# The structure of general relativity with a numerical illustration : the collision of two black holes

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## Abstract

The splitting of spacetime into space and time is analyzed by abstract methods, coordinate methods, and numerical computer methods. General relativity is assumed to be the correct theory of gravity and the Einstein field equations are studied in their canonical form. A comparison is made of the initial value and evolution equations for gravity in different formalisms (ADM, geometry of spacelike hypersurfaces, timelike congruences). These equations are then illustrated by a number of cosmological and black hole space-times. A study of coordinate conditions is undertaken. The axisymmetric (non-stationary and non-spherical) Einstein equations are discussed, and it is shown how to set up a numerical computer program to integrate these equations starting with a given initial data set. Various applications of this computer approach are discussed: collapse of rotating non-spherical stars, "runaway" collapse, and headon collisions of black holes. All of these situations would involve generation of gravitational radiation from the formation of black holes. The particular case of two nonrotating black holes colliding headon is chosen as a test case for the computer. The initial value problem is reviewed and then a computer program is written to generate the spacetime. This generation involves solving four coupled quasilinear hyperbolic (evolution) equations and one linear elliptic equation (maximal slicing) on each spacelike sheet. The numerical difficulties are discussed and graphical methods are used to present the result of the computations.