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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.

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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

If you're really serious about having an understanding of the underlying math, you can look at Barrett O'Neil's Semi Riemannian geometry. Chapter six is an introduction to special relativity from a very math-heavy perspective. –  Jerry Schirmer Jan 28 '14 at 8:34
Relativity, Groups, Particles by Sexl and Urbantke will work. –  NikolajK Jan 28 '14 at 8:44
I really enjoyed Tensor Geometry by Dodson & Poston; by rights it will give you a healthy foundation for General Relativity but treats the Special theory at the outset. books.google.co.uk/books/about/… –  Autolatry Jan 28 '14 at 11:46

3 Answers 3

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.

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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.

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This thread is a year old, but I felt the need to make this book known. The following book is perhaps what you seek:

Special Relativity in General Frames: From Particles to Astrophysics. Eric Gourgoulhon. Springer (Graduate Texts in Physics), 2013.

It is destined to be a classic text on the subject. It adopts a four-dimensional point of view from the very beginning. Gourgoulhon also treats accelerating and rotating observers aside from the usual treatment of inertial frames, which is something that you don't see in the average relativity textbook. In any case, I think the contents of the book can argue the case much better than I can. Selected contents are listed below:

  1. Minkowski Spacetime

    • Four Dimensions, Metric Tensor, Null Cone and Time Arrow, Spacetime Orientation, Vector/Linear Form Duality, Minkowski Spacetime
  2. Worldlines and Proper Time

    • Four-velocity, Four-acceleration, Langevin's traveller, Geometrical properties of a worldline
  3. Observers

    • Simultaneity, Spatial distances, Four-rotation of a local frame
  4. Kinematics I: Motion with respect to an observer

  5. Kinematics II: Change of observer
  6. Lorentz group
  7. Lorentz group as a Lie group
  8. Inertial observers and Poincare group
  9. Energy and Momentum 10.Anguar momentum
  10. Principle of least action

Etc. (22 chapters in total)

There is also an appendix that very briefly outlines the mathematical prerequisites for the book. A preview that includes the full contents and the first few pages are available on the amazon page for the book.

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