### Introducing the GR & QFT seminar

#### Posted by John Huerta

Here at the Australian National University, I’ve started running a seminar with Mathew Langford. Mat is a PhD student working on geometric analysis, specifically something called extrinsic curvature flow, but we both love mathematical physics, so we decided to teach some to each other, and whomever else wanted to listen. As it turns out, that’s a lot of people! I guess I shouldn’t be surprised; there are a lot of mathematicians who want to learn some more physics, and quite a few physicists who want to learn more mathematics. Now, through the magic of the Internet, I’d like to invite you to join in our discussion. So, over the next few weeks, as Mat lectures about general relativity and I lecture about quantum field theory, I’ll blog about it here. I’ll also keep a website for the seminar, complete with exercises:

In Mat’s opening lecture, about GR, he discussed three different views of spacetime, corresponding to the physics of Aristotle, Galileo, and Einstein. Bizarrely, it’s Galilean spacetime that’s the hardest to define: this turns out to be a fiber bundle, the bundle of spaces over times, equipped with a little extra structure to tell us what it means to be “inertial”.

You can find the notes for Mat’s lecture here:

Below the fold, I’ll give you a summary of these three kinds of spacetime, and how these mathematical ideas relate to the treatment you would meet in a physics class.

Physicists like to define ideas using *symmetry*, while mathematicians often
prefer explicit *structure*. For instance, the Lorentz transformations were
discovered by Lorentz, a physicist, well before Minkowski, a mathematician,
identified them as some of the symmetries of Minkowski spacetime. This
dichotomy is one way to understand why Mat’s approach to spacetime, as a
mathematician, differs from the usual treatment in a physics textbook: Mat is
describing structure, and getting the symmetries (coordinate transformations)
out as maps preserving structure.

Here, I’d like to summarize what Mat does in his lecture. First, I’ll describe the structure, then I’ll pass to the more familiar description in terms of symmetries. Since, however, I’m also a mathematician, I’ll tell you about the groups of transformations and eschew specific formulas. For that, you’ll need to read Mat’s notes!

So let’s get started, beginning with Aristotelian spacetime.

## Aristotelian spacetime

Aristotle believed in the idea of “absolute rest”: the Earth was the immovable center of the Universe, and all bodies on Earth tended to come to rest unless forces continued to push on them. From our more enlightened perspective, we may find this idea naive: it’s the forces of friction that bring moving bodies to rest. But let’s give Aristotle his due: since everything on Earth experiences lots of friction, Aristotle’s view is fairly accurate.

For the sake of our story, we’ll assume Aristotle also believed in “absolute time”: that every event in the Universe could be said to take place at a given time, the same for all observers. Again, since Aristotle’s focus was on the Earth, this is not unreasonable: when I, in Australia, call my friends in California, we could, on a whim, decide to synchronize our watches, and they wouldn’t go out of sync.

Thanks to having the Earth as our gold standard of absolute rest, we can then
label every point in the Universe with coordinates that are always at absolute
rest with respect to the Earth. Since Aristotle was a Greek, we’ll go ahead and
assume this space is 3-dimensional Euclidean space,
$S = \mathrm{E}^3 .$
This is just $\mathbb{R}^3$ with the Euclidean metric:
$g(X,Y) = X \cdot Y ,$
but we make a point of forgetting the origin in $\mathbb{R}^3$, so that $\mathrm{E}^3$ is an **affine space**. Likewise, having the Earth to set our gold standard of absolute time (say, by a clock in Greenwich), we can label every event with a time. Because we can quantify time differences, time is one-dimensional
Euclidean space:
$T = \mathrm{E}^1 .$
So, we’ll define **Aristotelian spacetime** $A$ to be the Cartesian product of
these affine spaces:
$A = S \times T .$
Points in spacetime are called **events**, and our ability to specify the
absolute position and time of each event comes from the projection maps:
$\Pi_S \colon A \to S, \quad \Pi_T \colon A \to T .$
An **Aristotelian observer** is a worldline in Aristotelian spacetime, which is a
function from time to space:
$O \colon T \to S .$

Aristotelian spacetime is not important enough for Mat to tells us about its
symmetries, but for the sake of our story, I’ll define the **Aristotelian group**
as the group of all bijections from $A$ to itself induced by isometries of $S$ and $T$. So, the Aristotelian group is just the Cartesian product:
$IO(3) \times IO(1)$
where $IO(n)$ denotes the **inhomogeneous orthogonal group**:
$IO(n) = \mathrm{O}(n) \ltimes \mathbb{R}^n .$
This is the group of isometries of $n$-dimensional Euclidean space:
$\mathrm{O}(n)$ provides rotations and reflections, and $\mathbb{R}^n$ the translations.

## Galilean spacetime

Strangely enough, Galilean spacetime is the hardest to describe. Galileo did
away with the Aristotelian notion of absolute rest by formulating the **principle
of relativity**: all observers moving at a constant velocity with respect to one
another experience the same laws of physics.

Mathematically, this means we want Galilean spacetime to be like Aristotelian
spacetime, but without the projection to space, $\Pi_S \colon A \to S$. We
will, however, hold on to the notion of absolute time by having a projection to
$T$. That is, we’ll define **Galilean spacetime** $G$ to be a fiber bundle over
time:

$\Pi_T \colon G \to T .$

As before, $T$ is $\mathrm{E}^1$, but now $G$ just has fiber $E^3$: at each moment of time, we live in $\mathrm{E}^3$. We define a **Galilean observer** to be a section:
$O \colon T \to G .$

Though the principle of relativity banishes absolute rest, something remains: the equivalence of observers with constant relative velocities. How do we express this mathematically? First, we need a way to compute the velocity of one observer with respect to another. Letting $O_1$ and $O_2$ be two observers, this means we want to compare displacement vectors at different times: $O_2(t_2) - O_1(t_2) \quad versus \quad O_2(t_1) - O_1(t_1) .$ Of course, we cannot do this without some kind of connection, because the affine spaces over $t_1$ and $t_2$ are not comparable, and nor are their vector spaces of displacements.

So let us equip $G$ with a very barebones connection, called a **parallelism**: for each pair of times $t_1$ and $t_2$, we specify an isometry between the vector spaces of displacements over those times:
$P_{t_1,t_2} \colon \mathbb{R}^3 \to \mathbb{R}^3 .$
This mathematical structure just says something physically reasonable: lengths remain the same as time passes. A meter stick stays a meter stick. But it allows us to compute the relative velocity of a pair of observers, because now we can compare their displacement vectors at different times. We say that $O_1$ and $O_2$ are in the same **inertial class** if the velocity vector:
$v(t) = \frac{d}{d t} (O_2(t) - O_1(t))$
is a constant, where we take the derivative using our parallelism.

Now we can define the **Galilean group** to be the group of all bundle
isomorphisms:
$\begin{array}{ccccc}
G & & \longrightarrow & & G \\
& \searrow & & \swarrow & \\
& & T & & \\
\end{array}$
which preserve inertial classes. A calculation now shows the
usual Galilean transformations:
$\begin{array}{rcl}
t' & = & t \\
x' & = & x - v t \\
\end{array}$
are in the Galilean group.

Given how easy the Galilean transformations are to rattle off, and how difficult the structure of Galilean spacetime was to describe, one might wonder why we’re doing this at all. But, as the example of Minkowski spacetime shows, it pays to think about both the symmetries and the structure they preserve.

## Minkowski spacetime

Our real goal in this lecture is Minkowski spacetime, along with its structure and symmetries. Minkowski spacetime is like taking Galilean spacetime and incorporating both the principle of inertia with the constancy of the velocity of light. When Einstein did this, he famously discovered that absolute simultaneity did not exist. Just as in moving from Aristotelian spacetime $A$ to Galilean spacetime $G$, we dropped the projection onto space: $\Pi_S \colon A \to S .$ Now, in moving from Galilean to Minkowski spacetime, we drop the projection onto time: $\Pi_T \colon G \to T .$

We define **Minkowski spacetime** to be a 4-dimensional affine space $M$ equipped
with the Minkowski metric on its space of displacements. That is, given any two
displacement vectors $X$ and $Y$ in $\mathbb{R}^4$, define their inner product
to be:
$g(X,Y) = -X^0 Y^0 + X^1 Y^1 + X^2 Y^2 + X^3 Y^3 .$
Why? There are so many explanations in books, on the Internet, and in Mat’s
lecture, that I’ll let you read those. Here, I’ll just note one of the more
well-known virtues: in units where the speed of light $c$ is 1, light rays have
vanishing length with respect to the Minkowski metric. Thus, isometries of
Minkowski spacetime take light rays to light rays, preserving the velocity of
light.

We define a **Minkowski observer** to be any **timelike curve**, that is a
curve whose tangent lies more in the time direction than in the space
direction:
$O \colon \mathbb{R} \to M, \quad g(O'(s), O'(s)) < 0 .$
The **Poincaré group** is the group of all isometries of Minkowski
spacetime alluded to above:
$O(3,1) \ltimes \R^4 .$
This includes, as Mat shows, the Lorentz transformations:
$\begin{array}{rcl}
t' & = & \frac{t - v x}{\sqrt{1 - v^2}} \\
x' & = & \frac{x - v t}{\sqrt{1 - v^2}} \\
\end{array}$
that originally started us on our journey.

## Re: Introducing the GR & QFT seminar

I’m looking forward to hearing about this seminar.

One minor quibble, well two actually if I mention that it’s usually ‘Aristotelian’ rather than ‘Aristotlean’, is that his universe is generally recognised as having a fixed centre, which coincides with the centre of the Earth. Were the Earth to be moved away from this centre, it would wish to return to that spot.

The four elements have characteristic tendencies to move in relation to this special place: earth - strongly toward, water - less strongly toward, fire - strongly away, air - less strongly away.

The planets move about on concentric rotating spheres, so there is very much a special place in the universe at their centre. Physics below the sphere of the moon is very different from physics beyond the moon.