Physical Sciences and Mathematics Commons^™

Geodesic Orbits About An Axisymmetric Mass Distribution, David J. Dean

Honors Theses

The General Theory of Relativity (GRT) gives rise to many interesting questions, one of which is a question about how test particles orbit various relativistic mass configurations. A test particle is defined as an electrically neutral particle small enough that its self gravitating energy, as calculated using standard Newtonian theory, can be neglected when compared with the particle's rest mass, ie. M/R << 1. (Here M is the mass of the particle in meters, and R is the radius of the particle again in meters. The mass of the sun in these units is 1.47 km.) For example, the earth, which can be considered a test particle for the sun, travels about the sun in an elliptic orbit. What would happen if the sun suddenly became a black-hole? How would the earth's orbit be affected? It is the purpose of this paper to answer these questions not only for the case of the spherical black-hole (which has been extensively studied), but also for the case of an axisymmetric mass distribution idealized by a long thin rod-like mass (which will henceforth be referred to as a line mass). The classical case and relativistic case concerning orbits about the line mass will be discussed, along with some interesting features of the relativistic line mass orbits. Some background information concerning GRT will be needed before the problem of line mass is approached.