[Paper Review] A nonlinear theory of tensor distributions

James Vickers, Julie Wilson|ArXiv.org|Jul 24, 1998

Mathematical and Theoretical Analysis6 references17 citations

TL;DR

This paper develops a nonlinear, coordinate-invariant theory of tensor distributions using Colombeau's generalized functions, extending the framework to ensure that tensor transformations and embeddings commute with diffeomorphisms. The key contribution is a consistent, diffeomorphism-covariant formulation of generalized tensor fields that preserves classical transformation laws and enables coordinate-independent calculations of distributional curvatures in general relativity.

ABSTRACT

The coordinate invariant theory of generalised functions of Colombeau and Meril is reviewed and extended to enable the construction of multi-index generalised tensor functions whose transformation laws coincide with their counterparts in classical distribution theory.

Motivation & Objective

• To resolve the lack of diffeomorphism invariance in Colombeau's original theory when applied to tensor distributions in general relativity.
• To extend Colombeau's algebra to define multi-index generalized tensor functions with transformation laws matching classical distribution theory.
• To construct a mapping $\tilde{\mu}^*$ that commutes with the embedding $\iota$ and preserves moderate and null ideals under smooth diffeomorphisms.
• To ensure that association (weak equivalence) to distributions commutes with diffeomorphisms, enabling consistent physical interpretation across coordinate systems.

Proposed method

• Adopt a coordinate-invariant definition of the smoothing kernel space $\mathcal{A}_k$ by weakening moment conditions, following Colombeau and Meril (1994).
• Define a pullback map $\tilde{\mu}^*$ on the base algebra $\mathcal{E}_s(\Omega)$ that preserves the moderate and null subalgebras $\mathcal{E}_{M,s}(\Omega)$ and $\mathcal{N}_s(\Omega)$.
• Construct the generalized tensor algebra $\mathcal{G}^p_q(\Omega)$ as a quotient of $\mathcal{E}_{M,s}(\Omega)$ by $\mathcal{N}_s(\Omega)$, ensuring tensor transformation laws are preserved.
• Use the covariant derivative associated with a background torsion-free connection to ensure compatibility with the embedding $\iota$, so $[\iota(X), \iota(Y)] = \iota([X,Y])$.
• Define association via weak limits: $[\tilde{T}] \approx S$ if the limit of integrals against test functions equals the distributional pairing.
• Verify that the pullback $\tilde{\mu}^*$ commutes with association: $[\tilde{T}'] \approx S'$ implies $[\tilde{\mu}^* \tilde{T}'] \approx \mu^* S'$.

Experimental results

Research questions

• RQ1 Can Colombeau's generalized function algebra be extended to define tensor-valued generalized functions that transform as tensors under smooth diffeomorphisms?
• RQ2 Does the embedding $\iota$ of classical distributions into generalized functions commute with diffeomorphism pullbacks in the generalized function framework?
• RQ3 How can the moment conditions defining the smoothing kernel space be modified to ensure diffeomorphism invariance in the generalized function construction?
• RQ4 Can the association relation between generalized functions and distributions be made compatible with diffeomorphisms?
• RQ5 Is it possible to perform coordinate-independent calculations of distributional curvatures, such as $\tilde{R}\sqrt{\tilde{g}}$, in general relativity using this extended framework?

Key findings

• The extended theory ensures that the pullback map $\tilde{\mu}^*$ on generalized functions commutes with the embedding $\iota$ of classical distributions, preserving diffeomorphism invariance.
• The generalized tensor fields transform under $\tilde{\mu}^*$ according to the same laws as classical tensor distributions, enabling consistent physical interpretation across coordinate systems.
• The Lie bracket of vector fields satisfies $[\iota(X), \iota(Y)] = \iota([X,Y])$, confirming compatibility with Lie algebra structure.
• For smooth metrics, the Levi-Civita connection's covariant derivative commutes with the embedding $\iota$, ensuring consistency in curvature calculations.
• The association relation commutes with diffeomorphisms: if $[\tilde{T}'] \approx S'$, then $[\tilde{\mu}^* \tilde{T}'] \approx \mu^* S'$, validating the physical consistency of the framework.
• The framework enables coordinate-independent evaluation of distributional curvatures, such as $\tilde{R}\sqrt{\tilde{g}} \approx 4\pi(1-A)\delta^{(2)}$ for conical singularities, as confirmed in prior work by Clarke et al. (1996).