Textbook on the Geometry of Special Relativity

Question

I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional "beginners" approach. Would anyone have a recommendation for such a textbook ?

I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do not need is another "and here is the formula for time dilation..." type of text, of which there are thousands out there, but something much more geometric and in-depth.

Thanks in advance.

Answer 1

I quite liked

The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity. Gregory L. Naber. Dover, 1992.

It has a very formal and very rigorous treatment of the geometry, and I still find it refreshingly in tune with the formal mathematicians' way I was taught linear algebra with (e.g. Friedberg, Insel & Spence). He does a brief physical motivation for why one can define the Minkowski spacetime $\mathcal M$ as a four-dimensional vector space equipped with a Lorentzian metric, and then rolls along with that and only that.

One curious aspect is his treatment of the electromagnetic field tensor, which he treats in the mixed-form, linear-transformation-like, $F^\mu_{\ \ \ \nu}$, which is natural for the phrasing of the Lorentz force in 'newtonian' (as opposed to 'lagrangian') relativity, i.e. $$\frac d{d\tau}p^\mu=qF^\mu_{\ \ \ \nu}q^\nu.$$ This forces one to think of tensor symmetry and anti-symmetry in quite different terms to the usual, and once you get your head around it it makes it much easier to deal with index raising and lowering later on.

He also includes sections on spinor representations of the Lorenz group, on de Sitter space, and certain weird causality-induced topologies on Minkowski spacetime, but I didn't really go into them at the time - they are a bit heavier than I would expect most physics undergraduates to handle. They look nicely formal, though.

Answer 2

I would encourage you to examine The Classical Theory of Fields, by Landau and Lifshitz.

You will need two years of calculus before opening the book. Four-vectors and tensors are introduced on page 14. There are occasional, specific sections that focus on giving you the needed mathematical tools needed for the subsequent physics.

The text covers special relativity, electrodynamics, and general relativity, in that order, from the ground up, and in under 400 pages. Its style is terse, precise, and compelling. Its contents are trustworthy and rigorous, and suited best to advanced students.

Landau likes to use action principles as a starting point for "deriving" the related differential equations.

This book opened my mind to many things when I was a young man.

Answer 3

One great option is Bernard Schutz's book, A First Course in General Relativity. It's technically a general relativity book, but it spends the first hundred pages on special relativity alone, introducing four-vectors and tensors clearly and methodically. This is all set up in a way that makes the generalization to curved spacetime quite natural.