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On Heraeus's Winter School Missing Tutorials/PDFs (Gravity and Light)

tags: others physics
date: 2022-12-21
update: 2024-02-17

Introduction

In 2015, a course (International Winter School on Gravity and Light), on the mathematical foundations of general relativity (manifolds, differential geometry, etc.) was held by Frederic P. Schuller (and others) at the Johannes Kepler University, Linz, Austria, 2015.

There are two main playlists on YouTube:

And, somewhat complementary, some unofficial notes (.pdf) by Richie Dadhley.

Finally, it seems that a considerable amount of time was dedicated to workshops (10h over two weeks, “where more advanced problems are solved in teams”, by comparison with the tutorials, 12h, “to practice the basics”), but unfortunately, nothing has been found pertaining to their content1.

Einstein, sanguine on paper

Einstein, sanguine on paper by M. Bivert through instagram.com

Tutorial PDFs & missing tutorial videos

There were official websites for the course, which are as of today (December 2022) unavailable; two of them are archived to some degree on archive.org:

I’ve taken the liberty to import from the latter all available tutorial .pdf files, and three tutorial videos (7, 8, 9/10) missing in the official YouTube playlist. We’re still missing the 14th and 17th video tutorials, which don’t seem available anywhere. I obviously don’t own any of this material, and am merely reuploading them to increase their visibility (archive.org pages aren’t directly google-able as far as I can tell).

Title .pdf
.zip with all sheets heraeus.zip
Topology Sheet_01.pdf
Topological manifolds Sheet_02.pdf
Multilinear Algebra Sheet_03.pdf
Differentiable Manifolds Sheet_04.pdf
Tangent Spaces Sheet_05.pdf
Fields Sheet_06.pdf
Connections Sheet_07.pdf
Parallel transport & Curvature Sheet_08.pdf
Metric Manifolds Sheet_09_10.pdf
Symmetry Sheet_11.pdf
Integration Sheet_12.pdf
Schwarzschild Spacetime Sheet_13.pdf
Relativistic Spacetime, Matter and Gravitation Sheet_14.pdf
Cosmology Sheet_15.pdf
Diagrams Sheet_16.pdf
Perturbation Theory Sheet_17.pdf

Here are the missing videos, and here’s a playlist containing all the available tutorials (14 and 17 are still missing then).

More courses by prof. Schuller

Frederic P. Schuller has two more courses on his YouTube channel:

The former seems to be a slower-paced version of the Heraeus course.

There’s a set of notes for the former, by written by Simon Rea. And there’s also one for the latter, originally written by Simon Rea, and later completed by Richie Dadhley.

Both contain elements of the course (e.g. some proofs) that have been relegated to the otherwise unavailable problem sheets.

Note: I’ve reworked the proof of the invertibility of the Fourier operator, in a hopefully clearer fashion (QM18). From the same lecture, here’s a proof for Parseval’s theorem.

Clarity

All three courses are remarkable, by comparison with similar material, for their mathematical precision (physicists can be mathematically “sloppy”), and for their clarity. By clarity, I mean that arguments are well-articulated (clear), and don’t assume too much from the students.

Bear in mind that clarity doesn’t mean easy: the subjects are technical, require a certain amount of work and background knowledge. Besides a (finite) linear algebra course, I guess a real analysis course (there are others; M. Penn’s courses tend to be well-articulated as well) is a great way to get a rigorous understanding of limits, convergence and related topics, as well as a feel for their generalization in more abstract settings (mainly metric/topological spaces).

Anyway, here’s a typical example of a well-articulated argument, from the lecture 4 of the quantum theory course (c. 43:17). Again, this assumes that you’re not completely oblivious to topological arguments, nor to “basic” mathematical reasoning2.

We’ve defined the orthogonal complement \(M^\perp\) of \(M\) in \(\mathcal{H}\), where \(M\) is a (not necessarily closed) linear subspace of a (separable) Hilbert space \((\mathcal{H}, +, ., <\cdot,\cdot>)\), as:

\[\begin{aligned} M^\perp := \{ \psi\in\mathcal{H}\ |\ (\forall\mu\in M),\ <\mu,\psi> = 0 \} \end{aligned}\]

There’s a property that \(M^\perp\) is then a closed linear subspace (the closedness implies, by a previous result, completeness, and thus that \(M^\perp\) forms a sub-Hilbert space). In Mathematical Methods in Quantum Mechanics by Gerald Teschl, this result is concisely justified en-passant in section \(1.3\):

By continuity of the scalar product it follows that \(M^\perp\) is a closed linear subspace […].

That’s it.

On the other hand, F. Schuller will quickly explain why it’s a linear subspace in the first place (similar results have been demonstrated more thoroughly earlier, hence why he’s lighter on details here), and then follows to prove the closed property more thoroughly (I’ll be omitting details here actually, so refer to the video for more): first by observing that \(M^\perp\) is actually:

\[\begin{aligned} M^\perp = \bigcup_{\mu\in M} \text{preimg}_{\omega_\mu}(\{0_{\mathbb{C}}\}); \qquad \text{with }\quad \omega_\mu := \begin{pmatrix} \mathcal{H} & \rightarrow & \mathbb{C} \\ \psi & \mapsto & <\mu,\psi> \\ \end{pmatrix} \end{aligned}\]

Then by recalling some notions and results from topology, including that a closed set is, by definition, one which has an open complement, and that an arbitrary (i.e. potentially infinite) intersection of closed set is closed, from which we could conclude, were we able to prove for an arbitrary \(\mu\in M\), that \(preimg_{\omega_\mu}(\{0_{\mathbb{C}}\})\) is closed. Which is then equivalent to prove that its complement is open; said complement can be reformulated as:

\[\begin{aligned} \mathcal{H}\backslash \text{preimg}_{\omega_\mu}(\{0_{\mathbb{C}}\}) = \{ \psi\in\mathcal{H}\ |\ <\mu,\psi> \neq 0_{\mathbb{C}} \} = \{ \psi\in\mathcal{H}\ |\ <\mu,\psi> = \omega_\mu(\psi) \in \mathbb{C}\backslash\{0_\mathbb{C}\} \} = \text{preimg}_{\omega_\mu}(\mathbb{C}\backslash\{0_\mathbb{C}\}) \end{aligned}\]

But, single-element sets are closed with respect to the standard topology on \(\mathbb{C}\)3, so their complements are open. In particular, \(\{0_\mathbb{C}\}\) is closed, and thus \(\{\mathbb{C}\backslash\{0_\mathbb{C}\}\}\) is opened.

Finally, because \(\omega_\mu\) is but an application of the inner-product, by continuity of the inner-product (established/mentioned in an earlier lecture, and used many times already) \(\omega_\mu\) is continous, and thus (recall that topologically-wise, a function is continuous if the pre-images of open sets are open) \(\mathcal{H}\backslash preimg_{\omega_\mu}(\{0_{\mathbb{C}}\}) \) must be open, and thus \(preimg_{\omega_\mu}(\{0_{\mathbb{C}}\})\) is closed, implying that \(M^\perp\) is closed. \(\Box\)

Note: This kind of situation is common in upper mathematical material: courses are typically taught to advanced students, which are expected to be able to fill the blanks with little effort, allowing teachers to be more concise.

Note: Teschl’s version is not to be dismissed either: Schuller’s is a great introductory material, but Teschl’s conciseness gives you plenty of room for strengthening your understanding.


  1. We’ve looked around a little with Thomas Walter, and there seems to be some inconsistencies between the planning and the available courses/tutorials; but nevertheless, as far as we can tell, the workshops really are missing. I’ve updated and clarified a few points regarding missing materials following Thomas’ return; thanks to him! ↩︎

  2. If you struggle with this, perhaps you’ll want to start with the Lectures on Geometrical Anatomy of Theoretical Physics↩︎

  3. F. Schuller doesn’t go far enough as to explain why this is, I guess because otherwise the lectures would be 5h long, but in case you’re wondering: in a metric space (which \(\mathbb{C}\) is, because it has a distance, the modulus), a subset \(M\) of \(\mathcal{H}\) is closed if the closure of \(M\) is included in \(M\); the closure of \(M\) consists of all points of \(\mathcal{H}\) which are arbitrarily close to the points of \(M\). So, if \(M = \{m\}\), the only point in \(\mathcal{H}\) arbitrarily close to \(m\) is \(m\) itself (think of an \(\epsilon\)-ball centered around \(m\), for \(\epsilon\) as small as we want). Note that I’m still omitting details on how exactly purely topological arguments translate to metric space. ↩︎


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