By John R. Rice
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Extra resources for Numerical Methods, Software, and Analysis
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. It may be preferable to construct CIVP coordinates in another way, for example from a 2-surface T = constant just inside R = R+ . Another possibility is to start with a Bondi sphere at r — oo, z = 1 and to construct null geodesies going inwards and backwards in time.
E12 = 0 is 0 = c2 + c3hy + cihyr + hyrr Eio = 0 is 0 = c5 + c6hu + hur EV = E22--g22(g22E22+g33E33) and leads to hy. and leads to hu. = 0 is 0 = c7+(rqiU)>r and leads to qtU (5) (6) (7) (8) Initial conditions for the integration are provided by equation (3). Once qu is found, q may be evolved to the "next" null cone, and the process repeated until the desired region of space-time has been covered. The numerical procedures used are summarised as they have already been reported [1,2]. The first step is to set up a finite grid.
1+r (26) V ; Making this transformation in the metric (2) leads to Einstein equations that are strongly singular at z — 1, in that they involve terms (1 — z)~n with n as large as 4. It is therefore clear that for progress to be made the metric will have to be written in another form. Inspired by the form of the Bondi metric at future null infinity [3,11] and after some trial and error, we write the metric as 28 Bishop: Some aspects of the characteristic initial value problem ds2 = -hunr2e2 n du2 - 2^—dudz z1 + 2hynr2e2qndudy +r2 (^-fdy2 + Fe-2"ndA .