Slashdot Mirror
Ask Slashdot: Math Curriculum To Understand General Relativity?
First time accepted submitter sjwaste writes "Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade."
I think Geometric Algebra (GA) has a better formulation than the traditional tensor way of doing General Relativity. It's not only easier to understand, but it's easier to use and the same math can also be far more easily applied in other areas of physics.
A capsule: There are 4 basic dimensions, (usually denoted "e_n" with n from 0 to 3) but let's call them: x,y,z and t. The squares of the first 3 are negative, but the square of t is positive. These basis vectors can be combined to create bivectors: the regular planes of rotation xy, xz, yz, as well as xt, yt, zt. The latter three are still planes of rotation, but due to the mixed sign of the squares, the rotation is hyperbolic rather than circular - calculations use sinh and cosh instead of sin and cos. The interesting thing is that these planes of rotation involving t are velocities (Lorentz boosts). Velocities are hyperbolic rotations, and the speed of light is a 90 degree rotation. GA has a simple way of handing multiple rotations which allows easy solution of problems that are seldom even attempted using the conventional approach.
"A Survey of Geometric Algebra and Geometric Calculus" by Alan Macdonald
Gives a good introduction to the basics and applications of GA, including relativity. You would need to at least get through the section on rotations before skipping down to the section on Spacetime Algebra. Also see "General Relativity in a Nutshell"from the same author, which gives a mathematical but not dense introduction to General Relativity in 100 pages, not using GA.
"Gravity, Gauge Theories and Geometric Algebra" by Anthony Lasenby, Chris Doran, Stephen Gull
General Relativity using GA - interestingly, curved space-time is not required using GA.
"Primer on Geometric Algebra for introductory mathematics and physics" by David Hestenes
Another good intro, much less dense than Macdonald's, with more diagrams and basic applications.
"Geometric Algebra Primer" by Jaap Suter
Gives a more gentle introduction and reference for the basic GA operations.
"3D Euclidean Geometry through Conformal Geometric Algebra (a GAViewer tutorial)" by Leo Dorst & Daniel Fontijne
Gives a hands-on, step-by-step tutorial using their free open-source GA visualization software, "GAViewer". This tutorial uses the conformal model which is more advanced than the regular 3-D model. (2 extra dimensions, of a very odd but useful type) Other tutorials are available at the same site. Their book Geometric Algebra for Computer Science, an Object Oriented Approach to Geometry" is also highly recommended, and can be previewed at Scribd. (The 2nd edition is worth getting on paper. It has some very useful reference pages not available online, and many corrected errata.)
"Is life so dear, or peace so sweet, as to be purchased at the price of chains and slavery?" - Patrick Henry
I think Geometric Algebra (GA) has a better formulation than the traditional tensor way of doing relativity. It's not only easier to understand, but it's easier to use and the same math can also be far more easily applied in other areas of physics.
A capsule: There are 4 basic dimensions, (usually denoted "e_n" with n from 0 to 3) but let's call them: x,y,z and t. The squares of the first 3 are negative, but the square of t is positive. These basis vectors can be combined to create bivectors: the regular planes of rotation xy, xz, yz, as well as xt, yt, zt. The latter three are still planes of rotation, but due to the mixed sign of the squares, the rotation is hyperbolic rather than circular - calculations use sinh and cosh instead of sin and cos. The interesting thing is that these planes of rotation involving t are velocities (Lorentz boosts). Velocities are hyperbolic rotations, and the speed of light is a 90 degree rotation. GA has a simple way of handing multiple rotations which allows easy solution of problems that are seldom even attempted using the conventional approach.
"A Survey of Geometric Algebra and Geometric Calculus" by Alan Macdonald
Gives a good introduction to the basics and applications of GA, including relativity. You would need to at least get through the section on rotations before skipping down to the section on Spacetime Algebra. Also see "General Relativity in a Nutshell"from the same author, which gives a mathematical but not dense introduction to General Relativity in 100 pages, not using GA.
"Gravity, Gauge Theories and Geometric Algebra" by Anthony Lasenby, Chris Doran, Stephen Gull
General Relativity using GA - interestingly, curved space-time is not required using GA.
"Primer on Geometric Algebra for introductory mathematics and physics" by David Hestenes
Another good intro, much less dense than Macdonald's, with more diagrams and basic applications.
"Geometric Algebra Primer" by Jaap Suter
Gives a gentle introduction and reference for the basic GA operations.
"3D Euclidean Geometry through Conformal Geometric Algebra (a GAViewer tutorial)" by Leo Dorst & Daniel Fontijne
Gives a hands-on, step-by-step tutorial using the free open-source GA visualization software GA Viewer. This tutorial uses the conformal model which is more advanced than the regular 3-D model. Other tutorials are available at the same site. Their book Geometric Algebra for Computer Science, an Object Oriented Approach to Geometry" is also highly recommended, and can be previewed at Scribd.
"Is life so dear, or peace so sweet, as to be purchased at the price of chains and slavery?" - Patrick Henry