Tensor calculus for general relativity

[Wenderness]

I'm taking a course on relativity, both special and general. According to my college, I have the required mathematical background (vector analysis, electromagnetics (though I can't recall more than a cursory glance at tensors) etc) to make sense of it. Special relativity I can handle, and I think I understand the general concepts of GR fairly well, but how to actually do the math eludes me.

The book I have is Ta-Pei Cheng's Relativity, Gravitation and Cosmology. It does an OK job of explaining the theory, but it tends to not do the calculations, instead calling them 'straight forward'. Math has, sadly, never came that naturally to me, and I don't follow. So what I need would be a guide to the mathematical framework, one that spells it all out explicitly. Does anyone have any recommendations?

 

[phyzguy]

Misner, Thorne, and Wheeler is the classic reference on GR, and it explains tensor calculus from several different viewpoints, so it might be a good reference.

 

[Naty1]

Some online sopurces...

Some Caltech notes:
http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html

And from Benjamin Crowell of this forum:
http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html

And from John Baez,
http://math.ucr.edu/home/baez/gr/gr.html

And from Hofstra,
http://people.hofstra.edu/Stefan_Waner/diff_geom/tc.html

And from mathpages, around 5.2:
http://www.mathpages.com/rr/s5-02/5-02.htm

Good luck..I collected some references but have not studied them due to time constraints so I can't recommend one over another.

 

[Rasalhague]

Sean Carroll's Lecture Notes on General Relativity can also be found here, along with a condensed version and some further resources.
http://www.pma.caltech.edu/Courses/ph136/yr2008/

Kip Thorne & Roger Blandford: Applications of Classical Phyisics
http://www.pma.caltech.edu/Courses/ph136/yr2008/

Kip Thorne also has a series of video lectures online about gravitational waves, which include an introduction to tensor analysis.
http://elmer.tapir.caltech.edu/ph237/CourseOutlineA.html

I found the following book, online in PDF format, helpful in getting a handle on some of the basic mathematical concepts relating to tensors: vector spaces, dual spaces, etc.

Ray M. Bowen and C. C. Wang:
Introduction to Vectors and Tensors, Vol 1: Linear and Multilinear Algebra
http://repository.tamu.edu/handle/1969.1/2502

Ray M. Bowen and C. C. Wang:
Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis
http://repository.tamu.edu/handle/1969.1/3609

Part two of this series of video lectures from MIT has an introduction to tensors, from the second half of lecture 15 onwards, although it only deals with orthonormal coordinate systems.
http://ocw.mit.edu/OcwWeb/Materials-Science-and-Engineering/3-60Fall-2005/CourseHome/index.htm

 

[kakarotyjn]

Are you a Chinese?There is a series books on Differential and General Relativity written by 梁灿彬,it's nice!

 

[dulrich]

but how to actually do the math eludes me.

If you are completely lost on the math, one of my favorite intro to tensor calculus books is "A Brief on Tensor Analysis, 2nd ed." by James G. Simmonds (ISBN 0-387-94088-X). His approach is very physical, so you can let your intuition guide you until the math starts to sink in. This won't take you very far, but this is a step down to get your feet wet. It's about 100 pages long.

 

[im_hammer]

There is a book which talks about the tools for practical computation,
"A relativist's toolkit:
the mathematics of black-hole mechanics "

I am now reading it, and I think it is very helpful~