Introduction to General Relativity: A Math and Physics Perspective

[andytoh]

I just finished creating 33 exercise questions (with my full solutions, taking up 22 pages of typing, some easy, some hard) for the first section of the first chapter of Hawking's "Large Scale Structure of Spacetime." The section is Differential Manifolds, from his Differential Geometry Chapter (which itself has nine sections, preceding his next chapter that finally begins general relativity). Let me polish up my work (I think I might need some more submanifold questions) before I give out the link for those who want to do some differential manifolds exercises for general relativity studies.

These exercises only cover 4 and a half pages in the book, so at this rate of thoroughness, I could come up with about 3300 exercises (and about 2200 pages of work) for the book!

 

[robphy]

Suggestion: It might be more digitally-accessible [now and in the future] if you use \LaTeX rather than Microsoft Word. There are editors available if you need to use that pictorial menu system. Here is a supplementary program made by the makers of the Equation Editor: http://www.dessci.com/en/products/texaide/

 

[andytoh]

I've been using MathType 5, but I have problems drawing shapes for my differential manifold questions. For drawing shapes and mappings, I've been using only MS Paint. Here is an example from my work:

http://www.sendspace.com/file/eav3p2

What program should I use to make the drawings I want, including for later on when I draw Minkowski timegraphs and Penrose diagrams?

 

[robphy]

Whatever you choose, you should pick a format that you can easily edit later. I'd suggest something http://en.wikipedia.org/wiki/Vector_graphics" )

For something that does \LaTeX, which can be adapted to be used in this forum, is http://jpicedt.sourceforge.net/ . Another option is xfig http://www.xfig.org/ (for windows, there's winfig [but the site seems down now] ).

Unless you're ready to publish, I'd suggest using pencil-and-paper-and-scanner (or a https://www.physicsforums.com/blog/2006/05/20/tabletpcs-for-science-and-science-teaching/" ) and drawing something "good-enough" for now... and focus on the subject content rather than the visual appearance. [I'd make a similar suggestion regarding the "typesetting"... focus on the content first.]

 

[Chris Hillman]

There's a great big wide world out there, operatingsystemwise

And you should avoid assuming that everyone is as Microsoft-centric as you are!

LaTeX is the universal standard for papers in physics and math, as well as many other fields; LaTex (and other Tex-based document formating systems) is so superior to any other document formating system that it makes little sense to use anything else, in fact using anything else tends to brand you as a bit of a bumpkin.

(Don't worry, I'm not entirely serious. But I'm not entirely joking either.)

Chris Hillman

 

[Daverz]

Judging from the table of contents only, Relativity on Curved Manifolds might be another choice. The chapter on physical measurement looks particular interesting (well, I just ordered a copy, because that subject interests me).

I just got this book and after spending a few hours with it am very impressed. I'd recommend it for anyone who is looking for a mathematically sophisticated supplement to more introductory books on GR. No exercises, though.

So far I'm particularly impressed with the way they explicitly work out the math of parallel propagation using a connector, which is a linear transformation from one tangent space to another on a curve.

 

[andytoh]

The MathType 5 I use is convertable to LaTeX, if I ever decide to make the switch.

I just ordered perhaps one of the most thorough books on the mathematics of general relativity: "Semi-Riemannian Geometry With Applications to Relativity" by O'Neil. Here is the review from amazon that convinced me to order the $130 book:

"If you want to engage in a serious study of general relativity, then you must master the mathematical language of semi-Riemannian manifolds in which it is cast. Sadly, the development of classical Riemannian geometry as studied by pure mathematicians only parallels the development of semi-Riemannian geometry in the early stages; eventually, the two subjects diverge rather drastically. For example, the famous Hopf-Rinow Theorem, one of the cornerstones of modern Riemannian geometry, simply has no Lorentzian analogue at all; every single equivalence in the theorem fails in Lorentzian geometry. Thus, one could master all five volumes of Spivak's definitive treatment of Riemannian geometry and still be unprepared to deal with light cones, timelike, null and spacelike geodesics, and the multitude of other uniquely semi-Riemannian constructs that appear in general relativity. O'Neill's wonderful book, which first appeared in 1983, provides the well-prepared reader with a mathematically rigorous, thorough introduction to both Riemannian and semi-Riemannian geometry, showing how they are similar and pointing out clearly where they differ. After developing the mathematical machinery in the early chapters, the last part of the book turns to general relativity by offering lucid introductions to the Robertson-Walker cosmological models (Big Bang singularities), the Schwarzschild model for a single non-rotating star (including black holes), and a brief introduction to Penrose-Hawking causality theory. ...It is not an "easy" text to read, but then, I have never found the "easy" introduction to differential geometry and general relativity. The reviewer who says this is not a suitable first text is simply in error; there is no better first text on the subject. If you have studied linear algebra, advanced calculus, and a little topology, then with dedication and hard work, you can learn more from O'Neill's text than from many of the far more popular recent texts, written by physicists, which attempt to circumvent the mathematics insofar as is possible while introducing general relativity. This is a perilous course for which the serious student will pay dearly later on, when she/he wants to study any of the many areas of modern physics in which differential geometry (differential forms, bundle theory, connections on a principle fiber bundle, gauge theory, etc.) plays an essential role."

This book might beat my current favourite book that focuses solely on the mathematics of relativity, which is "Tensors and Manifolds, with Applications to Relativity" by Wasserman. I also ordered "The Topology of Fibre Bundles" by Steenrod. There should be little mathematical obstacle in my studies of general relativity now.

 

[andytoh]

And you should avoid assuming that everyone is as Microsoft-centric as you are!

LaTeX is the universal standard for papers in physics and math, as well as many other fields; LaTex (and other Tex-based document formating systems) is so superior to any other document formating system that it makes little sense to use anything else, in fact using anything else tends to brand you as a bit of a bumpkin.

(Don't worry, I'm not entirely serious. But I'm not entirely joking either.)

Chris Hillman

I've mastered using keyboard shortcuts with MathType 5.2 (copying and pasting onto with MS Word) and with LaTeX. I don't need to use a single mouse click with either method. I am definitely a faster typer with MathType 5.2 and am also much more confident with what I am actually typing with the WYSIWYG interface of MathType. In fact, I type faster with MathType then I write out a solution by hand, simply because I can copy and paste long expressions and often repeat themselves in a solution. However, if I were to publish a work, I would have to use LaTeX. On the other hand, I could always convert MathType to LaTeX with WordtoLaTeX.