Let us continue this brief introduction into relativistic cosmology.
An important notion is the so-called cosmological principle.
It states that universe is same, at a given time,
it's same in all directions and at all locations.
This is what's described as homogeinity and isotropy.
In a sense, cosmological principle is the generalization of Copernican principle,
but on a global, universal scale. Now, there is a perfect cosmological
principle proposed by Hoyle and collaborators which says that universe
should also be same at all times and that was the basis of this steady state
cosmology, which unfortunately did not survive experimental tests.
So, what do we mean by isotropy and homogeneity?
The picture on the left is homogeneous. You can be anywhere in the infinite plain
covered with the stripes and it will look the same,
but it's not isotropic. The stripes have a given direction.
So it's not the same along the stripes or orthogonal to it.
The picture on the right is isotropic from the point in the middle,
same in all radial direction, but it's clearly not homogeneous.
And, here, we have an example, it would be homogeneous and isotropic texture that
is same everywhere in all directions in all places and the one on the right which
is neither homogeneous nor isotropic. Why is this important?
Well, is it really true? Well, it is true on scales larger than
about 100 megaparsecs and there are a couple of experimental
pieces of evidence for that. The map on the right shows positions of
radio sources in the sky. Those tend to be very far away and so
there are a fair sampler or of [INAUDIBLE] universe's volume.
As you can see, it is, it does look pretty homogeneous.
The ellipse on the lower left is indicative of what cosmic microwave
background sky looks like, very uniform. There are fluctuations in it, but they
are parts in a million, so that tells us that indeed the universe
of very large scales is homogeneous and isotropic.
Not so on scales smaller than about hundred megaparsecs where we see all the
large scale structure, filaments, and voids, and so on.
Turns out that doesn't really matter so much as far as cosmological tests are
concerned and cosmology really operates on scales of gigaparsecs and larger.
So, it's true that locally, universe is not homogeneous, isotropic,
you are standing on a planet after all, but on large enough scales it is a good
approximation. Let's go back to general activity.
Remember, the most important notion is that presence of mass and energy, the
term is geometry and geometry determines where mass and energy go.
And the two have to be consistent, so getting a consistent solution to that
is essentially Einstein's equations. We will not derive them in here in any
shape or form. Just need to show you a little bit of
what they look like. So we start first with Poisson equation
which describes gravitational potential phi as a function of density rho and this
is true in Newtonian physics as well. Now, general relativity says that
potential, gravitational potential can be also made equivalent to geometrical
description through a metric transfer. And, there is the plane density, there is
so-called matter energy density tensor. And that is a matrix four by four and we
need not go into its details. So a short connotation, so using these
two indices, mu and nu, which go from zero to three, this represents 16 partial
differential equations of this form and those are the Einstein equations.
So basically, thing to remember. On the one side, there is a term that is
all about spacetime geometry. And on the other side, there is a term
that is all about mass and energy and for the move.
So this is what Einstein equations are really all about.
So there are 16 of them, but we made those important assumptions
that universe is homogeneous and isotropic and that means that only one
coordinate would matter, radial coordinate.
So that makes things much simpler, instead of 16 equations we only have one.
And that is what's called the Friedmann equation which is the basic staple of
cosmology. Just one more thing.
Gravity as we know it is an attractive force and this is certainly true in
[INAUDIBLE] as well. But Einstein's equations allow for
introduction of an additional term in potential.
And you can think of it, as say integration constant or a new face of
gravity just like electricity and magnetism are two faces of
electromagnetic interaction, but something that's only apparent of
cosmological scales. Essentially, this corresponds to constant
energy density term and that is what's called a cosmological constant.
Now, if it is seen as, say integration constant, we have no idea what it is.
The theory doesn't give us its value and we don't even know its sign.
It can be attractive force or repulsive force.
The result, it is a repulsive force and it is actually very important, but we'll
cover that later in the class. So, those are the basic notions of
relativistic cosmology. Next time we'll talk more about the
expanding universe and what that means.