Computer algebra in gravity research

Abstract

The complicated nature of calculations in general relativity was one of the driving forces in the early development of computer algebra (CA). CA has become widely used in gravity research (GR) and its use can be expected to grow further. Here the general nature of computer algebra is discussed, along with some aspects of CA system design; features particular to GR’s requirements are considered; information on packages for CA in GR is provided, both for those packages currently available and for their predecessors; and applications of CA in GR are outlined.

Curvature Invariants for Charged and Rotating Black Holes

“…Related expressions for an earlier set of invariants due to Carminati and McLenaghan [18] have been explored using GRTensor [30]. Other aspects of invariants have been discussed using specially designed software [31], and a general review of the application of symbolic computing to problems in general relativity was given by MacCallum [32]. The thirteen nonzero invariants in Equation 10are not all algebraically independent, as the choice of a specific metric uses up additional degrees of freedom.…”

Spacetimes with continuous linear isotropies I: spatial rotations

“…CLASSI was developed by Jan Åman and collaborators, using and extending Inge Frick's SHEEP software, and is described in [31] and more briefly in [28] (see also the manual [1] supplied with the software). To carry out the Cartan-Karlhede procedure CLASSI uses the Newman-Penrose formalism as set out in Chapter 7 of Stephani et al…”

Symbolic tensor calculus on manifolds: a SageMath implementation

“…In 1969, R.A. d'Inverno developed ALAM (for Atlas Lisp Algebraic Manipulator) and used it to compute the Riemann and Ricci tensors of the Bondi metric. According to [22], the original calculations took Bondi and collaborators 6 months to finish, while the computation with ALAM took 4 minutes and yielded the discovery of 6 errors in the original paper by Bondi et al Since then, numerous packages have been developed; the reader is referred to [14] for a recent review of computer algebra systems for general relativity (see also [13] for a review up to 2002), and to [10,4] for more recent reviews focused on tensor calculus. It is also worth to point out the extensive list of tensor calculus packages maintained by J. M. Martin-Garcia at http://www.xact.es/links.html.…”

“…The latter invokes a chain of simplifying functions, which depends on the symbolic backend. 17 Let us now discuss the second case in the __add__() method of Element, namely the case for which the parents of both operands are different (lines 14…”