Learning differential geometry related to special relativity

Question

I apologize if math.stackexchange is the wrong place for this question -- I cannot currently create an account on physics.stackexchange.

I am currently taking a graduate course on Differential Geometry. Over the weeks, I have discovered that the later parts of the course focuses more on physics, especially content concerning spacetime/relativity. I have no background in physics, and am struggling to motivate the examples given as I have no experience with the physics theory to relate to.

So, my question is, are there any recommended resources for a person wishing to learn (i) about (semi-)Riemannian manifolds (ii) in a way that also talks about the physics-related motivations of the subject, along with some exposition of said physics theory?

I have found Besse's book on Einstein manifolds, the basic material chapter of which seems to coincide pretty well with the course material.

Answer 1

• Barrett O’Neill: Semi-Riemannian Geometry With Applications to Relativity

This is a wonderful book on (as the same suggests) semi-Riemannian manifolds with special and general relativity as applications. Chapter 6 is on special relativity (after introducing the basics: manifolds, tensor fields, covariant derivatives, submanifolds, exponential map etc). Much later on, you have GR in the form of the FRW and Schwarzschild spacetimes, and finally causality theory in Lorentzian manifolds and the Hawking and Penrose incompleteness theorems.

But be warned, these chapters only briefly introduce SR/GR, covering the basic concepts only. However, since you say you’re taking a course in differential geometry, I think this is a wonderful text (everything is clearly defined, explained, and I find the motivation to be sufficient).

• W.D Curtis, F.R Miller: Differential Manifolds and Theoretical Physics

This is also a very nice book about manifolds and their use in physics, particularly classical mechanics (Lagrangian and Hamiltonian), SR (a nice treatment), electromagnetism (a nice enough overview), Lie groups and rigid bodies. Towards the end, there are some more advanced topics as well: Principal bundles, a brief survey of some quantum effects, and electromagnetism as a gauge theory. The only thing is they’re not aiming at SR/GR specifically, but still it’s a very nice book (and since you say you don’t have much physics background, you might find that reading the introductory sections of the physics chapters to be helpful).

• Bamberg and Sternberg: A Course in Mathematics for Students of Physics

This book discusses special relativity very briefly in sections 4.6-4.8, but it’s a nice discussion. Even if they don’t go in depth into SR, they do have lots of other nice topics in physics (optics, circuits and how they motivate algebraic topology, thermodynamics). So as far as motivation goes, this book is great.

• Gregory L Naber: The Geometry of Minkowski Spacetime

This book is all about the geometry for special relativity, and is explained in a nice mathematical way (I’ve only read ~80% of chapter 1)

• Anadijiban Das: The Special Theory of Relativity, A Mathematical Exposition

I’ve only read parts of this book, and it’s fine I think for the most part, but sometimes the font and notation makes me a little dizzy… interpret that however you will.

• Sachs and Wu: General Relativity for Mathematicians

This is a very ‘definition-theorem-proof’ based book with clearly defined concepts.

• Misner, Thorne, Wheeler: Gravitation

This is a physics book on GR, but I think it’s the only book which tries extremely hard to give geometric motivation for everything (even if once in a while the pictures are easy to misinterpret)

• Robert Geroch: Mathematical Physics

Chapter 15-16 are about Minkowski space and the Lorentz group, and this is a nice connection between math and physics (but I already knew some of the physics, so…). Besides, he covers lots of topics so it’s a fun read.

• Roger Penrose: The Road to Reality

The whole book has a ‘conversational’ feel, and Chapters 17-19 are about spacetimes as a whole (our various developments of this concept from history), and introducing special and general relativity. This is like a ‘bedtime story’ book, and you can gain lots of context from here (I really enjoyed it), but you won’t get too deep into the theory here.

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My favourite book out of all the ones above is by O’Neill because it’s very detailed in the semi-Riemmanian geometry aspect, and he shows how that applies to SR and GR. My next favorite is the one by Curtis-Miller. All the books have their pros and cons. Also, for the most part, they’re aimed (i) of your request, i.e semi-Riemannian geometry, or at the very least, at the linear level, describing the geometry of Minkowski space.

Right now I can’t think of a book which does both (i) and (ii) in detail, because math books won’t go deep into the physics and won’t really motivate the physics, while physics books won’t go deep into the math. So, to get a balance of things, I would just read the introductory remarks of each chapter of Physics books on SR/GR. If you want to learn physics (or atleast understand what questions they’re interested in/how they think about things) it’s best to learn straight from the masters. A very demanding but rewarding book like this is Course of Theoretical Physics, Vol II- The Classical Theory of Fields, by Landau and Lifshitz (here the first chapter is on SR, then they cover electromagnetism, and finally end with GR). Warning: a modern math student will struggle with their style of writing, so if you’re pressed for time, just read things and move on rather than going through things line-by-line.

Answer 2

There are a huge number of books introducing special relativity, but I thought I might be able to help you with some of the physical meaning in an answer.

The fundamental example of special relativity is Minkowski spacetime, $M$ which is $\mathbb{R}^4$ with coordinates $x^0,x^1,x^2,x^3$ and the metric $$g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right)=\begin{cases} 1, & i=j>0\\ -c^2, & i=j=0\\ 0, & i\ne j \end{cases}$$

where $c$ is the speed of light, $x^0$ is a time coordinate, and $x^i$ for $i>0$ are spatial coordinates. The metric encodes the idea of a speed of light, namely a tangent vector $v\in T_pM$ is at the velocity of light if $g(v,v)=0$. Otherwise, it is slower than light-speed if $g(v,v)<0$ and faster than lightspeed if $g(v,v)>0$. If you wish to measure the time experienced by a clock traveling along a curve $\gamma$ then it is given by

$$t=\frac{1}{c}\int_\gamma \sqrt{-g(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))}\, d\lambda$$

while if the curve lies at a constant time, then the length of the curve is

$$l=\int_\gamma \sqrt{g(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))}\, d\lambda$$

In this notion of spacetime, one has many surprising phenomena such as relativity of simultaneity, length contraction, and time-dilation. In order to get a good understanding of the physical meaning of all this, make sure these concepts are clear. Doing so can take some time though!

Most of what I've said above is still the correct interpretation for other manifolds with a pseudo-metric, but those are usually the subject of General Relativity.