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 easytouse program for tensor component calculations. A large number of tensor expressions commonly used in gravitation theory and theoretical physics are predefined. The userfriendly highlevel 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 threedimensional 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 matterdominated 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?
