What are the mathematic prerequisites for learning relativity?

[lizzie96]

Hello,

I am interested in self-teaching special and general relativity at an undergraduate level, but was wondering what the main mathematical prerequisites are. I currently have almost no idea of GR and a very mathematically basic idea of SR, and have studied maths and further maths to A-level standard (UK). Could anyone recommend any good introductory textbooks/arxiv documents/in-depth online rescources?

Thank you very much!

 

[George Jones]

A-level maths includes single-variable calculus, right? If so, try Exploring Black Holes by Taylor and Wheeler.

 

[Meir Achuz]

SR is relatively simple mathematically. You just need algebra and calculus to do it (and just algebra is enough for much of SR.) What matrix and tensor math is required is more easily learned while learning SR than from a math book. GR requires much more sophisticated math. Even Einstein had to rely on mathematicians for GR.

 

[bcrowell]

Both SR and GR can be presented mathematically at a variety of levels. An example of a presentation of GR with almost no math is General Relativity from A to B by Geroch. Exploring Black Holes by Taylor and Wheeler is also a good book. Neither Hartle nor Geroch's GR from A to B presents the main current of GR, more like a specific little corner of it. There are GR textbooks designed for upper-division undergrads, such as Hartle, that present more of a survey of GR in general. My own free GR book, which you can find by googling, is also aimed at about your math level.

 

[WannabeNewton]

Could anyone recommend any good introductory textbooks/arxiv documents/in-depth online rescources?

Check out these notes bud:

http://www.physics.ox.ac.uk/users/iontrap/ams/teaching/rel_A.pdf
http://www.physics.ox.ac.uk/users/iontrap/ams/teaching/rel_B.pdf

And here's the full text: https://www.amazon.com/dp/019966286X/?tag=pfamazon01-20

The text presents the physics of SR in great detail.

 

[bcrowell]

WannabeNewton -- what do you like about the Steane book?

 

[WannabeNewton]

Hey Ben, while I can't say this is unique to the Steane book, I like the fact that it goes into mathematical and pictorial detail on things like the headlight effect, visual appearances and super-snapshots, radar coordinates, circular and anharmonic motion, motion under a central potential, Thomas precession (and a heuristic discussion of the Lorentz group), and most importantly rigidity and internal stresses all in the context of SR. It also follows the Purcell exposition of EM which is one of my most favorite things in physics. All the "advanced" material like precession of the spin of a charged particle in an electromagnetic field, relativistic fluids, and classical field theory are added bonuses.

So to sum up, I think it goes into detail on a lot of insanely cool topics that various other SR books don't spend much (if any) time on.

EDIT: In fact this book reminds me a lot of Rindler's book: https://www.amazon.com/dp/0198567324/?tag=pfamazon01-20, especially the material on visual appearances and super-snapshots.

 

[HallsofIvy]

It's old but "The Mathematical Theory of Relativity", by Sir Arthur Eddington, is still good- and available from Dover Books.

 

[bcrowell]

The Eddington book is available online here: https://archive.org/details/mathematicaltheo00eddiuoft

 

[robphy]

EDIT: In fact this book reminds me a lot of Rindler's book: https://www.amazon.com/dp/0198567324/?tag=pfamazon01-20, especially the material on visual appearances and super-snapshots.

Steane does cite Rindler as one of his influences
http://www.physics.ox.ac.uk/Users/smithb/website/coursenotes/SR_CH1-5.pdf (see pg. 14)

 

[Naty1]

Lizzie, for an introduction, you can skim through Einstein's own book,
RELATIVITY The Special and General Theory...you can find it online, free.

one source: http://www.gutenberg.org/files/30155/30155-pdf.pdf


Also, Some good online resources here:

http://www.einstein-online.info/

 

[Arsenic&Lace]

Most physics students enter into GR with no math prerequisites beyond typical mathematical physics courses and elementary calc. I just finished a graduate course in GR at the level of Caroll. I have a decent background in pure mathematics courses, as did a friend of mine who took the course.

The perspective on the mathematics is so different that neither of us felt that the actual knowledge gained from the math courses was terribly benificial. However, I can't really say for sure what effect the experience of taking hard math classes had on my performance.

I didn't personally find the mathematics of GR to be challenging to understand, I struggled with the quantity of material that was covered (especially cosmology, there were oodles of formulas and relations to remember, and that, I confess, is my weakness), which is an artefact of taking a course, which you are not doing. Tensors confused me for a bit, but that was mostly just notation (I found the index manipulations to be mysterious for the first few weeks of the course), but once I got past that hurdle, everything seemed quite straightforward. The idea behind tensors is simple but challenging to explain, and the mathematicians don't make things any easier with their high-powered, super precise definitions, so conversing with somebody in person about them or doing lots of simple exercises might be necessary.

If you are ambitious and have a background in elementary calculus (especially vector calculus, n-dimensional generalized Stokes theorem crops up everywhere for instance), linear algebra (GR is built on a abstraction of typical undergrad linear algebra), and differential equations, you can probably achieve a reasonable understanding of the subject.

 

[TomServo]

The book The Geometry of Special Relativity by Tevian Dray is good, but it does use some calculus.

The SR section of Halliday and Resnick might be what you're lookin fer.

 

[WannabeNewton]

Steane does cite Rindler as one of his influences
http://www.physics.ox.ac.uk/Users/smithb/website/coursenotes/SR_CH1-5.pdf (see pg. 14)

Ah nice, I hadn't noticed that acknowledgment of Rindler upon a first reading of Steane.

As a side note, my most favorite textbook on special relativity is the one by Woodhouse: https://www.amazon.com/dp/1852334266/?tag=pfamazon01-20

I didn't mention it before only because I couldn't tell if the mathematical level of Woodhouse posed a problem for the OP but come to think of it, SR is probably the best place to learn tensor algebra, tensor calculus, and index manipulations because the geometric structure of Minkowski space-time is trivial; this way once you get to GR, the computational machinery will be entirely familiar to you and you will just breeze through the formalities. The mathematics of GR, as far as local analysis goes, is thankfully extremely simple; global analysis of space-times does require more sophisticated machinery but you won't have to worry about that for a while so it's no biggie

 

[George Jones]

I couldn't tell if the mathematical level of Woodhouse posed a problem for the OP

In the OP, lizzie96 wrote

have studied maths and further maths to A-level standard (UK).

which is the last year (advanced) before university in the UK. I would rate it as somewhere between grade 12 at a standard US high school and first-year university, probably closer to first-year university.

Posters should keep this in mind when replying and thus should suggest books that: present relativity based on first-year calculus; or present relativity at a slightly higher level of maths and that also present the the extra math in a readable way suitable for self-study; or suggest math books that are readable and suitable for self-study, and that can be used as input for relativity.

 

[Seydlitz]

For the mentors who would like to view what A Level Math looks like, and also Further Math. These are full past papers.

http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20%289709%29/9709_s12_qp_31.pdf

http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20-%20Further%20%289231%29/9231_s12_qp_13.pdf (This one is more advanced)

Having done the examination myself this year together with several years of experience learning with A-Level Math, I should also add that the syllabus covers nothing whatsoever in terms of proof and rigorousness. (There's only proof by induction in A Level Further Math)

 

[bcrowell]

Hey Ben, while I can't say this is unique to the Steane book, I like the fact that it goes into mathematical and pictorial detail on things like the headlight effect, visual appearances and super-snapshots, radar coordinates, circular and anharmonic motion, motion under a central potential, Thomas precession (and a heuristic discussion of the Lorentz group), and most importantly rigidity and internal stresses all in the context of SR. It also follows the Purcell exposition of EM which is one of my most favorite things in physics. All the "advanced" material like precession of the spin of a charged particle in an electromagnetic field, relativistic fluids, and classical field theory are added bonuses.

So to sum up, I think it goes into detail on a lot of insanely cool topics that various other SR books don't spend much (if any) time on.

I spent some time looking at the abridged online version of Steane you linked to as well as the Amazon peep-show view (which actually has nearly everything viewable). My impression is that the coverage is much too broad for a one-semester upper-division undergrad course. Also, anyone who hasn't already completed a solid upper-division E&M course isn't going to be able to understand the treatment of E&M. The breadth of coverage looks especially extreme when you contrast it with Taylor and Wheeler's Spacetime Physics. I would not recommend Steane as a first introduction to SR. It might make a good second or third book on the subject, or a good reference book. I would still recommend Taylor and Wheeler to someone who's had a freshman physics survey and wants to learn some SR.