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.