Constructing a space from the geodesic equations

Abstract

Given a space with a metric tensor defined on it, it is easy to write down the system of geodesic equations on it by using the formula for the Christoffel symbols in terms of the metric coefficients. In this paper the inverse problem, of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor explicitly from the Christoffel symbols. The procedure is extended for determining if a second order quadratically semi-linear system can be expressed as a system of geodesic equations, provided it has terms only quadratic in the first derivative apart from the second derivative term. A computer code has been developed for dealing with large systems of geodesic equations. Program summary Program title: geodesicCOMMENTED.nb Catalogue identifier: AEBA_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBA_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 373 No. of bytes in distributed program, including test data, etc.: 3641 Distribution format: tar.gz Programming language: MATHEMATICA Computer: Computers that run MATHEMATICA Operating system: MATHEMATICA runs under Linux and windows RAM: Minimum of 512 kbytes Classification: 1.5 Nature of problem: The code we have developed calculates the space when the geodesic equations are given. Solution method: The code gives the user the option of selecting a subset of the metric tensor required for constructing the Christoffel symbols. This system is over-determined hence the results are not unique. Running time: Dependent on the RAM available and complexity of the metric tensor.