7.1. Introduction
In this chapter, we solve problems taken from topics in special relativity, general
relativity
and cosmology. The special relativity problems are similar to those found in Classica
Electrodynamics, by John D. Jackson, John Wiley and Sons, Inc. and Classical
Mechanics
by Herbert Goldstein, Addison-Wesley. The general relativity and cosmological
problem;
were taken from topics covered in Gravitation and Cosmology, by Steven Weinberg,
Johr
Wiley &: Sons, Inc., and Gravitation, by Misner, Thorne, and Wheeler, W. H. Freemar
and Company. This chapter is divided into three sections:
1. Introduction
2. Problems
3. Unsolved Problems
The Introduction is divided into three subsections: (1) Special Relativity, (2) Genera
Relativity and Cosmology, and (3) Mathematica Commands. The first two subsection;
include a short review of the foundations of special relativity, general relativity, anc
cosmology and explain those concepts and definitions needed to understand the
problems
The third subsection lists the Mathematica packages, user-defined rules, and user
defined procedures for boosts, relativistic velocity parameters, Christoffel symbols, cur
vature tensor, Ricci tensor, Killing equations, Einstein tensor, and geodesic equations.
W(
have also included rules and functions that define the metrics and Christoffel symbol'
for the Schwarzschild and Kerr metrics. These user-defined procedures are not the mosi
efficient; we were primarily interested in clarity, not speed, so the algebraic symmetric;
of the geometric operators were not considered. Several sophisticated packages are
avail
able to handle such geometric manipulations efficiently. Also, the use of differentia
forms was omitted because of space limitations. You are encouraged to take advantage
of these symmetries by modifying these procedures and to use differential forms in the
calculations.
The Problems section is divided into subsections on special relativity problems and or
general relativity and cosmological problems. The methods used to solve these
problems
are not unique; you are encouraged to find procedures that will illuminate the physics
and to find other approaches that make the calculations faster.
7.1.1 Special Relativity
7.1.2 General Relativity and Cosmology
Spacetime metric
Field equations
Free-falling test particles and light trajectories
Robertson-Walker cosmology
7.1.3 Mathematica Commands
Packages
User-defined metric, boost, and velocity parameters
Metric
Rule for relativistic velocity parameters
Boost along the x-axis
User-defined geometric procedures
Christoffel symbols
Example: Christoffel symbols for a pseudo-Euclidean metric
Curvature tensor
Example: Curvature tensor for pseudo-Euclidean metric
Ricci tensor
Example: Ricci tensor for Godel metric
Killing's equations
Example: Killing vector equations in pseudo-Euclidean space
Einstein tensor
Example: Einstein tensor for wave metric
Geodesic equations
Example: Geodesies for a pseudo-Euclidean metric
User-defined metrics and Christoffel symbols
Schwarzschild metric
Kerr metric
Protect user-defined operators