimension.
It has finite volume and finite surface, but it can grow.
So there is no edge and it can expand. If you will,
you know, the fourth dimension into which it's expanding is time.
Now, flat or negative curvature universe is simply infinite in all directions and
you can think of a grid stretching out everywhere.
So, in, in either case, there is no edge and there is no center, because of
homogeneity and isotropy. So the universe is expanding into itself
in the sense and there is no, nothing outside of the universe.
Oh, well how does it expand. The function describes that is called the
scale factor which we mentioned earlier in [INAUDIBLE] function inside
Robertson-Walker matrix, usually called R(t) or a(t).
You take any two distance points and measure how their distance changes in
time, that's essentially what scale factor is.
And because universe is homogeneous and isotropic, it same exact same function
will apply to all points in space and time.
So if we knew what that function was then we can tell exactly how the universe is
going to evolve. Being able to do this is exceptionally
what cosmological models are. And we do this by solving the Friedman
equation, which is what Einstein equations are reduced to after assuming
homogeneity and isotropy. But first, let's think about cosmological
redshift. This is the usual balloon analogy,
inflate the balloon and points on it go further apart.
Well, if you glued little buttons on the surface of the balloon, buttons will not
be expanding as balloon expands, those would be like galaxies.
But if you drew wiggle lines on, on the balloon, those will stretch as the
balloon expands. Those will be like photons.
So we talk about the redshift since the wavelength tends to increase, moves to
the red. And familiar notion is the Doppler shift
which in nonrelativistic case is just the ratio of speed by which something is
moving to the speed of light or speed of sound in case of sound Doppler shift.
In special relativity, there is little bit of relativistic correction, shown
here. So,
we do think of cosmological redshift often as being a Doppler shift.
Galaxies are moving away from us therefore there is a Doppler shift in
wavelength stretching, but really more correct explanation or more correct way
of thinking of it is that it corresponds to the stretching of space.
Well, if any distance stretches this according to the scale factor, r(t), then
the wavelengths will do the same. And our definition of redshift is change
in wavelength divided by wavelength itself.
And so, if we do either these two, we get that the scale factor.
Ratio between scale factored now, and some different time, is one plus ratchet.
So, we essentially use one + red shift as the measure of cosmic expansion.
This turns out to be exactly equivalent to the Doppler shift expan, explanation,
but this is probably a better way of thinking about it.
Consider an observer at a radius of zero and some radial light rays.
since they move at [INAUDIBLE] six, that means that some of the spatial
displacement is equal to the temporal placement.
And the following equation must apply according to Robertson-Walker metric.
So if we emit our array of light at time t1, and its observed at t of zero, then
integral over those two paths has to match, and according to this formula.
So if a co-moving source is some distance R and its coordinates is fixed, well
then, the relative displacements of, or relative increments in time, scaled by
the scale factor must be the same. So therefore, there's going to be a
redshift in expanding universe. You can divide it to and it's exact same
result as we've gotten before. Well, lets consider Hubble's law in the
light of this explanation. For nearby source we can write this as a
following expression, R dot is the time derivative of scale factor of this time.
So, there are actually is essentially time derivative of the scale factor
divided by the log of scale factor itself.
In other words, it is dimensionless fractional change in scale factor and
that is what Hubble constant is. So we can derive Hubble's law directly
from Robertson-Walker metric and Hubble constasnt is not really a constant,
it changes in time. But at any given time, this is a constant
obviously and so, it's an instantaneous expansion rate.
Todays values usually do not get h not, little zero subscript.
Well, finally here is something interesting to think about.
The holy cow of physics is a ration of energy,
but turns out energy is not conserved in an expanding or contracting universe.
The couple simple examples you, those protons that stretch with redshift, but
where does their energy go? A proton may be emitted, able to ionize a
hydrogen atom say, but when it's received it's energy's much
less because wavelength is less. Where did that energy go?
Or, the other way around. Take two galaxies, there not
gravitationally bound but they still have some potential energy relative to each
other. Expansion of universe carries them apart,
the potential energy drops so the system has gained energy, from where.
So, the upshot is that energy's actually not conserved, in an expanding universe.
It's something to think about. So next time we'll talk about Friedmann
Equation which is the actual way in which we compute cosmological models.