# Mathematics prerequisite to that of relativity

• fuzzy127

#### fuzzy127

Hi I am interested in studying special and general relativity at a mathematically rigorous level, and I need to know what it is necessary to understand as far as math goes prior to studying it. I think I need the following:

Single and multivariable calculus and analysis
Differential Equations
Partial Differential equations

but I would like it if someone could tell me if there is anything else. Thank you!

I would also like to know...

Advanced calculus and differential geometry for GR. Ofcourse, its not necessary to know these before you start. Many texts cover the necessary geometry. So I guess if you know advanced calculus, you'll be prepared to start with a standard introductory text like Schutz for GR.

Any standard texts for all of these subjects for a beginner? If you can suggest something easy for self study?

Thanks and regards,

Mitesh

You can study special relativity in a rigorous sense with little more than AP calculus under your belt. Unsurprisingly, however, you'll be able to study only a little bit of it with that level of mathematics.

On a more serious note, your question doesn't have a specific answer. Certainly, most of the essential structure of Minkowski space and the behaviour of systems within Minkowski space can be put on a rigorous footing with just an understanding of the existence and uniqueness theorems of ODEs. The more powerful PDE techniques, particularly those involving Sobolev spaces and the sub- and super-solution methods become crucially important when looking at something like existence of solutions to the constraint equations of general relativity, but aren't immediately applicable to special relativity. General relativity also would require, of course, a good knowledge of differential geometry (at least to graduate level) if you were interested in looking at non-trivial mathematical results; again, this isn't really an issue in special relativity.

That's not to say that special relativity is trivial: if you're interested in studying something along the lines of the semiclassical stability of Minkowski space, a PhD in mathematics and a tenured position would probably be sufficient. (cf. Christodoulou's book on semiclassical stability.) :-)

@shoehorn

Thanks for your reply, however, as of now, I am actively involved in chemistry, and wish to study SR (and GR too), by self learning. My uni doesn't provide any courses in any of the subjects sited above, and hence, I would be dependent largely on internet. As it may be, thanks again for the reply.

Regards,

Mitesh

Thanks!

One word - Overwhelming!

Regards,

Mitesh

@shoehorn

Thanks for your reply, however, as of now, I am actively involved in chemistry, and wish to study SR (and GR too), by self learning. My uni doesn't provide any courses in any of the subjects sited above, and hence, I would be dependent largely on internet. As it may be, thanks again for the reply.

Regards,

Mitesh

Well, I was responding to the OP, but okay.

As pointed out by others, the mathematics required really depends on how you approach and study Special Relativity. Needless to say, you have to study SR before GR to make any sense of GR. And its not clear from your post whether you have already been exposed to SR at the level of a freshman physics course. If so, and if you're fairly confident about the basics of SR (i.e. consequences like length contraction, time dilation and some idea of the connection between SR and electrodynamics) then you can simultaneously read things like tensors and differential equations and proceed to classical field theory (e.g. Landau's Classical Theory of Fields)

However, if you're starting out, you should definitely consider looking at Spacetime Physics (https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20) and 'Introduction to Special Relativity' by Robert Resnick (https://www.amazon.com/dp/0471717258/?tag=pfamazon01-20).

After reading special relativity at the level of these books, you can refer to books on general relativity (some of which will also introduce special relativity from a more mathematically rigorous standpoint) such as the books by Jim Hartle and Tai Pei Cheng. Books are more a matter of personal taste, but the reason I mentioned books in my response was to give you some idea of the content and presentation style. Books on GR for instance, will generally not dwell much on the basic consequences of SR but will assume that readers are already familiar with them.

PS--When you start out, the level of mathematics is really limited to a little bit of differential and integral calculus, trigonometry and geometry. After that, as OP have pointed out, you will need PDEs and differential geometry.

The most important mathematical tool needed for studying relativity is tensor calculus. I believe any mathematical book on tensor should do the job. But I still recommend a book, Schaum's outline series Theory and Problems of Tensor Calculus by David C. Kay. I studied this book and found that it is easy to follow, full of examples and I learned all the materials in it in just a few weeks. With a little bit knowledge of relativity and vector calculus will definitely help you learn tensor. I hope my suggestion can help you.

You should wait until you get through lower division math as you suspected. The familiarity with vectors and matrices will make tensors easier, the multivariable and vector calc is necessary because in someways you will be learning a logical extension of that.

I wouldn't worry about classes in DEs because Einstein's equations are nonlinear, the DEs classes usually show you how to solve linear equations, and when they're not they're still not as hard as what you see in GR. The only two solutions you'll see in a textbook on GR are the vacuum solution and the wave equation, where you don't really need to know a lot about solving DEs to get how they're solved.

One other thing on the physics side, you should study Newtonian gravity because you can get it back out in the appropriate limits, and if you know it well you might appreciate better when the theories act the same way and when they are different.