Generalized algorithm of computation of coefficients of the covariant Taylor series

I. M. Potashov

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 2 (2023), pp. 51-66

Résumé

In this article we deal with the algorithm for computing covariant power expansions of tensor fields in Fermi coordinates, introduced in some neighborhood of a parallelized $m$-dimensional submanifold (${m=0, 1,\ldots$, $m=0$ corresponds to a point), by transforming the expansions to the corresponding Taylor series. The method of covariant series can be useful in solution of some problems of general relativity, in particular, it is convenient for computing of the spacetime metric components, and also it is used in the theory of quantum gravity. The algorithm under consideration is the generalization of [14]. For an arbitrary real analytic tensor field, the coefficients of such series are expressed in terms of its covariant derivatives, the connection components and covariant derivatives of the curvature and the torsion. The algorithm computes the corresponding Taylor polynomials of arbitrary orders for the field components and is applicable to connections that are, in general, nonmetric and not torsion-free. It is shown that this algorithm belongs to the complexity class LEXP.

Détail

Keywords: pseudo-Riemannian manifolds, tensor fields, covariant Taylor series, computational complexity of algorithms.