Using algebraic programming to teach general relativity

The authors' department considers general relativity (GR) teaching a necessity for midlevel graduate physics students, so we introduce a GR course at the third-year undergraduate level. During the last two years, we have used several packages in the course (Reduce plus Excalc for algebraic programming, Mathematica and Maple for graphic visualization). Even when the students were real beginners in computer manipulation, we obtained visibly good results. Algebraic programming systems (such as Reduce) that contain differential geometry packages can become an important tool for learning. Using a computer, students can quickly and comfortably learn the important notions of differential geometry, tensor calculus, and exterior calculus (for example, using Excalc). We also introduce some aspects of GR using computer algebra systems. For example, we can easily obtain the Schwarzschild solution and the Reissner-Nordstrom solution as exact solutions of the Einstein equations. We also use computer algebra to find and treat other exact homogeneous solutions of the Einstein equations containing the cosmological constant (de Sitter and anti-de Sitter metrics). Using some simple Excalc procedures, the article illustrates how to teach Riemannian geometry as well as how to use computer algebra to obtain some exact solutions of Einstein equations.