Regge Calculus: A Unique Tool for Numerical Relativity

Abstract

The application of Regge calculus, a lattice formulation of general relativity, is reviewed in the context of numerical relativity. Particular emphasis is placed on problems of current computational interest, and the strengths and weaknesses of the lattice approach are highlighted. Several new and illustrative applications are presented, including initial data for the head on collision of two black holes, and the time evolution of vacuum axisymmetric Brill waves. I. NUMERICAL RELATIVITYThe complexity of the Einstein equations, combined with the sparsity of relevant analytic solutions, necessitates a range of other tools with which to explore complex physical scenarios. These include series expansions and perturbation techniques, together with numerical solutions of the fully non-linear equations. Unfortunately, the numerical solution of Einstein's equations has proved to be an exceedingly difficult problem. Over the last three decades numerous schemes have been developed, or adapted, to tackle the vast range of problems which fall within the purview of numerical relativity.The classic approach, involving a three-plus-one dimensional split of space and time, was first expounded by Arnowitt, Deser and Misner (ADM) [1]. This approach is natural in the context of our Newtonian intuition, and is also directly applicable to computer simulations. Whilst the traditional ADM approach has dominated numerical relativity, questions regarding its long-term stability have lead to the development of many other formulations of the Einstein equations in three-plus-one dimensions.Techniques of current interest include those developed by Sasaki and Nakamura, and later by Baumgarte and Shapiro, known generically as Conformal ADM (CADM) formulations. Incorporating insights from York's analysis of the initial value problem, these algorithms have shown superior stability properties compared with ADM in some applications. Other recent formulations are based on symmetric-hyperbolic forms of the Einstein equations, where it is hoped that the mathematical proofs of stability and well-posedness confer numerical advantages compared with techniques whose mathematical structures are more uncertain. The recent review by Lehner [2] discusses many of these issues.Despite the enormous effort invested in these techniques (and many others, including characteristic formulations), most modern numerical relativity codes continue to suffer from problems with long-term stability and lack of accuracy. Lack of resolution is only a partial solution; new insights or techniques seem to be required to overcome many of these problems.

Einstein equation from covariant loop quantum gravity in semiclassical continuum limit

“…The leading contributions to Z N ,δ in Eq. (27) have been organized into disjoint sectors associated to different σ. Each sector σ has its own partition function ℓ e iS σ /ℓ 2 P +···…”

“…Although the general mathematical proof for the convergence of Regge solutions to Einstein equation solutions is not available in the literature, extensive studies of the Regge calculus provide many analytical and numerical results, which all support the convergence, and demonstrate the Regge calculus as a useful tool in numerical relativity (see e.g. [26,27] for reviews). Among the results, there has been a rigorous proof of the convergence in the linearized Regge calculus and linearized Einstein equation [28][29][30].…”

Counting a black hole in Lorentzian product triangulations

“…In [27] a Regge calculus version of black-hole geometry was constructed. (For reviews of Regge calculus see [28].) It should be kept in mind that unlike in Regge calculus we are working with simplices of fixed squared edge lengths a 2 and −αa 2 for the spacelike and timelike edges, respectively.…”

Deriving the ADM3+1evolution equations from the second variation of arclength

“…This method is known as smooth lattice relativity and is closely related to the Regge calculus [8,9,10]. Both methods use a lattice to describe the metric but they differ most notably in the way they treat the curvatures.…”