∞ Ad Infinitum AS

Tensors in Physics 1.8

Harald H. Soleng

This is a combined ebook and software package. The software is a Mathematica package for tensor computations. The ebook is a guide to differential geometry in general and to the rigid frame tensor analysis technique in particular and a user manual for the software package.

Cartan was developed in part at the European Centre for Nuclear Research (CERN) in Geneva and at the Nordic Institute for Theoretical Physics (NORDITA) in Copenhagen.

Tensors in Physics and Cartan can be purchased through Wolfram Research or their dealers or directly from our internet shop. The package comes with a pdf e-book user manual. A printed edition of the User Manual for Tensors in Physics 1.2 is available. While dated in some aspects, all the mathematics and most of the User's Guide is still valid for the latest version.

General product information

Cartan is an easy-to-use tensor component package for interactive tensor calculations in general Riemann-Cartan spaces of arbitrary dimensions and signatures. The program employs the powerful formalism of rigid frames (e.g., orthonormal frames or vielbeins) and can return results both in the rigid frame or in the coordinate basis. Tensors such as Riemann, Ricci, Weyl, Einstein, Lanczos, and Cotton-York are predefined. It is also possible to extend the program by adding your own functions and variables.

The Cartan User's Guide and Reference Manual gives a detailed exposition of the use of the program with all its functions and variables. It contains examples of how to find solutions to Einstein's field equations or to the field equations of the Einstein-Cartan theory, and also of how to calculate the renormalized stress-tensor in curved space-time. The manual also gives an overview of general tensor calculus and the formalism used by Cartan.

Cartan turns Mathematica into a handy tensor computation tool and expert system. The package adds a large number of geometrical functions and variables to Mathematica with a a special view towards the needs of general relativists, but the program is useful in all fields working with tensor expressions.

The tensor concept is important in physics and has wide applications in such diverse fields as relativity theory, cosmology, high energy physics, field theory, thermodynamics, fluid dynamics and mechanics. The package includes examples and references from the works of major mathematicians and scientific researchers such as Euler, Lagrange, Riemann, Levi, Civita, Brans, Dicke, Ricci, Cartan, Christoffel, Weyl, Cotton, York and Einstein.

The program allows the user to compute tensor calculations with hundreds or thousands of components and may, moreover, be extended by the addition of user-defined functions. The program will function on all computers in which the Mathematica version 2.0 or upgrades has been installed.

About the power

The final form of Einstein's field equations were published in 1915. It took almost 50 years until in 1963 Kerr solved these equations for the case of a rotating black hole. With Cartan and Mathematica for Windows 3.11 on an old PC (486/66 MHz with 16Mb RAM), the Kerr solution can be verified in 2 ½ minutes while playing Mozart on the CD ROM at the same time!

Software requirements

Cartan has been tested and developed using Mathematica 7.0 and 8.0. Cartan is compatible with the Notebook interfaces.

Cartan is presently available for all platforms running Mathematica and with the ability to down-load and unpack a gz-compressed tar file or a zip file.

A sample of built-in functions

A small subset of the Cartan functions are listed below:

CConnection

computes the connection from the vierbein (and torsion).

CRiemann

computes the Riemann tensor.

CRicci

computes the Ricci tensor.

CEinstein

computes the Einstein tensor.

CDerivative[Tensor]

returns the covariant derivative of "Tensor."

CDivergence[Tensor,n]

returns the covariant divergence of "Tensor" with respect to index number "n."

OuterProduct[Tensor1,Tensor2]

returns the outer product of two tensors.

Contract[Tensor,n,m]

returns the contraction of "Tensor" with respect to index number "" and "m."

Cartan does all its calculations using rigid frames, e.g., orthonormal bases.

A sample of built-in variables

A small subset of the geometrical variables of Cartan are listed below:

Connection

the connection coefficients.

Riemann

the Riemann tensor.

Einstein

the Einstein tensor.

Ricci

the Ricci tensor.

Weyl

the Weyl tensor.

Cotton-York

the Cotton-York tensor.

Kretschmann

the Kretschmann invariant ("square" of Riemann).

SpinCoeff

the Newman-Penrose spin coefficients.

NPCurvature

the Newman-Penrose curvature scalars.

All the built-in expressions are represented as components relative to a rigid frame, e.g., orthonormal frames.