Algebraic approach to the characteristic initial value problem in general relativity  pp. 41-49

Algebraic approach to the characteristic initial value problem in general relativity

By Jörg Frauendiener

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Abstract. Penrose has described a method for computing a solution for the characteristic initial value problem for the spin-2 equation for the Weyl spinor. This method uses the spinorial properties in an essential way. From the symmetrized derivatives of the Weyl spinor which are known from the null datum on a cone one can compute all the derivatives by using the field equation and thus one is able to write down a power series expansion for a solution of the equation. A recursive algorithm for computing the higher terms in the power series is presented and the possibility of its implementation on a computer is discussed.

INTRODUCTION

Due to the nonlinear nature of general relativity it is very difficult to obtain exact solutions of the field equations that are in addition of at least some physical significance. Prominent examples are the Schwarzschild, Kerr and Friedmann solutions. Given a concrete physical problem it is more often than not rather hopeless to try to solve the equations using analytical techniques only. Therefore, in recent years, attention has turned towards the methods of numerical relativity where one can hope to obtain answers to concrete questions in a reasonable amount of time given enough powerful machines. However, it is still a formidable task to obtain a reliable code. There is first of all the inherent complexity of the field equations themselves when written out in full without the imposition of symmetries or other simplifying assumptions.