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Manuscript Title: The Maple package NPTOOLS: a symbolic algebra package for tetrad formalisms in general relativity.
Authors: S. Cyganowski, J. Carminati
Program title: NPTOOLS
Catalogue identifier: ADJM_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 115(1998)200
Programming language: Maple.
Computer: Sun Sparc 5 workstation.
RAM: 380K words
Keywords: Computer algebra, Maple, General relativity, Nptools, Tetrad formalisms, Newman-penrose, Orthonormal tetrad, Formalism, Lorentz group, Petrov-plebanski, Astrophysics, Gravitational systems, Nulltran, Inv, Nullor, Ornull, Pptype.
Classification: 1.5, 5.

Nature of problem:
There are three main sections to the NPTOOLS program. The first of these provides a translation between the physical quantitites of the orthonormal tetrad formalism of Ellis [2,3] and MacCallum [5] and the null tetrad quantities of the Newman-Penrose [6] formalism. The second provides an implementation of the 6-parameter Lorentz group and has a facility for checking the invariance of Newman-Penrose equations under subgroups of the Lorentz group. This part of the program proved useful as an aid in the proof of the theorem that general relativistic, Petrov type III, shear-free perfect fluids which obey a barotropic equation of state p = p(w) with w+p not equal 0, are non-expanding (see Carminati and Cyganowski [1]). The third section of the program is an implementation of the Petrov-Plebanski classification scheme [4,7] for the Ricci tensor.

Solution method:
Each section of the program relies on equations from differential geometry theory and Einstein's theory of general relativity. In particular, equations derived from the Newman-Penrose null tetrad formalism [6] and from the orthonormal tetrad formalism of Ellis [2,3] and MacCallum [5] are used.

Running time:
The main focus of the program is on algebraic expressions and thus the running time typically depends on the size and complexity of the expressions. For example, when the command "nulltran" is used to apply a null rotation leaving the basis vectro n**mu fixed, to each of the Newman-Penrose quantities, a Maple CPU Time reading of 2.9 seconds is observed. The most time consuming command is "inv". When this command is used to test for independence of equations under null rotations the complexity and size of the equation is a major determining factor. For, example, when we tested for invariance under a null rotation leaving l**mu fixed, an expression consisting of 42 terms resulted in a Maple CPU Time reading of 5.4 seconds whilst applying the same null rotation to an expression having 4 terms and testing for invariance resulted in a Maple CPU Time reading of 3.8. seconds. It should be noted that the terms present in the expression play a key role in the length of computation time. The command "pptype" which is the main feature of the third section of the package produced a Maple CPU Time of 0 seconds when applied to the Ricci tensor for a perfect fluid. These CPU time measurements were taken using the in-built Maple CPU Time facility of Maple V release 3 running on a Sun Sparc 5 workstation with a single 85 MHz microSPARC II CPU and 64MB of main memory.

References:
[1] J. Carminati, S.O Cyganowski, Class. Quantum Grav. 14 (1997) 1167.
[2] G.F.R. Ellis, J. Math. Phys. 8 (1967) 1171.
[3] G.F.R. Ellis, Relativistic cosmology, General Relativity and Cosmology: Proc. Int. School of Physics 'Enrico Rermi' (Course XLVII, 1969), R.K. Sachs, ed (Academic Press, London, 1971).
[4] C.B.G. McIntosh, J.M. Foyster, A.W.C. Lun, J. Math. Phys. 22 (1981) 2620.
[5] M.A.H. MacCallum, in: E. Schatzmann, Cargese Lectures in Physics, Vol. 6, Lectures at the International Summer School of Physics, Cargese, Corsica, 1971 (1973) p. 61.
[6] E.T. Newman, R. Penrose, J. Math. Phys. 3 (1962) 566.
[7] J. Plebanski, Acta Phys. Polon. 26 (1964) 963.