Special Relativity and Classical Field Theory

For quite a while Leonard Susskind has been giving some wonderful courses on physics under the name “The Theoretical Minimum”, pitched at a level in between typical popularizations and standard advanced undergraduate courses. This is a great idea, since there is not much else of this kind, while lots of people inspired by a popular book could use something more serious to start learning what is really going on. The courses are available as Youtube lectures here.

Book versions of some of the courses have now appeared, first one (in collaboration with George Hrabovsky) about classical mechanics, then one (with Art Friedman) about quantum mechanics. I wrote a little bit about these here and here, thought they were very well done. When last in Paris I noticed that there’s now a French version of these two books (with a blurb from me for the quantum mechanics one).

The third book in the series (also with Art Friedman) is about to appear. It’s entitled Special Relativity and Classical Field Theory, and is in much the same successful style as the first two books. Robert Crease has a detailed and very positive review in Nature which does a good job of explaining what’s in the book and which I’d mostly agree with.

The basic concept of the book is to cover special relativity and electromagnetism together, getting to the point of understanding the behavior of electric and magnetic fields under Lorentz transformations, and the Lorentz invariance properties of Maxwell’s equation. Along the way, there’s quite a lot of the usual sort of discussion of special relativity in terms of understanding what happens as you change reference frame, a lot of detailed working out of gymnastics with tensors, and some discussion in the Lagrangian language of the Klein-Gordon equation as a simpler case of a (classical) relativistic field theory than the Maxwell theory. Much of what is covered is clearly overkill if you just want to understand E and M, but undoubtedly is motivated by his desire to go on to general relativity in the next volume in this series.

At various points along the way, the book provides a much more detailed and leisurely explanation of crucial topics that a typical textbook would cover all too quickly. This should be very helpful for students (perhaps the majority?) who have trouble following what’s going on in their textbooks or course due to not enough detail or motivation. Besides non-traditional students in a course of self-study, the book may be quite useful for conventional students as a supplement to their textbook.

One of the most annoying things someone can do while reviewing a book is to start going on about their own different take on the material, criticizing the author for not writing a very different book. So, the rest of this posting is no longer a review of the book, it’s now about the very different topic of what I think about this material, nothing to do with Susskind’s valuable and different approach.

This semester I’m teaching a graduate level course on geometry, and by chance the past week have been discussing exactly some of the same material about tensor fields that Susskind covers. The perspective is quite different, starting with trying to explain a coordinate-invariant point of view on what these things are, only then getting to the formalism Susskind discusses. I can’t help thinking that, with all the effort Susskind (and pretty much every other physics textbook…) devotes to endless gymnastics with tensors in coordinates, they could instead be providing an understanding of the geometry behind this story. It’s unfortunate that many if not most of those who study this material in physics don’t ever get exposed to this point of view. Thinking in geometrical terms, the vector potential and field strength have relatively simple interpretations, and using differential forms the equations needed for the part of E and M Susskind covers are pretty much just:

F=dA, dF=0, and d*F=*J

Similarly, for the special relativity material, there’s a danger of the basic simplicity of the story getting lost in calculations of how things appear in coordinates with respect to different reference frames. What you fundamentally need is mainly that objects are described by a (conserved in the absence of forces) energy-momentum p, which satisfies p2= -m2, with Lorentz transformations taking one such p to another. The wider principle is that things are described by solutions to wave equations, with special relativity saying that the Lorentz group takes solutions to solutions.

I’d like to believe that such a very different course and very different book would be possible, quite possibly am very wrong (I’ve never taught special relativity to anyone). Maybe some day someone, inspired by Susskind’s project, might try to do something at a similar level, but from a more geometric point of view.

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30 Responses to Special Relativity and Classical Field Theory

  1. Anon says:

    MIT open course ware, has a complete course on Quantum physics complete with lecture videos assignments and exams. Just started (probably won’t finish). Its very good.

  2. ilovecats says:

    Do you think Penrose’s book (e.g. Road to Reality) covers these topics from a more geometric angle? In any case, as a student who has been following your blog for the last 3 years, I’d love it if you sometime wrote just a couple of short articles on what you mean (books can come later :).

  3. Jim Holt says:

    Does Susskind derive EM as a gauge theory? It seems to me that conceptually this is the right way to go: getting EM from SR (instead of the other way around) by quantizing the SR equation E^2 – p^2 = m^2 to get an equation for a (complex) scalar field, and then applying the gauge principle (which can be motivated by SR) to get the EM vector potential. Fringe benefit of this top-down approach to EM: you are not shocked by the Aharanov-Bohm effect.

  4. Peter Woit says:


    Penrose’s book is kind of the opposite of Susskind’s. It’s deeply geometrical, but at the same time really is not appropriate for beginners, many professional mathematicians and physicists find it challenging to follow.

    The main reason I haven’t written up some of these things myself it that there are many other places that this has been done. For some suggestions, see


    Jim Holt,

    He does use gauge symmetry for the coupling of EM to matter, although for particles coupled to EM (gauge potential changes the action for a particle trajectory). In this book everything is classical, so he can’t get coupling to EM via gauge symmetry of a wave-function. The Klein-Gordon equation is discussed, but treated as an example of a Lorentz invariant classical field (somewhat of a motivation for the classical EM Maxwell equations, not something used to describe matter).

  5. vmarko says:

    Usually, if you want a thorough geometric approach to SR, EM and beyond, the main reference is the MTW book. Though differential forms as such are never beginner material.

    Btw, coupling EM to matter vis gauge symmetry is an entirely classical concept, i.e. classical field theory. Dirac equation is as classical as the Klein-Gordon equation, as long as you understand them as equations for fields, as opposed to relativistic QM. In fact, the whole Standard Model action is classical, together with the Higgs mechanism and all… Quantization only builds on top of that, leading to QFT.


  6. Tim May says:


    I’ve been a daily reader of your blog for about 8-10 years, or whenever it started, since after reading your book. I’ve been generally skeptical of string theory for a very long time, maybe 30 years or more. (I joined Intel Corp. in 1974 as a device physics guy, working on DRAM memory chips in the early days. Took classes from Jim Hartle and Douglas Scalapino at UCSB, but am by no means a graduate-level physicist! Still, I follow it.)

    Iroinically, two of my favorite physicists currently speaking are Leonard Susskind and Nima Arkani-Hamed. Though both are associated in some way with string theory, both are excellent speakers. (I could add that both are speaking a lot about things that are not directly associated with string theory. I don’t think either has given up on string theory, just that their interests seem to be in other areas. While one can criticize string theory, it seems to have directly or tangentially led to some interesting other theories and some new math. A point I recall you have also made._)

    I am glad that your criticisms of some aspects of string theory have not extended to criticisms of either of these two, or of Witten, Maldacena, or others. (I am not happy that a Czech blogger, whose lengthy explanations I often like, is so prone to personalizing his criticisms.)

    Living only about 45 miles southeast of Stanford, outside of Santa Cruz, I have been to half a dozen or so of Susskind’s talks, plus have seen him hosting a bunch of talks. Never made it to Arkani-Hamed’s talks, but have greatly enjoyed his videos. Have seen the Cornell 5-part series about 2.5 times (the final two more than three times.)


    –Tim May

  7. S says:

    I’ve had a bit of success* teaching SR to undergrads using Chapter 6 of Barrett O’Neill’s wonderful book “Semi-Riemannian Geometry with Applications to Relativity.” The earlier chapters do exactly what you’re asking with tensors (and differential geometry), and chapter 6 is a really lovely geometric discussion of SR. (I also want to check out Gregory Naber’s book, “The Geometry of Minkowski Spacetime.” I know from other books that he’s a wonderful writer, but I haven’t had the chance to read this one yet.)

    *”bit” = small sample size, not low success rate.

  8. Thomas Larsson says:

    Let me disagree about index-free notation being simpler than tensor calculus. Sure, you can eliminate a few indices, but OTOH you must add definitions (of d and *), and if you want to express more complicated things (upper and lower indices, symmetrized or anti-symmetrized, etc.) more definitions are needed. In the end you risk being swamped by definitions instead of indices, which does not seem like such a great advantage to me.

    Moreover, finding the right definitions usually involves index manipulation. The reason why the exterior derivative is interesting is that it is a group homomorphism, which intertwines between modules of the group of coordinate transformations (i.e. if A is an antisymmetric tensor field, so is dA). If you are interested in manifolds with some extra structure, e.g. a volume, symplectic, or contact form, the relevant group preserves these structures (the algebras S_n, H_n, K_n in Sophus Lie’s notation), and there are new homomorphisms, “exterior derivatives”, which only need to intertwine with the relevant subalgebra of W_n.

    You could perhaps look up these homomorphisms in the literature, but at least the classification of binary and higher homomorphisms is rather recent and not readily available in textbooks. Moreover, I once worked out the exterior derivatives in a case that was not yet published. Some fifteen years ago I was briefly interested in the exceptional Lie superalgebras E(3,6) and E(3,8), because there is a correspondance between these superalgebras and SU(3)xSU(2)xU(1). Kac and Rudakov had described the homomorphisms for E(3,6) but had not yet published the result for E(3,8) when I worked it out.

    The point is that this was done using techniques from tensor calculus (and I proud myself of being good at index gymnastics). It could not have been done with index-free notation, because none was available for this superalgebra at the time, and probably still isn’t.

  9. Peter Woit says:

    Thomas Larsson,
    I’m not claiming that index-free notation is always the best way to calculate things. Often you need to pick a well-chosen set of coordinates and calculate using those. My point is that it is worthwhile to understand the geometrical, coordinate independent, significance of the objects one is calculating with. A good example is cases in GR where you find solutions with singularities, need to realize these are not physical singularities, but coordinate singularities.

  10. NoGo says:

    Have you seen “It’s About Time” by N. David Mermin?
    It is (IMHO) very accessible and very geometrical introduction to Special Relativity for non-scientists. No tensors there… But it’s just SR, no electromagnetism etc.

  11. Doug McDonald says:

    Its clear to me after 40 years of seeing things like this piece by Woit that the problem
    with all the “geometry” methods is that they are not taught as standard fare to
    undergraduates. I can’t understand a word of them (with the words meaning their
    specific mathematical meanings of course, not say “manifold” as its ordinary one.)
    I’ve not found a usable intro to that stuff.

    One needs at least a couple of years of digesting the math at a pliable age
    before “just using the math” for the physics.

  12. Robinson says:

    I wrote Leonard Susskind some years ago to thank him for these lectures. I learned just enough to be able to vaguely follow what’s going on in modern physics at the moment. Allan Adams Open Courseware (MIT) on foundations of Quantum Mechanics is also really good.

  13. Anon says:

    Perhaps of interest for index free definitions (Thorne & Blandford, Modern Classical Physics)


  14. Jeff M says:

    Well, I learned special relativity out of Spacetime Physics, thereby no doubt dating myself. Perhaps even worse than one might think, since when I first read it Styx was a big band. Differential forms are wonderful, but I didn’t see them until I was a grad student in math. Learned GR the classical way, from Rindler. Lot’s of indices that I remember.

  15. Thomas Larsson says:

    Peter Woit,
    I did not talk about choosing coordinates for a specific problem, but rather about expressing the same equations with or without indices. The geometrical meaning can be understood in both cases, but index-free notation requires that somebody has already worked out the right operators like d and *. In contrast, tensor calculus can be applied also in unchartered territory. The geometrical meaning of E(3|8)-invariant supergeometry might not be clear, but it was still possible to figure out the analogues of the exterior derivative using index notation.

  16. cthulhu says:

    Feynman’s “Six Not-So-Easy Pieces” is mostly about Special Relativity and I liked it, but I’m looking forward to this new book too; I’ve enjoyed the other two.

    On the tensor index issue, I remember my professor for tensor calculus + calculus of variations (taught out of the engineering department for the engineering majors, not the math department) extolling the geometric interpretation for many problems – grads, divs, and curls instead of index gymnastics – at least until you have to calculate.

  17. Manfred Requardt says:

    Concerning the index free notation. There exists a combination of both methods, called the abstract index notation (see e.g. the nice book of Wald). I think, at least for physicists the abstract index notation has certain advantages, in particular if you have to do some calculations.

  18. S says:

    I had the opposite experience of some. I learned the index notation first, from a physicist, as an undergrad, and I had no idea what in heck he was talking about. (None of us did). It can be taught well, I’m sure, but it certainly wasn’t to us, it was only mystifying. Learning the geometric interpretation (from O’Neill, in my own case) brought clarity where there had been only mystification.

  19. Luca Ambrogioni says:

    I think that the best example of a deeply geometrical textbook suitable for beginners is “Gauge Fields, Knots and Gravity” by John C. Baez. It starts by giving an insightful but somehow informal explanation of differential and Reimanian geometry and then uses it for introducing EM and GR from a purely geometrical (and topological) point of view. It is a really beautiful book.

  20. former mathematician says:

    I can’t find a link, but there was a limerick circulating in the 50s, titled “LPE to DSC.” It began “A connection is simply outre when expressed in an intrinsic way.” The response, “DSC to LPE,” ended “The devil take you and Christoffel.”

  21. Jeff Moreland says:

    Very Special Relativity by Sander Bais gives a good geometric introduction to SR for interested laymen.

  22. Bobito says:

    “Iā€™m not claiming that index-free notation is always the best way to calculate things. Often you need to pick a well-chosen set of coordinates and calculate using those. My point is that it is worthwhile to understand the geometrical, coordinate independent, significance of the objects one is calculating with. A good example is cases in GR where you find solutions with singularities, need to realize these are not physical singularities, but coordinate singularities.”

    If one uses Penrose’s abstract indices, indices are merely labels that indicate tensor type and symmetry (it works for spinors too), and their use requires no choice of coordinates/frames. This is all explained in detail in volume one of Penrose/Rindler. It is a mistake to confuse the use of indices with the use of coordinates. They are two different things.

  23. Mozibur Ullah says:

    I first learnt about general relativity from Gerochs semi-popular book General Relativity from A to B just before I left for university (to study mathematics); it’s a wonderful book that uses just basic arithmetic to get to the essence of the subject.

    When it came to index notation, although I could see how it worked, I couldn’t see why it worked. It’s only when I came across Lees book on smooth & topological manifolds that some light dawned. I also found Michors book on Natural Operations (which is freely available) an eye-opener on categorical methods to help organise the material.

    Although index-free thinking is more geometric I think both skill sets are useful, otherwise physics written in index notation would not be accessible; I also think it shouldn’t be forgotten that this route requires a serious investment of effort (the tangent bundle at a point p is the set of all derivations at that point!) and probably the optimum book explaining this material hasn’t been written yet. It took me some time to learn, for example, that bundle and sheaf methods were, roughly speaking, dual to each other.

    I don’t think it’s beyond the bounds of the possible to teach differential forms in high school if the right approach was taken – a practical approach; after all, in the UK, advanced students learn about the cross product in vector geometry, it’s only later that I realised it was the Lie algebra of SO(3)! And of course calculus itself was once only for the cognoscenti whereas every schoolboy knows about it.

  24. I’d also second Ambrogionis suggestion on Baezs ‘Gauge fields, knots and gravity’ as a beautifully written introductory book to differential forms in physics.

  25. Milkshake says:

    Are there videos of your lectures on modern Geometry?

  26. Peter Woit says:

    Definitely not. Keep in mind that the lectures in that course are mostly the usual material and just about every math department has a similar course (with likely a more industrious and more talented lecturer…). The course webpage
    shows what I’m covering and give reading suggestions. There are a lot of good textbooks on the subject at this level.

    I do think a course at a bit lower level, more aimed at physicists, would be a good idea, and is not something that is so common, maybe some day I’ll try and teach one. But that’s not this course.

  27. Mark Weitzman says:

    Another interesting book on special relativity is Special Relativity in General Frames: From Particles to Astrophysics (Graduate Texts in Physics) – Eric Gourgoulhon, which doesn’t just focus on inertial observers, and treats things like the Thomas precession, and Sagnac effect in an easy manner.

  28. Manfred Requardt says:

    As to so-called Abstract Index Notation. It is usually attributed to R.Penrose. But actually one can find it already in Laugwitz, Math.Zeitschr. 61, (1954) 100, and as far as I remember already Schouten introduced it.
    But anyway, as it is not so deep an observation, it is perhaps not so important who introduced it (‘invented’ is perhaps to great a word) .

  29. Luca Ambrogioni says:

    To Manfred

    I would say that the abstract index notation gets reinvented everytime someone who understands differential geometry opens a physics book. I think that most physicists writing indices have a geometrical object in mind.

    You will never understand relativity if you think that $R_{abcd}$ is “just a number”.

  30. Manfred Requardt says:

    Dear Luca, you are absolutely wright. After all, an alternating differential form over the cotangential space is nothing but an antisymmetric tensor. On the other hand, not all geometric objects are antisymmetric, that is, are forms.

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