Abstract
We present a new algorithm for computing covariant power expansions of tensor fields in generalized Riemannian normal coordinates, introduced in some neighborhood of a parallelized k-dimensional submanifold (k = 0, 1, . . .< n; the case k = 0 corresponds to a point), by transforming the expansions to the corresponding Taylor series. For an arbitrary real analytic tensor field, the coefficients of such series are expressed in terms of its covariant derivatives 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. We show that this computational problem belongs to the complexity class LEXP.
Highlights
Covariant series expansions in Riemann normal coordinates [1, 2], in Fermi normal coordinates [3, 4], and in some other geodesic coordinate systems find important applications in general relativity [6,7,8,9])
Explicit formulas and recipes for coefficients of the series are given in Refs. [10,11,12,13,14,15], so that, from a formal point of view, the problem is solved completely
It is essential to note that the problem, in the context of calculating a covariant series, can be set in different ways depending on the explicit form of the expansion
Summary
Covariant series expansions in Riemann normal coordinates [1, 2], in Fermi normal coordinates [3, 4], and in some other geodesic coordinate systems (e.g., see Ref. [5]) find important applications in general relativity [6,7,8,9]). Explicit formulas and recipes for coefficients of the series are given in Refs. It is essential to note that the problem, in the context of calculating a covariant series, can be set in different ways depending on the explicit form of the expansion. [15] where the problem is considered with a sufficient degree of generality: one needs only a connection to form the series; the connection is not assumed to be metric or torsion-free; the covariant series are defined in some normal neighborhood of an arbitrary submanifold. In our setting, having established a connection (or spacetime metric), one needs to compute the covariant derivatives of the curvature (and, in general, of the torsion), to collect like terms, and to compute coefficients of coordinate monomials.
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