Tensor Operations on Degenerate Inner Product Spaces

TL;DR

This paper extends tensor operations to degenerate inner product spaces, overcoming key obstructions to define covariant contraction, thereby laying algebraic foundations for invariants in Singular Semi-Riemannian Geometry with applications to General Relativity.

Contribution

It introduces a natural way to perform tensor operations on degenerate inner product spaces, enabling the construction of geometric invariants in singular settings.

Findings

Defined covariant contraction for certain degenerate cases

Established algebraic framework for invariants in Singular Semi-Riemannian Geometry

Applied to study singularities in General Relativity

Abstract

Well-known operations defined on a non-degenerate inner product vector space are extended to the case of a degenerate inner product. The main obstructions to the extension of these operations to the degenerate case are (1) the index lowering operation is not invertible, and (2) we cannot associate to the inner product in a canonical way a reciprocal inner product on the dual of the vector space. This article shows how these obstructions can be avoided naturally, allowing a canonical definition of covariant contraction for some important special cases. The primary motivation of this article is to lay down the algebraic foundation for the construction of invariants in Singular Semi-Riemannian Geometry, especially those related to the curvature. It turns out that the operations discussed here are enough for this purpose (arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646). Such invariants can be applied to the study of singularities in the theory of General Relativity (arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099, arXiv:1112.4508).