# Mathematically rigorous introduction to special relativity

Surprisingly this precise question doesn't seem to have been asked somehow but please correct me if I'm wrong.

I'm looking for a rigorous introduction to special relativity. I have fairly limited physics background, just the two introductory physics courses I took for as part of my degree, but I have a lot of math background and practice working with rigorous mathematics. I've only learned a bit of differential geometry though self study so I'm not really looking to jump straight into general relativity (for which great answers to this question already exist). My understanding of the math involved in special relativity is that its exceedingly basic in comparison, more on the level of Newtonian mechanics with some twists than anything. Still I've found most introductions to Newtonian mechanics incredibly lacking in the rigour department and rather hard for me to understand (not analytical mechanics, I find that more rigorous typically). I'd like to avoid such references for studying special relativity.

Is there a treatment of special relativity that rigorously defines the systems it considers and doesn't throw punches on the math or hide mathematical details for special relativity?

The problems are not mathematical. SR is just affine $${\bf R}^4$$ equiped with a non-positve definite quadratic form of the sort covered in any undergraduate linear algebra class. The hard bit is understanding how the quite simple algebra relates to the world around us. This requires realizing how much our day-to-day experience prejudices our thought processes.

It seems "obvious" that if you and your friend synchronise identical watches, and he goes to town for a day and you stay home, then when you compare watches on his return they should read the same. There is no logical reason why this should be so, and indeed it is not so, but people find this hard to accept even in the face of repeated verifications. SR is psychologically challenging not mathematically challenging. Excess mathematical language will confuse more than it illuminates,

• I can't help think that what the OP is suggesting is like learning carpentry without ever working on wood. Not completely impossible, just missing the point - the point in this case being, as Mike Stone indicates, the physics. May 19, 2021 at 1:44

I would read a book on general relativity. The introductory chapter(s) of a typical GR book will formulate special relativity in the language of differential geometry to prepare for the leap to curved space-times. Wald is at a fairly high level of mathematical rigor. Carroll is a more gentle introduction but still covers the most important points.

Having said that, I 100% agree with the following points in the answer by mike stone:

(a) this approach will obscure a lot of the physics, since a GR book will assume you understand the physics of special relativity,

(b) special relativity is somewhat trivial from a mathematical point of view, the interesting part of it is the physics.

• I second this suggestion. I found the first few chapters of Bernard Schutz's "A first course in general relativity" to be an excellent introduction to 4 vectors and the mathematics of special relativity. He also then goes on to do a fairly gentle introduction to differential geometry and GR. May 19, 2021 at 1:51
• @EricSmith I 100% agree, Schutz is a great book. May 19, 2021 at 2:26

The stage on which special relativity is set is Lorentzian space (an Euclidean space of indefinite signature); that is, $$\mathbb{R}^d$$ with a metric that is semi-definite. Most of special relativity is devoted to exploring the physical consequences of this stage, whereby physical concepts like causality emerge naturally. For this, I would recommend looking into classical electrodynamics (the books by Griffiths, Zangwill, or Jackson are classics).

You mention having familiarity with differential geometry. The modern way to frame electrodynamics is through the language of differential forms, or sections of a tensor bundle. The effect of boosts and rotations on matter is then captured through studying the associated $$\text{SO}(1,d-1)$$ principal bundle over $$\mathbb{R}^d$$. You will see through this construction that again the theory of electrodynamics emerges naturally.

In addition to the standard text books, the following two articles may be of some interest to you:

(A) There is a close relationship between the theory of electrodynamics and special relativity. The principles of special relativity are supported by Maxwell equations as follows:

Principle 1. The speed of light in vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source:

The speed of light also appears in the Maxwell equations and is inversely related to the square root of the two constants. The two constants, namely dielectric constant and magnetic permeability are treated as properties of free space (vacuum) and therefore should be invariant under transformations between two frames. Therefore, the speed of light term in Maxwell equations remain same in all the inertial frames, moving at various velocities with each other.

Principle 2. The laws of physics are the same for all observers in any inertial frame of reference relative to one another:

The Maxwell equations, representing fields created by a charge in space, written by observers in any two inertial frames are covariant (they have same form in all the frames). Thus, the laws of electrodynamics are same in all the inertial frames. These two properties of the Maxwell equations require that the theory of electrodynamics is consistent with the special relativity. A charge can be seen by observers from various frames moving at different velocities with each other. For the results to be consistent, when we compare observations made by observers in these different frames, the three spatial coordinates and time needs to follow the Lorentz transformation rules. Even for accelerating charges, it is necessary to follow the relativistic transformation rules.

This is discussed in detail in the article published in the Canadian Journal of Physics, May 2017: https://tspace.library.utoronto.ca/handle/1807/78885

(B) Ultimately the basic particle has to follow the rules of Special Relativity. There should be some characteristics in the basic particle which makes it obey the rules of various theories. We may suggest a mathematical model for the basic particle which can give the desired results as suggested by these theories in a mathematical analysis. We can call this model as the logically and mathematically probable picture of the basic particle.

The AIP Advances, March 2011 article discusses such an attempt. https://aip.scitation.org/doi/10.1063/1.3559461