Tensors in Physics

Doing black holes and cosmology with Mathematica

Harald H. Soleng

Norwegian Computing Center


Tensors in Physics is a tensor component package published by the Scandinavian University Press. It comes with a book containing a brief course in differential geometry, a user's guide and reference manual for the Mathematica package CARTAN. CARTAN is an easy-to-use program for tensor component calculations. A large number of tensor expressions commonly used in gravitation theory and theoretical physics are predefined. The user-friendly high-level commands of CARTAN makes it an ideal tool for interactive tensor calculations.

In the demonstration it is first shown how CARTAN can be used as a tool in three-dimensional vector calculus in general coordinates. Then the package is used to verify the Schwarzschild solution of Einstein's field equations describing the gravitational field outside a static, spherical symmetric mass distribution. This is also the solution describing a static black hole.

Then I solve Einstein's field equations for a homogeneous, isotropic and spatially flat matter-dominated Universe. It is shown that the Kretschmann curvature invariant diverges at the origin of time. This is the Big Bang. Finally we check that the Kerr metric really solves Einstein's gravitational field equations in vacuum. This solution describes a rotating black hole. It took almost fifty years from the field equations were published until this solution was published. One can only guess how fast the solution had been found had Einstein had access to Mathematica.

Background

Loading and initialization

Warming up with vector analysis

Veryfying Schwarzschild's solution of Einstein's field equations

Exercise in relativistic cosmology

Veryfying Kerr's solution

Implementational issues

How to get CARTAN?