Tensors in Physics. User's Guide for the Mathematica Package CARTAN Version 1.2
Marc Mars · Classical and Quantum Gravity · 1998
Despite its general name ` Tensors in Physics ' is a package containing the Mathematica-based program CARTAN and the corresponding User's Guide and Reference Manual. CARTAN is a computer algebra package running on Mathematica 2.0 or later versions which allows for explicit calculations of geometrical objects in metric compatible manifolds (with or without torsion). The input (metric, torsion, etc) must be given using rigid frame bases (i.e. with constant inner scalar products). The output tensors can, however, be written either in the original rigid frame or in the coordinate basis. The program has built-in definitions of most of the tensor and connection objects in a metric manifold, including the Newman - Penrose formalism and spinor calculations. The program is user-friendly and easy to learn, although some knowledge of Mathematica is obviously required. A comparison of the CPU time consumed by CARTAN in a typical calculation shows that it is substantially slower than special purpose programs for tensor calculus such as SHEEP. Regarding the User's Guide and Reference Manual, it is composed of two different parts. The first part is a review of tensor calculus in metric-compatible manifolds. According to the author, its main intention is to fix the notation and conventions used by the program, but it is far too long to just accomplish that and too short and concise to provide a self-contained introduction to differential geometry. The second part concerns the program itself, describing how to use it, which functions and variables it contains and how new functions can be defined by the user. It also includes a few examples of interactive sessions with CARTAN to show how problems can be solved using this program. The manual is well organized and clearly written. The widespread use of Mathematica for teaching and research purposes has triggered the development of several Mathematica-based packages for tensor manipulation. Some of them are free of charge, such as TTC, GRTensorM, EinS or Ricci (which is for symbolic tensor calculus rather than explicit tensor calculations) and some are commercial, such as CARTAN and MathTensor. CARTAN is substantially cheaper than MathTensor which, on the other hand, has a much larger range of capabilities. In my opinion, ` Tensors in Physics ' is a valuable tool for research and teaching institutions which already use Mathematica and need a professional, user-friendly tensor package specifically designed to calculate connections and curvature tensors in metric-compatible manifolds.