Antisymmetric tensor generalizations of affine vector fields

TL;DR

This paper explores antisymmetric affine tensor fields as spacetime symmetries, establishing their properties, relations to parallel transport, and bounds on their quantity in n-dimensional spaces.

Contribution

It introduces and analyzes antisymmetric affine tensor fields, extending the concept of affine vector fields and deriving their integrability conditions and bounds.

Findings

Antisymmetric affine tensor fields relate to parallelly transported antisymmetric tensors along geodesics.

The number of independent antisymmetric affine tensor fields is bounded by a combinatorial formula.

Conditions for the existence of these fields vary across different spacetimes.

Abstract

Tensor generalizations of affine vector fields called symmetric and antisymmetric affine tensor fields are discussed as symmetry of spacetimes. We review the properties of the symmetric ones, which have been studied in earlier works, and investigate the properties of the antisymmetric ones, which are the main theme in this paper. It is shown that antisymmetric affine tensor fields are closely related to one-lower-rank antisymmetric tensor fields which are parallelly transported along geodesics. It is also shown that the number of linear independent rank-$p$ antisymmetric affine tensor fields in $n$ dimensions is bounded by $(n+1)!/p!(n-p)!$. We also derive the integrability conditions for antisymmetric affine tensor fields. Using the integrability conditions, we discuss the existence of antisymmetric affine tensor fields on various spacetimes.