So welcome everybody to the last lecture of the decade. It's true. And so I think the subject of

this lecture is a pretty good one in the sense that the plan for today is to build the universe,

and I think then we have done enough for the time being. So I already mentioned yesterday that when

we build world models what we have to do is we construct a homogeneous and isotropic universe

by building up a metric and then putting energy in there and that together with an evolutionary

equation will allow us to build what we call a world model which is essentially a description

on how the universe evolves as a function of time on a very very high meta level. So what we do today

is we'll be looking at how we build world models. And we have to start, you'll see that this lecture

will be fairly abstract, so it's quite good that we have vacation after this because then you'll

forget how abstract it was. And we have to think a little bit on how we describe space and so on.

What we do today was until about 20-30 years ago kind of the core of cosmology lectures. So

cosmology lectures were mainly busying themselves with building different world models. We don't

have to do this anymore because as we'll see next year we have a fairly good understanding on what

our world looks like, how our universe develops, so we won't look in all of the different world

models that are possible. So we have to jump over a bit of the physics but I'll hope I'll give you

enough background that you can understand what is going on and that you can fill the

blanks by yourself. So we have to start with something that is something that you probably

haven't seen so much in lectures and that is assumptions of the theory of general relativity,

GR for short. I'm not assuming that you had lectures on GR and for everything that we do here

you do not need to understand GR but it is important to have a certain gut feeling for

what's going on. So I'll give a very brief introduction to what we do and how we describe

space in GR to give you a feeling for what's going on. If you want the details go into the

lectures on general relativity, we have quite good ones here at FAU or check the literature.

Depending on how you feel about things if you are less mathematical take the old books Weinberg's

cosmology is a good one or gravitation by Miesner Thorne-Bieler usually abbreviated MTW or take any

of the modern GR books which generally have a much better background in terms of the mathematical

formalism but because of that probably sometimes make it less clear and how you think how you apply

things to to general problems that you encounter. So what are the assumptions of GR? The assumption

of GR is that space is four-dimensional so there are three spatial dimensions and a time dimension

but how you build up these four dimensions and go back between different reference frames is

fairly complicated. The presence of matter or the presence of energy if you want because energy

and matter in that sense is the same modifies space in a way that's described by the so-called

Einstein field equation okay which gives you an equation that connects the behavior of space so

the way how space behaves in terms of its curvature as a function of this presence.

And then there are three formal requirements to GR. One is what's called covariance which is more

philosophical but very important point that says that the laws of physics must be formulated in a

way that is independent of the coordinate system. That is a very big thing or was a very big thing

a hundred years ago when GR was was first promoted or discovered. It's far more profound than many

people think because right now if you are a student of physics you generally see the laws

phrased in a way that's covariant so you generally don't even notice that our theories really have

been constructed in a way so that they are coordinate system independent. Then an assumption

of GR is what's called the strong equivalence principle that's less important for us but that

essentially says there's no difference between an inertial system and a free-falling system.

And finally there's an assumption built into GR and that says that locally space is such that the

theory of special relativity holds that means space is locally Minkowski. So this all boils down

to that in GR once you've described space in principle you've described everything that's

there. So it does not make a lot of sense to immediately start with a GR description and

thankfully we will never have to really go through full GR for the rest of this lecture.

But what's mentally quite good although sometimes a bit misleading is to get a feeling for how we

describe space in general and we do this by looking at two-dimensional space to give you