Journal: March 31, 2009

Bryan Newbold, bnewbold@mit.edu
http://web.mit.edu/bnewbold/thesis/

Some more papers from trolling around the net, some are great!

Algorithms for computer algebra calculations in spacetime: I. The calculation of curvature by D. Pollney et al, 1996, Class. Quantum Grav.
Haven't read through this yet but looks nice, compares other packages and presents algorithms. I don't know what any subsequent articles were...
Computer Algebra Systems in General Relativity Salah Haggag, (Conference Paper) 2007
Just a review, some good history and excellent bibliography.
Numerical Experimentation within GRworkbench Andrew Moylan, Undergrad Thesis at ANU, 2003. Also GRworkbench: A Computational System Based on Differential Geometry and Functional Programming Framework for GRworkbench.
Wow! Great stuff, too bad I can't find the software actually hosted anywhere (links are all dead). Written in C++ with templates for common calculations, GUI interface, functional (combinator-based) scripting language hacked on, etc. The intro paper is really clearly written from a CS standpoint, explicitly lists all the geometric concepts and the software equivalents.

Last night I started working through the anomalous precession calculation (see anomalous_precession.scm). I generated a new coordinate system for the 2+1 polar spacetime with a Schwartzchild metric and was able to derive the Christofel symbols (though in a frustratingly "simplified" form):

(down
 (down (up 0 (+ (/ (* -2 (expt M 2)) (expt r0 3)) (/ M (expt r0 2))) 0)
       (up (/ (* -1/2 M) (+ (* M r0) (* -1/2 (expt r0 2)))) 0 0)
       (up 0 0 0))
 (down (up (/ (* -1/2 M) (+ (* M r0) (* -1/2 (expt r0 2)))) 0 0)
       (up 0 (/ (* 1/2 M) (+ (* M r0) (* -1/2 (expt r0 2)))) 0)
       (up 0 0 (/ 1 r0)))
 (down (up 0 0 0) (up 0 0 (/ 1 r0)) (up 0 (+ (* 2 M) (* -1 r0)) 0)))

I got stuck when I tried to take a covariant derivative of a dummy test path to derive the geodesic equations of motion; there are a lot of dt dummy variables floating around so I installed new coordinates for the-real-line to use 'k' instead of 't'; but I still get:

(((((covariant-derivative polar-Cartan-over-path) d/dk)
   ((differential test-path) d/dk))
  (orbital-plane-polar '->coords))
 ((the-real-line '->point) 'k))

;Bad selectors -- DERIV:EUCLIDEAN-STRUCTURE #[compiled-closure 13 (lambda "mathutil" #x1b) #xed0 #xbbbb4c #x1fdb064] (0) k

Fiddling about with this expression frequently results in an assertion error when the orbital-plane-polar coordinate system is passed a the-real-line point.

I should go through and derive some more pedestrian geodesics and sure i'm doing things right.

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