Hi everyone so today I wanted to take a little bit of a look at a topic in
physics which is notoriously difficult and that is the topic of general
relativity. It's a topic that builds on a lot of the work from Einstein back from
1915 in which gravity is described as being a geometric property of space-time.
There's a lot of math involved in the topic and a lot of really cool ideas as well.
I was recently watching a YouTube
lecture by Leonard Susskind about general relativity and he starts off by
saying that... general relativity has a reputation for being very difficult
I think the reason is that it's very difficult.
So today in this video I just
wanted to take a little bit of a peek at the topic to see what gives it this
reputation and for people who have never really encountered general relativity
give you an idea of what it's all about. You can see if it lives up to its
reputation and maybe this video will even give you a little bit of
inspiration to want to go and learn the topic yourself. Of course I don't have
enough time in this video to really teach you general relativity so we're
just going to have a little glance at it and the way that we're going to do that
is by just having a look through some lecture notes that I have here. In fact
this is actually a summary of lecture notes it's called A No-Nonsense
Introduction to General Relativity by Caltech professor Sean Carroll and it's
a summary of Sean Carroll's lecture notes for when he would teach this as an
entire course and those notes themselves have been adapted into a full textbook
but I think that this little summary of notes will be more than enough for us today.
I think these notes will be a great
place to start because Sean Carroll is a great science communicator, he's been
like the scientific adviser for the Avengers films and done lots of work
in that space. I wanted to take a moment to mention that Sean Carroll is actually
coming to do his first ever tour of Australia and New Zealand in late
February this year and I'm able to offer my viewers 15% off tickets
so I'll leave the link down in the description if you would like to come
along and if you do you can enter the code tibees on the website to get that
discount that is thanks to Think Inc the company that Sean is touring with and
they have toured other people around Australia in New Zealand such as Brian
Green, Lisa Randall and Neil deGrasse Tyson. Sean's doing this tour on the
back of releasing his latest book and he is going to be talking about the
multiverse and the nature of reality so it sounds pretty interesting, and with
that let's have a little bit of a look at general relativity. So the first page
starts off with a bit of an introduction that immediately addresses that general
relativity or GR is difficult and it says it's mainly for two reasons one is
that tensors are everywhere and two is that space-time is curved. These two
facts force people in GR to use a different language than everyone else
which makes the theory pretty inaccessible however it claims that it
is possible to grasp the basics of the theory. It sums up GR in two statements
one is that space-time is a curved pseudo-riemannian manifold with a metric
signature, and two, the relationship between matter and the curvature of
space-time is contained in this equation which has become a bit of a famous
equation, Einstein's equation, although it is very difficult to understand unless
you understand the notation and most of the rest of these notes is doing just
that, trying to get people to a point where they can actually understand this
equation and I would agree it's not easy. The notes go on then to cover special
relativity which it's more likely that you have heard of and encountered
already. Special relativity is often taught at the end of high school or in
undergraduate because the math that's involved is actually not too hard
you need some pythagoras and the ability to grasp things like time dilation and
length contraction. It says that special relativity stems from considering the
speed of light to be invariant in all reference frames.
Usually when you encounter special relativity for the first time
you don't encounter it in the form of four vectors but that's what we go on to
have to learn here. See we have four components of our X vector including t
X,Y and Z so we have three spatial components and one time component. All
four of these components making up space-time. All of our vectors in space-time
will be four vectors including a special one, the metric, which gives us a way of
taking the dot product of two vectors. The next few pages discuss ideas such as
the space-time interval, proper time, four velocity which is velocity using four
vectors and likewise a momentum four vector. We discuss the idea of time-like
and space-like with respect to light cones. In the light code the edges of the
cone here represent traveling at the speed of light. Once you get to the topic
of tensors it kind of becomes make or break for you because they're really
important. It says here that tensors may be thought of as objects like vectors
which transform under a change of coordinates according to the following
rule called tensor transformation law. So tensors are really useful when you need
to do changes of coordinates but why would we be doing changes of coordinates
in general relativity? sort of you know what are we up to here? Well one of the
big ideas in general relativity is the equivalence principle and I guess it
says that in some situations you won't be able to tell the difference between
gravity and acceleration for example if you're in an elevator both the effects
of gravity and the effects of acceleration would feel the same to you.
So sometimes we have to wonder if space-time has a real gravitational
field or if we're just experiencing the effects of acceleration and the question
of whether space-time has a real gravitational field is similar to asking
whether a geometry is flat. It was the problem
worked on in mathematics by Riemann so hence Riemannian geometry that we
mentioned in the introduction. Essentially if you're unable to do a
coordinate transform that could make your geometry flat then it is curved so
we need to understand how things transform when you make coordinate
transformations and that is why we care about tensors and tensor analysis. It's a
difficult topic and I don't have much time to really explain tensors more but
we'll just have a little flick through the pages so at least you can see what
we're dealing with. On this page tensors are used to shrink Maxwell's equations
into just two relations which generalize to curved space-time and
are the equations which govern the behavior of electromagnetic fields in
general relativity. GR is a theory of gravity and of geometry so it's
important to understand what we mean when we talk about curvature and
manifolds. Here it says basically a manifold is a possibly curved space
which in small enough regions looks like flat space the important thing about
manifolds is that they have the same dimensionality everywhere. Dimensionality
would be like 2D, 3D, 4D something like that.
So a piece of string attached to a plane would not be a manifold since it's
partly one-dimensional and partly two-dimensional. If we had a real
gravitational field around an object say this is some kind of planet and this is
the gravitational field then we would expect to experience tidal forces on an
object that is out here say this is a person, then because of the direction of
the gravitational force is not truly horizontal but has a downwards vertical
component here and a slightly upwards vertical component here, the object in
here would actually be slightly squished or maybe a lot squished I don't know. So
you'd be squished or stretched by tidal forces and that can indicate that you do
have a true gravitational field. We could help to identify tidal forces by
calculating a curvature tensor and defining our geometry as flat where the
curvature tensor is zero everywhere. Gravity is essentially equal to having a
curvature. Towards the end here we encounter the Schwarzschild solution it
says here that the Schwarzschild metric is the unique solution to Einstein's
equation in a vacuum with a spherically symmetric matter distribution so this is
perhaps a slightly more physical interpretation that we can use to kind
of understand what's going on out there in space. There are a couple of places
where this solution does have singularities they would be at R equals
to 0 and R equals to 2GM. From what I know
modern research in general relativity seems to be doing things like trying to
quantize general relativity or to find and apply numerical solutions to things
like black holes. Numerical solutions mean you basically put in a lot of
numbers and try to find out what would happen from certain initial conditions,
try to play the equations forward in time. Quantum field theory or QFT is
another very hard and notoriously difficult area of physics and of course
the interface between GR and QFT is very difficult and I guess attracts people
who are really interested in some of those fundamental connections. Black
holes become relevant to general relativity because of these
singularities that we saw before in the solutions to Einstein's equation.
The notes finish off with a little bit of a mention of Hubble and cosmology and
how you can use GR equations to talk about an open, flat or closed universe.
So that's my little summary I hope you didn't find it too intimidating the
intention is not to try and teach you the ins and outs of GR I wouldn't be
able to do that as I'm no expert on it myself but maybe you
found a little bit of interest in having a look at the topic and what goes
into it maybe you even feel like learning some of it for yourself. I'll
try and leave plenty of resources to do that down in the description and a
little reminder that if you're interested in going to see Sean Carroll
talk on his tour and you live in Australia or New Zealand then head along
to Think Inc and use the code tibees to get 15% off of some tickets. Thanks for
watching.