By Ray d'Inverno

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**Additional info for Approaches to Numerical Relativity: Proceedings of the International Workshop on Numerical Relativity, Southampton, December 1991**

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And d'Inverno, R. A. (1990). Class. Quant Grav. Lett, 7, L23-L27. [2] Bishop, N. , Clarke, C. J. S. and d'Inverno, R. A. (1990). A. Jnl. , 86, 64-65. [3] Isaacson, R. , Welling, J. S. and Winicour, J. (1983). J. Math. , 24, 1824-1834. , Isaacson, R. , Welling, J. S. and Winicour J. (1986). M. ), Cambridge University Press, Cambridge. [5] Hultquist, P. F. (1988). Numerical methods, Benjamin/Cummings, California, 246. [6] Misner, C. , Thorne, K. S. and Wheeler, J. A. (1973). Gravitation, Freeman, San Francisco, 608-611.

The metric is denoted by Ha(3 : N is the lapse function, N%(i = 1,2,3) is the shift vector; A and B are components of the spacelike part of the metric defined by A2dR2 + R2{B~2dQ2 + B2 sin2(0)d$2) (39) K+ is a component of the extrinsic curvature. The Cauchy calculation is carried out in a domain R < i2+, and it is assumed that matter is present only in a central region 0 < R< Rm with Rm < JR+. 1 Construction of the CIVP coordinates The construction that will be described here is based upon the CIVP coordinates of sections 2 and 5, and coordinate lines are null geodesies starting from R = 0.

As is well-known gravity can bend light and so refocus initially diverging light rays. This caused many-valuedness and hence singularities in coordinate systems based on light rays. A detailed study of caustics in Minkowski spacetime was made in Friedrich and Stewart (1983), Stewart (1991) and algorithms to evolve them were described in Corkill and Stewart (1983) and Stewart (1986). , a perfect fluid. In this paper we show how this can be done, by changing both the set of dependent variables and by changing the foliation used.