(1) a list of symbols
(= coordinates),
(2) a symmetric
matrix of functions of these coordinates (= metric tensor) and
(3) a list
of simplification rules (optional).
The main routine in the package -- RGtensors[metric_, coordinates_] -- then computes explicit expressions for all common Riemannian Geometry tensors (Riemann, Ricci, Einstein, Weyl) and tests if the space belongs to any of the following categories: Flat, Conformally Flat, Ricci Flat, Einstein Space or Space of Constant Curvature. Each tensor is stored as a nested list (of its components) under an appropriate global name.
The following functions for operating on these tensors are defined: Raise/Lower indices, Contract (multiple) indices, Covariant and Lie Differentiation and Covariant Divergence. These functions, together with the built-in functions Outer (giving tensor products) and Transpose (index rearrangement), provide the necessary tools for performing all common tensor operations on the computer. Several examples of the use of these functions on tensors computed using different metrics are given. A more detailed description of the capabilities of RGTC can be found in this article.
In addition to the code and the examples, the notebook RGTC.nb contains Instructions and Usage Tips. Some more complicated examples are given here.
Versions after (3.6.7) have a new routine for classifying the Weyl and Ricci tensors in any frame.
The package requires Mathematica 3.0 or later.
Beginning with version 2.5, tensor components can be calculated with respect to an arbitrary frame, and approximate calculations (series expansions) can be carried out. Version 2.7.8 significantly improves these capabilities.
Version 3.1.8 introduces
definitions for the NP quantities, three new auxiliary functions, and two
palettes for entering the operators/symbols used. It also has minor speed
improvements, corrects a rare bug that can affect calculations with series,
and gives more informative messages when something goes wrong.
Version 3.2.5 improves the handling of complicated expressions and corrects an error in HStar when acting on 0-forms.
Version 3.2.8 corrects another error in HStar when acting on certain Series expressions.
Important Note: A change in coding introduced in version 3.2.5 (and 3.2.8)
prevents testing for conformal flatness of 3-dimensional spaces.
Thus versions 3.2.5 and 3.2.8 of RGtensors will not print "Conformally Flat" when
appplied to a conformally flat 3-dim metric, possibly leading to erroneous conclusions.
This was corrected in version 3.3.0.
Version 3.5.1 is a Mathematica 6 - compatible update. In addition, it introduces three new functions for calculating Lie derivatives
and the Laplacian of tensors as well as the norm of the gradient of scalars. It also has several minor improvements.
Version 3.5.3 corrects an error affecting certain complete contractions of equal rank tensors using multiDot
(complete contractions of the Riemann tensor in the examples are unaffected -- that's why this error was not discovered earlier!).
Also, Contract and multiDot can now be used with arbitrarily defined nested lists,
different levels having different dimensions, provided, of course, that the indices to be summed have the same dimension.
Version 3.5.8 corrects an error that appears when covDiv acts on some series expansions.
Version 3.5.9 changes the definition of LieD so that it gives correct results
in any frame (original definition was valid in the coordinate frame only).
Version 3.6.0 corrects a typing error in 3.5.8 that caused HStar and eta in an NP frame to give results
with an extra overall factor of i .
Version 3.6.7 introduces a new routine (Classify) for classifying the Weyl and Ricci tensors in any frame.
It also introduces new functions (metric and Plebanski) for constructing the metric from the line-element
and the Plebanski tensor from any symmetric tensor.
The present version 3.7.0 extends the definition of Plebanski to include the case of the Weyl-type tensor
constructed from two symmetric tensors.
For comments, questions or suggestions please contact the author at
sbonano@inp.demokritos.gr
Last
modified 2009-07-21