Which math skills to learn for understanding relativity?

[Lowland]

Summarized: Which mathematical fields should you focus on if your goal is to gain a better understanding of relativity and related subjects?
(My apologies if there are already other threads asking this question, but 'math' and 'relativity' aren't really useful search terms on this forum.)

Hi everyone,

I've been interested in relativity and related fields like quantum physics for quite some time, but lately I've been running into a problem: I've reached the limit of how much you can understand with only a high school background in mathematics. There's only so much popscience can explain to you on these subjects.

Since I'm 31, I simply don't have time to focus on learning all-round math full time. So I was wondering which mathematical fields would be most useful to focus on if my goal is understanding physics like relativity. My math skills themselves are at high school level, but I do have a university level background that includes formal logic, so I'm confident I will be able to advance my understanding as long as I'm able to narrow my studies down to just a few fields. What approach would be best for someone with only a limited amount of time?

Thank you in advance.

 

[Dishsoap]

When you say relativity, do you mean general or special relativity?

Put simply, when you study special relativity, you'll learn all about length contraction and time dilation, which are some cool effects resulting from the speed of light being a constant. Some might disagree, but I think that some basic algebra and geometry skills (think high school level) are all that is needed to have a first-order understanding of how it works, and it's one of the most fun topics a physicist will ever encounter :)

On the other hand, general relativity describes how gravity plays into the "relativity" picture, and is based on the premise that space and time are one and the same (sort of), and describes space-time curvature. The math that goes into GR is usually at a much higher level.

For most physics curricula, students will usually be exposed to special relativity in their first or second year of undergrad (at a basic level), but general relativity is either an upperclassman course or one for graduate students. For an intro to the math of special relativity, any "intro to college physics"-type book will likely help you out, for instance those by Serway & Jewett or Halliday & Resnick.

 

[m4r35n357]

My math skills themselves are at high school level

That should be sufficient for special relativity (if you can do algebra and pythagoras you have all you need). The hard part is understanding the fundamental concepts.

BTW early on you will hear about a thing called $\gamma$. Do not be tempted to start plugging it into random non-relativistic formulas (this approach is the root of approximately 99.999999% of learning failures). Don't take anything for granted. Learn about the Lorentz Transform, Minkowski space and spacetime diagrams (if an article only talks about time dilation and length contraction, bin it!).

For general relativity it is very different, you will need to understand calculus, pseudo-riemannian geometry, tensors, differential forms and abstract geometry and topology amongst other things (search for these terms and you will see what I mean!).

 

[Lowland]

Thank you for these answers. It's very hard to guess what a good starting point is before you already have a decent foundation on a subject. And I did indeed google 'mathematical skills for relativity' without differentiating between general and special relativity. Seems like I can already jump into first-year university books on special relativity (I just learned from popscience thus far, thinking my math wasn't good enough) and hopefully by the end of that have a better understanding of what I need for general relativity. Thank you again for giving me a proper starting point!

 

[PeroK]

Thank you for these answers. It's very hard to guess what a good starting point is before you already have a decent foundation on a subject. And I did indeed google 'mathematical skills for relativity' without differentiating between general and special relativity. Seems like I can already jump into first-year university books on special relativity (I just learned from popscience thus far, thinking my math wasn't good enough) and hopefully by the end of that have a better understanding of what I need for general relativity. Thank you again for giving me a proper starting point!

My recommendation for an SR textbook that would ideal for your situation is:

https://www.goodreads.com/book/show/6453378-special-relativity

There is also a free text online at:

http://web.stanford.edu/~oas/SI/SRGR/notes/srHarris.pdf

I don't think it's as good as Helliwell, but it is free. It's definitely achievable to learn SR properly without advanced mathematics. That is your best bet.

For QM and GR, as others have said, you do need to learn a significant amount of undergraduate level mathematics.

PS I think the difference between learning SR from popular science and from a proper textbook is this: you can read a pop science book and think you understand SR, when in fact you may have misunderstood everything! Text books have exercises and problems that test your understanding of the material.

PPS another suggestion from me that you may not like. If you do start to learn SR from an undergraduate text, to be honest, I would forget everything you think you've learned from popular science. That knowledge can be as much a hindrance as a help.