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Visualization of the General Relativistic Disk of Dust

Introduction

Einstein's general theory of relativity describes gravitation as a geometric property of the four-dimensional manifold of space and time. The properties of the gravitational field of the rigidly rotating disk of dust can be investigated by using a visualization technique which is called ray tracing in four-dimensional curved spacetime. This method generates images as seen by a realistic observer. The main idea is to use null geodesics to probe the properties of the gravitational field. This visualization technique provides a very compact presentation of a vast number of light rays. Furthermore, it allows a geometric and intuitive approach. Since the pictures are observables, they are independent of the chosen coordinate system, which is an important feature and advantage in the realm of general relativity.

The general relativistic gravitational field created by a rigidly rotating disk of dust was first studied numerically in 1971 by Bardeen and Wagoner. Einstein's field equations for a rigidly rotating disk of dust can be reduced to a single non-linear complex partial differential equation - the so-called Ernst equation - for which a boundary value problem has to be solved. Neugebauer and Meinel succeeded in solving this problem by means of the inverse scattering method, which is a technique known from soliton theory, in 1995. This way, they could provide the global analytical solution of Einstein's field equations for this object. Their explicit expressions for the metric coefficients allow a direct numerical implementation of the geodesic equation.

Results

rigid disk

The general relativistic rigidly rotating disk of dust as 192*144 pixel movie (Mpeg, 7.1MB) or as 384*288 pixel movie (Mpeg, 15.4MB).