In my experience, one of the biggest problems in learning relativity (whether special or general) is becoming comfortable with tensors. Unfortunately, most of the textbooks I've come across do an inadequate job of teaching the fundamentals, including Hartle's Gravity, An Introduction Einstein's General Relativity, Schutz's A First Course in General Relativity and D'Inverno's Introducing Einstein's Relativity, three of the most popular and highly recommended introductory to intermediate level textbooks on general relativity.
Having investigated more than a dozen textbooks in the course of my own struggle with this problem, I've concluded that it's wise to study tensors independently either before or in conjunction with reading your favorite relativity book.
The good news is, a number of exceptionally well-written materials are available for free on the internet. Although I've found a number of online sources not particularly helpful because they're geared toward classical mechanics, assume only Euclidean spaces or focus exclusively on the older coordinate approach, I have found the following excellent:
1. For the absolute beginner, the chapter on tensors by Prof. James Nearing (University of Miami), available on the internet for free at
James Nearing's Mathematical Tools for Physics
and for purchase at a truly great price at, e.g.,
Amazon book link for Nearing's book
Nearing clearly explains in a very concise but rigorous fashion both tensor algebra (tensors as multi-linear functions forming a particular kind of algebra) and tensor calculus (components approach: tensors are objects that transform in a certain manner; calculating with components). He also provides a lot of physics motivation. And, his book also provides chapters on matrix algebra, vector calculus, the calculus of variations and other basic mathematics required for physics.
2. For the more advanced student, the lecture notes by Prof. Edmund Bertschinger (MIT) available on the internet: here's his introduction:
Bertschinger's Introduction to Tensor Calculus for General Relativity
but there's more advanced material available at: Bertchinger's General Relativity Notes
(And while you're at it, don't forget that MIT makes available as free downloads, entire courses on many subjects, including special and general relativity!)
There are also a number of excellent books (besides Nearing's) for sale, including my favorites:
- Zafar Ahsan, Tensor Analysis with Applications
- David Kay, Outlilne of Tensor Calculus
Cf. my Amazon reviews of Kay's book and James G. Simmonds' book Simmonds, A Brief on Tensor Analysis or even check out all my reviews on Amazon: my book reviews on Amazon
Having investigated more than a dozen textbooks in the course of my own struggle with this problem, I've concluded that it's wise to study tensors independently either before or in conjunction with reading your favorite relativity book.
The good news is, a number of exceptionally well-written materials are available for free on the internet. Although I've found a number of online sources not particularly helpful because they're geared toward classical mechanics, assume only Euclidean spaces or focus exclusively on the older coordinate approach, I have found the following excellent:
1. For the absolute beginner, the chapter on tensors by Prof. James Nearing (University of Miami), available on the internet for free at
James Nearing's Mathematical Tools for Physics
and for purchase at a truly great price at, e.g.,
Amazon book link for Nearing's book
Nearing clearly explains in a very concise but rigorous fashion both tensor algebra (tensors as multi-linear functions forming a particular kind of algebra) and tensor calculus (components approach: tensors are objects that transform in a certain manner; calculating with components). He also provides a lot of physics motivation. And, his book also provides chapters on matrix algebra, vector calculus, the calculus of variations and other basic mathematics required for physics.
2. For the more advanced student, the lecture notes by Prof. Edmund Bertschinger (MIT) available on the internet: here's his introduction:
Bertschinger's Introduction to Tensor Calculus for General Relativity
but there's more advanced material available at: Bertchinger's General Relativity Notes
(And while you're at it, don't forget that MIT makes available as free downloads, entire courses on many subjects, including special and general relativity!)
There are also a number of excellent books (besides Nearing's) for sale, including my favorites:
- Zafar Ahsan, Tensor Analysis with Applications
- David Kay, Outlilne of Tensor Calculus
Cf. my Amazon reviews of Kay's book and James G. Simmonds' book Simmonds, A Brief on Tensor Analysis or even check out all my reviews on Amazon: my book reviews on Amazon
