Learning General Relativity

Math blogger Joseph Nebus does another A – Z series of posts, explaining technical terms in mathematics. He asked readers for their favorite pick of things to be covered in this series, and I came up with General Covariance. Which he laid out in this post – in his signature style, using neither equations nor pop-science images like deformed rubber mattresses – but ‘just words’. As so often, he manages to explain things really well!

Actually, I asked for that term as I am in the middle of yet another physics (re-)learning project – in the spirit of my ventures into QFT a while back.

Since a while I have now tried (on this blog) to cover only the physics related to something I have both education in and hands-on experience with. Re General Relativity I have neither: My PhD was in applied condensed-matter physics – lasers, superconductors, optics – and this article by physicist Chad Orzel about What Math Do You Need For Physics? covers well what sort of math you need in that case. Quote:

I moved into the lab, and was concerned more with technical details of vacuum pumps and lasers and electronic circuits and computer data acquisition and analysis.

So I cannot find the remotest way to justify why I would need General Relativity on a daily basis – insider jokes about very peculiarly torus-shaped underground water/ice tanks for heat pumps aside.

My motivation is what I described in this post: Math-heavy physics is – for me, that means a statistical sample of 1 – the best way of brazing myself for any type of tech / IT / engineering work. This positive effect is not even directly related to math/physics aspects of that work.

But I also noticed ‘on the internet’ that there is a community of science and math enthusiasts, who indulge in self-studying theoretical physics seriously as a hobby. Often these are physics majors who ended up in very different industry sectors or in management / ‘non-tech’ jobs and who want to reconnect with what they once learned.

For those fellow learners I’d like to publish links to my favorite learning resources.

There seem to be two ways to start a course or book on GR, and sometimes authors toggle between both modes. You can start from the ‘tangible’ physics of our flat space (spacetime) plus special relativity and then gradually ‘add a bit of curvature’ and related concepts. In this way the introduction sounds familiar, and less daunting. Or you could try to introduce the mathematical concepts at a most rigorous abstract level, and return to the actual physics of our 4D spacetime and matter as late as possible.

The latter makes a lot of sense as you better unlearn some things you took for granted about vector and tensor calculus in flat space. A vector must no longer be visualized as an arrow that can be moved around carelessly in space, and one must be very careful in visualizing what transforming coordinates really means.

For motivation or as an ‘upper level pop-sci intro’…

Richard Feynman’s lecture on curved space might be a very good primer. Feynman explains what curved space and curved spacetime actually mean. Yes, he is using that infamous beetle on a balloon, but he also gives some numbers obtained by back-of-the-envelope calculations that explain important concepts.

For learning about the mathematical foundations …

I cannot praise these Lectures given at the Heraeus International Winter School Gravity and Light 2015 enough. Award-winning lecturer Frederic P. Schuller goes to great lengths to introduce concepts carefully and precisely. His goal is to make all implicit assumptions explicit and avoid allusions to misguided ‘intuitions’ one might got have used to when working with vector analysis, tensors, gradients, derivatives etc. in our tangible 3D world – covered by what he calls ‘undergraduate analysis’. Only in lecture 9 the first connection is made back to Newtonian gravity. Then, back to math only for some more lectures, until finally our 4D spacetime is discussed in lecture 13.

Schuller mentions in passing that Einstein himself struggled with the advanced math of his own theory, e.g. in the sense of not yet distinguishing clearly between the mathematical structure that represents the real world (a topological manifold) and the multi-dimensional chart we project our world onto when using an atlas. It is interesting to pair these lectures with this paper on the history and philosophy of general relativity – a link Joseph Nebus has pointed to in his post on covariance.

Learning physics or math from videos you need to be much more disciplined than with plowing through textbooks – in the sense that you absolutely have to do every single step in a derivation on your own. It is easy to delude oneself that you understood something by following a derivation passively, without calculating anything yourself. So what makes these lectures so useful is that tutorial sessions have been recorded as well: Tutorial sheets and videos can be found here.
(Edit: The Youtube channel of the event has not all the recordings of the tutorial sessions, only this conference website has. It seems the former domain does not work any more, but the content is perserved at gravity-and-light.herokuapp.com)

You also find brief notes for these lectures here.

For a ‘physics-only’ introduction …

… I picked a classical, ‘legendary’ resource: Landau and Lifshitz give an introduction to General Relativity in the last third of the second volume in their Course of Theoretical Physics, The Classical Theory of Fields. Landau and Lifshitz’s text is terse, perhaps similar in style to Dirac’s classical introduction to quantum mechanics. No humor, but sublime and elegant.

Landau and Lifshitz don’t need manifolds nor tangent bundles, and they use the 3D curvature tensor of space a lot in addition to the metric tensor of 4D spacetime. They introduce concepts of differences in space and time right from the start, plus the notion of simultaneity. Mathematicians might be shocked by a ‘typical physicist’s’ way to deal with differentials, the way vectors on different points in space are related, etc. – neglecting (at first sight, explore every footnote in detail!) the tower of mathematical structures you actually need to do this precisely.

But I would regard Lev Landau sort of a Richard Feynman of The East, so it takes his genius not make any silly mistakes by taking the seemingly intuitive notions too literally. And I recommend this book only when combined with a most rigorous introduction.

For additional reading and ‘bridging the gap’…

I recommend Sean Carroll’s  Lecture Notes on General Relativity from 1997 (precursor of his textbook), together with his short No-Nonsense Introduction to GR as a summary. Carroll switches between more intuitive physics and very formal math. He keeps his conversational tone – well known to readers of his popular physics books – which makes his lecture notes a pleasure to read.


So this was a long-winded way to present just a bunch of links. This post should also serve as sort of an excuse that I haven’t been really active on social media or followed up closely on other blogs recently. It seems in winter I am secluding myself from the world in order to catch up on theoretical physics.

11 Comments Add yours

  1. Peter Mander says:

    I wonder if GR will turn out to be a special case of a more general conception.

    1. elkement says:

      It should turn out as the classical / large scale limit of a unified theory of quantum gravity (… as far as I’ve understood current challenges in theoretical physics…)

  2. Erik Brown says:

    Great Elke, now you have me reading general covariance. Just what I needed, more thought provoking reading….aargh. But his explanations are really good. I like his comment, tensors are ‘the things mathematicians get into when they figure vectors just aren’t hard enough.’

    1. elkement says:

      Thanks, Erik – have fun :-)

  3. It sort of terrifies me that I know what you’re writing about. Recently, I did a bit of work as a writer/editor again and was immersed in a feeling of competence, noticing that I have a long way to go before I begin to feel even a little like that in mathematics. Yet, I notice here that progress IS being made!

    If you had to pick between permanently doing either social media or physics studies, which would you choose?

    1. elkement says:

      Great to hear from you, Michelle! Are you writing about anything STEM-related? I always imagine you would make for such a great science writer given your two backgrounds!
      I think in technical studies one learns the same things again and again, at deeper levels and from different perspectives – first the different perspectives make everything more confusing and patchy, and suddenly all falls into place! GR is a prime example to illustrate this…

      That question is rhetorical, right? ;-) If I would be forced to choose I’d pick physics without hesitating for a moment :-) But seriously: It seems that I am only ‘doing’ social media now when I want to mull about something in physics or engineering and spelling it out in public serves a purposes… or I tweet some tech / science stuff I found interesting … so there is not much ‘other’ (really social ;-)) part of social media left anyway!

      1. No, the writing now is simpler stuff, like helping science geeks with non-science writing tasks. I enjoy it, to be honest. I think I have a lot more work to do before I could begin any kind of responsible transmission of science knowledge. Still have lots to learn. :)

        I (obviously) can relate to choosing study over social media. I don’t do a lot of reading on-line any more. To be honest, after reading hundreds of pages of calculus, statistics, and whatever else, in a term, I’m too tired to read for pleasure. But it is always a treat to drop in on your blog and see what you’re doing! :)

        1. elkement says:

          Thanks for still reading here, despite your demanding schedule! I really appreciate it!!

          I think your writing help for science geeks is much needed … I know from experience how hard some non-science writing can be for a geek. For example, it took a long time and many versions until I was really satisfied with our company website. The hard part was to ‘stick to my own voice’ and not to feel obligated to regurgitate hackneyed phrases about how businesses should present themselves. Seems so obvious and easy to do, just let go of the artificial marketing BS stuff and write as you talk (or blog :-)), but for some reason it was quite a process.

          1. I think the difficulty you describe with maintaining your own voice is something all people struggle to do, regardless of a literary, science or business background. I suppose that is self-evident in the fact that there are so many genres of writing! You do perceive correctly about what ‘help’ is actually helpful to science geeks. This was the issue that I was discussing with one of these writers. He expressed some frustration at having had previous professional help that changed his writing in such a way that he didn’t feel it was sounding like him at all, and at the same time didn’t address the problem he was having with the text. I am lucky in that I have some experience that helped him achieve what he needed, and was useful. This is probably why I enjoyed getting back into the editor’s role for a few hours.

            One thing I enjoyed about reading your blog–and other science-minded people writing about every-day life–is you collectively have a certain style, a preference for certain phrases or rhythms, which locates you as a professional among others in a field, but you also have your particular personality, which is compelling and engaging. It is very difficult to work past those hackneyed phrases, but also so important. You make yourself a person others can know, and then can choose to do business with. You always have such an interesting variety of interaction on the blog, which I think is a strong indicator of how successful you’ve been at achieving your own writing personality.

            1. elkement says:

              What an interesting comment, thanks, Michelle!!

              Sometimes I think I started blogging and experimented so much with different websites to finally tackle the unwanted business phrases (And in two languages – I found the German version much more of an ordeal than the English one…).

              Maybe we started our very peculiar not-really-business-y looking German blog in order to finally transfer that style also to other ‘publications’ … because the blog is the first ‘website’ people find anyway, and the feedback on our style has been heartening (by the clients who finally show up, perhaps others had been ‘deflected’ before). It’s exactly as you say – people get to know you, and this is important if you offer services for private persons selling ‘from human to human’, rather than ‘pitching your professional services to a corporation’. It might have also been difficult for me, as I had done the latter for a long time …

            2. We always get so far off topic in these conversations! :)

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