On the Linearization of Regge Calculus
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Dept. of Ma th./CMA University of Os lo Pure Mathematics No 13 ISSN 0806–2439 May 2008 On the linearization of Regge calculus Snorre H. Christiansen ∗ May 9, 2008 Abstract We study the linearization of three dimensional Regge calculus around Euclid ean metric. We provide an explic it formula for the corres ponding quadratic form and relate it to the curl t curl operator which appears in the quadratic part of the Einstein-Hilbert action and also in the linear elasti city complex . We insert Regge metrics in a discrete versio n of this complex, equipped with densily defined and commuting interpolators. We show that the eigenpairs of the curl t curl operator, approximated using the quadr ati c par t of the Regge action on Regg e met rics , con ve rge to their continu ous counterp arts, inter preting the comput ation as a non- conforming finite element method. 1 Introduct ion Regge calculus [31] is an approach to general relativity [39] which to some extent predates the explosion of numerical methods in the natural sciences. Spacetime is represented by a simpl icial comple x. Given this simplicial complex a finite dimensional space of metrics is defined. We call such metrics Regge metrics. A functional defined on this space of metrics and mimicking the Einstein-Hilbert action, is provided. We call this functional the Regge action. A critical point of the Regge action on the space of Regge metrics is generally believed to be a good approximation to a true solution of Einstein’s equations of general relativity [26]. Regge calculus is quite popular in studies of quan tum gravit y [32]. Its dis- crete nature also makes it a natural candidate for the construction of efficient algorithms to simulate the classical field equations, a possibility expressed al- ready in Regge’s paper. In this direction we are aware of [6] and [20]. However it seems that the bulk of numerical relativity computations are performed using other methods [25]. In part icular recent progress on the simu lation of merging black holes [29] appear to be carried out in sophisticated but in some sense tradi tiona l finite differe nce schemes. We hope that this paper can contribute to establishing Regge calculus as a good alternative. Its geometric “coordinate free” nature would make it a structure-preserving method in the sense of [21] and could be decis ive in particular for simulating cosmologies, where global qualitative effects are important. ∗ CMA, Universi ty of Osl o, PO Box 1053 Blinder n, NO- 0316 Oslo , Nor wa y . ema il : [email protected] 1
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