## September 13, 2012

### 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.

Posted at September 13, 2012 8:25 AM UTC

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### Re: Introducing the GR & QFT 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.

Posted by: David Corfield on September 13, 2012 1:58 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

it’s usually ‘Aristotelian’ rather than ‘Aristotlean’

I’m glad you’re here to supplement my fuzzy knowledge of Aristotle’s worldview. I guess, mathematically, this amounts to Aristotelian space having an origin. But maybe one cannot fully compare.

Posted by: John Huerta on September 14, 2012 4:14 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Looks like it’s time for me to convert to Aristotelian’. I like your comments about the Earth at the centre of the universe. I might incorporate that into the notes.

Posted by: Mat Langford on September 15, 2012 4:26 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Bizarrely, it’s Galilean spacetime that’s the hardest to define:

I guess I can imagine why you say that this is “bizarre”. But let me ask anyway: “Why is this bizarre?”

I sense the prejudice that naïve ideas have to be easy to define, while the deeper the idea, the more complicated it becomes. Could that be? Maybe I am misreading what you mean.

I think more often than not it’s the other way around: the deep idea that, as they once thought about GR, “only 3 people on the planet can understand” turns out to be much simpler than an idea that every man on the street thinks he understands. People often say “easy” for “familiar”.

For instance when it was understood that the sun is the center of rotation in the solar system, the theory of planetary motion became vastly easier to define. Even though it was apparent to the man on the street that it is instead the earth that is the center of rotation. The theory of epicycles that grows out of this apparently easy idea is complicated.

These days, to give a more recent example, it has been discovered that when you get to the bottom of it, sets are harder to define than homotopy types. Another Copernican revolution, which puts the right concepts into the center, doing away with the bizarre logical epicycles that we had all been so familiar with.

I have just argued in another post that in this context many other central aspects of GR become much simpler than what they used to look like.

For reasons like this I usually tend to think that it is reasonable to expect the opposite: the further we get to the bottom of the fundamental rules of the universe, the simpler these rules become. Down there somewhere is a tautology waiting for us to discover it, and everything else will be a consequence of it.

Posted by: Urs Schreiber on September 13, 2012 3:20 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Maybe a more historically respectful thing to say would be that it’s not surprising that it’s difficult to describe older ideas accurately when we insist on using language, technology, and expectations of rigor that have co-evolved with newer ideas and are adapted for describing those.

Posted by: Mike Shulman on September 14, 2012 4:09 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

What I mean by that is that Aristotle and Galileo didn’t remotely have the requisite language and technology in order to appreciate the context in which an Einsteinian universe can be seen to be simpler, nor would they have felt the need to describe their universes in this way. I believe that historically, an appreciation for the importance of describing the universe in this way only developed alongside relativity. Before Einstein, physicists had never heard of smooth manifolds (and mathematicians barely had). So while I do agree that deeper understandings are often simpler when expressed in the right language, at least part of the reason we view that language as ‘right’ is because it expresses in a simple way the deep understandings that we have come to about our universe, after a laborious process of trial and error.

Posted by: Mike Shulman on September 14, 2012 6:33 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Something to be expected as a theory becomes empirically successful is that even when it’s replaced by a conceptually very different theory, the new theory’s going to have to explain that old success.

E.g., one way GR can do this for Newtonian cosmology is by showing how it is possible to characterise the latter as an example of a 4d Riemannian manifold with geodesics, even if this outlook would be very foreign to the Newtonians.

Posted by: David Corfield on September 14, 2012 8:33 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Thanks for the comments Urs (and everybody),

there’s actually a more straightforward reason for why my exposition of Galilean spacetime is, bizzarrely’, more complicated than, e.g. Minkowski spacetime: I didn’t wish to presuppose a preferred inertial class, preferring rather a bare bones approach. We could have built this in from the beginning, and just assumed that Galilean physics’ lives on an affine space. I’m not sure how to approach SR in the same bare bones way, but I suspect we would just begin with a manifold, and then add an inertial’ (i.e. flat) connection, which tells the space to be Minkowski (if we insist on the connection also being Levi-Civita).

At the moment, I’ve been thinking a lot about Newtonian/Einsteinian gravity, and how the equivalence principle suggests a modification of the special observers (inertial -> freely falling), and, hence, the parallelism/connection. Haven’t quite figured it all out yet, though. Any suggestions would be appreciated :)

Mat

Posted by: Mat on September 15, 2012 12:43 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

So while I do agree that deeper understandings are often simpler when expressed in the right language, at least part of the reason we view that language as ‘right’ is because it expresses in a simple way the deep understandings that we have come to about our universe,

If I understand you correctly you are raising the point which I might paraphrase like this:

The reason that our deep theories in the end have a simple expression is because we eventually adopted a language designed to make them have a simple expression.

But I would say: it’s certainly a necessary condition that we re-adjust our language as we see deeper into the structure of the world around us. But it is not in general sufficient to adopt the language in order to make any old theory become simple.

In other words, I don’t agree (if this is what you were suggesting) that it is no surprise that GR is conceptually simpler than pre-Einsteinian thinking, just because we have now adjusted to this point of view.

John H. here is giving a nice proof of this claim of mine: here he is, with all the tools of modern mathematics at his disposal. He can use anything that seems suitable – it need not be the language which is well-adapted to GR – and see if there is a formulation of Galilean physics that is conceptually simple. But it doesn’t work out.

This is not about whether Galileo knew the concept of smooth manifolds or not. This is about whether with the best formal language available, there is a simple formulation of his theory. And that’s crucial: the reason why Galileo never thought of his theory as “bizarre” is because he never tried to formalize it as fully as we are discussing it here. Maybe if he had, he would have noticed what is indeed intrinsically bizarre about it, independently of the language used to talk about it: the intrinsic bizarreness lies in the fact that two kinds of translations that appear in the theory are treated very differently. Two concepts that are clearly similar to some extent are forced by the theory to be entirely incomparable. I don’t think any reasonable choice of language can “define that away” and make it look nice and elegant.

And I guess one might argue that indeed Einstein to a good extent solved this theoretical mismatch more than being driven by experimental insight. Special relativity was directly driven by experimental insight, but general relativity much more by pure thought and strive for conceptual naturalness.

I think the other examples that I mentioned follow this pattern. Maxwell’s equations which in the original form filled two full pages (I keep saying this, at some point I need to dig out a url that proves this) admit a language in which their statement reduces to 6 symbols $\d \ast F_{\nabla} = j$ only because they were on the right track. Change just one prefactor anywhere on these two pages of Maxwell’s and you get a theory that superficially looks like entirely of the same complexity as the original theory, but you will never find a coherent language in which it reduces to a 6-symbol expression. Because it turns out that all those prefactors on these two pages actually satisfy a subtle joint coherence condition – which is the necessary condition that we can absorb them into an elegant language of exterior calculus.

For these reasons I hold that there is something like intrinsic conceptual simplicity. It is true that it does go together with a subtle interplay with the natural language and both theory and language are part of the same process, clearly. But I don’t believe that any random theory has a simple expression if only you adopt the language appropriately

(Not for languages that satisfy some very basic requirements of what a “sensible language” is, at least. I mean, we can of course always introduce a language that contains one new symbol which by definition expresses whatever weird statement we want. But that wouldn’t be a “sensible theory” by some measure which maybe we could make precise, but which hopefully we need not, for the sake of the present discussion.)

Posted by: Urs Schreiber on September 15, 2012 10:50 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

it’s certainly a necessary condition that we re-adjust our language as we see deeper into the structure of the world around us. But it is not in general sufficient to adopt the language in order to make any old theory become simple.

I agree. This is why I added the words “part of”. (-:

Posted by: Mike Shulman on September 20, 2012 7:40 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

John, your overview and Mat’s notes are wonderful! They do a terrific job of focusing on the critical physics fundamentals and the mathematical techniques for describing and analyzing them. It’s not often that one can find such accurate, clear descriptions that include explanations, both mathematical and physical, that don’t assume a great deal of expert knowledge. It would be most appreciated if you could continue to write your overviews summarizing each lecture. It is extremely difficult to explain/teach at this level. Keep up the good work!

Posted by: Charlie Clingen on September 13, 2012 3:35 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Thanks, Charlie! We’ll try to live up to your praise.

Posted by: John Huerta on September 14, 2012 4:02 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Thanks Charlie! It’s been a lot of fun so far. Hopefully we can keep it up.

Posted by: Mat Langford on September 15, 2012 4:21 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

This is very nice, thanks for posting it! I look forward to future posts. I also have a couple of quibbly questions about this one…

Firstly, in Mat’s notes, he says that the Galilean group is composed of the Galilean transformations between relatively moving frames, the three-parameter “rotation group”, and the four-parameter group of translations. I presume that this “rotation group” refers to $O(3)$ (which includes reflections as well as rotations), by analogy with the groups you describe in the other two cases. What about time reflection? If I understand correctly, the Aristotelian group and the Poincare group both include time reflection (in $I O(1)$ or $O(3,1)$), but Mat’s description of the Galilean group doesn’t mention it. And your definition of the Galilean group as consisting of bundle automorphisms over $T$ seems like it would exclude not only time reflection, but also time translation.

And secondly, is it really okay to describe a parallelism as consisting of isometries $\mathbb{R}^3 \to \mathbb{R}^3$? Isn’t the point that we *don’t* have a canonical identification of the vector space of displacements in each affine fiber with $\mathbb{R}^3$ (or any other fixed 3-dimensional vector space)?

Posted by: Mike Shulman on September 14, 2012 7:01 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

I guess we have some freedom in what we consider to be part of the Galilean group, but you’re right, I definitely don’t want to exclude time translations. So, really, the group ought to consist of the bundle isomorphisms:

$\begin{array}{ccc} G & \longrightarrow & G \\ \downarrow & & \downarrow \\ T & \longrightarrow & T \\ \end{array}$

that preserve inertial class.

You’re also right that matching the vector spaces of displacements up with $\mathbb{R}^3$ isn’t quite right, but I did it to ease of exposition. I guess, in my mind, I was shoving an isometry of the vector space of displacements over a time $t$ with $\mathbb{R}^3$ into the parallelism. Some kind of diagram like this:

$\begin{array}{ccc} & P'_{t_1,t_2} & \\ V_{t_1} & \longrightarrow & V_{t_2} \\ \downarrow & & \downarrow \\ \mathbb{R}^3 & \longrightarrow & \mathbb{R}^3 \\ & P_{t_1,t_2} & \\ \end{array}$

I bet that’s OK (ultimately, the parallelism allows us to trivialize the bundle, once we pick an observer to be the origin in each fiber), but it’s certainly not the most invariant way to work.

Posted by: John Huerta on September 14, 2012 11:52 AM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Aristotle may have been either ignorant or dismissive of friction but in the Epicurean school exemplified in the writings by Lucretious it is not, he says that atoms ‘are in perpetual motion at enormous speed, since in the void they get no resistance from the medium, and when they collide they can only be deflected, not halted’.

(I think its also very interesting that they supposed that atoms must have a minimal amount of indeterminancy (which they called ‘swerve’) to allow for free-will in the observable world.)

Posted by: mozibur ullah on September 14, 2012 10:09 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Aristotle may have been either ignorant or dismissive of friction but in the Epicurean school exemplified in the writings by Lucretious it is not, he says that atoms ‘are in perpetual motion at enormous speed, since in the void they get no resistance from the medium, and when they collide they can only be deflected, not halted’.

Could you give me a precise reference for this quote? That’s remarkable.

Posted by: Urs Schreiber on September 15, 2012 10:17 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

Of course, its in the lines 80-332 in Book 2 of ‘De Rerum Natura’, here’s an unabridged translation by William Leonard, and a commentary for these lines are in the entry for Lucretious in the Stanford Encyclopedia of Philosophy, scroll down to section 4 where they discuss Physics.

he shows that atoms are in constant motion:

For far beneath the ken of senses lies
The nature of those ultimates of the world;
And so, since those themselves thou canst not see,
Their motion also must they veil from men

and that they are constantly colliding with another:

Inveterately plied by motions mixed,
Some, at their jamming, bound aback and leave
Huge gaps between, and some from off the blow
Are hurried about with spaces small between.

But they’re able to be stay together to form compounds, but if I’m reading Lucretious right they’re still in motion:

And all which, brought together with slight gaps,
In more condensed union bound aback,
Linked by their own all intertangled shapes,-
These form the irrefragable roots of rocks
And the brute bulks of iron

and that the motion must be unpredictable, (Empson uses the word swerve here), for otherwise they’d never collide and nothing would be then created.

For were it not their wont
Thuswise to swerve,
down would they fall, each one,
Like drops of rain, through the unbottomed void;
And then collisions ne’er could be nor blows
Among the primal elements; and thus
Nature would never have created aught

The SEP says that Epicurus holds that atoms have internal structure consisting of partless magnitude, called ‘minima’, (Lucretious pretty much leaves this out in his polemics); the Nyaya-Vaisesika atomic theory also held atoms to be without extension but at the same time spherical. The suggestion is that they were using a non-classical form of logic to get around the obvious contradiction here (Nyaya is known as the indian school of logic).

Posted by: Mozibur Ullah on September 16, 2012 1:36 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

its in the lines 80-332 in Book 2 of ‘De Rerum Natura’,

Thanks for digging this out! I didn’t know and find it fascinating that Lucretius gives such a surprisingly accurate idea of what we know is actually going on.

I don’t quite have the time to further follow up on this right now, but since this is something worth recording in context I have taken the liberty of moving this into an $n$Lab-entry-to-be: friction.

Maybe I find the time later to turn that into an actual entry. Or maybe somebody else feels inspired to do so.

Posted by: Urs Schreiber on September 20, 2012 6:35 PM | Permalink | Reply to this

### Re: Introducing the GR & QFT seminar

I should point out that the quote in my comment was from SEP and not directly from Lucretious Poem :)

Posted by: Mozibur Ullah on September 16, 2012 9:31 PM | Permalink | Reply to this

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