[Paper Review] Generalized pseudo-Riemannian geometry

Michael Kunzinger, Roland Steinbauer|ArXiv.org|Jul 7, 2001

Mathematical and Theoretical Analysis29 references4 citations

TL;DR

This paper develops a nonlinear distributional pseudo-Riemannian geometry using Colombeau's algebra of generalized functions, enabling the rigorous treatment of singular metrics in general relativity. It introduces generalized metrics, connections, curvature, and geodesics, proving a generalized Fundamental Lemma and showing that geodesics exhibit physically consistent refracted behavior across impulsive gravitational waves.

ABSTRACT

Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-)Riemannian geometry'' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.

Motivation & Objective

• To establish a consistent framework for pseudo-Riemannian geometry in the setting of Colombeau's algebra of generalized functions.
• To define generalized metrics, connections, and curvature tensors that extend classical differential geometry to singular, distributional settings.
• To provide a rigorous foundation for studying geodesics and curvature in spacetimes with impulsive gravitational waves and conical singularities.
• To unify and generalize previous heuristic or partial treatments of singular metrics in general relativity using nonlinear distributional methods.
• To demonstrate the consistency of the generalized framework with both smooth geometry and linear distributional approaches in key physical cases.

Proposed method

• Employs the special (simplified) version of Colombeau's algebra of generalized functions to ensure diffeomorphism invariance and handle nonlinear operations on singular objects.
• Derives characterizations of invertibility in the algebra of generalized functions, essential for defining generalized metrics and connections.
• Introduces generalized sections on generalized mappings to define geodesics of a generalized metric, extending the classical geodesic equation to singular settings.
• Defines generalized connections and curvature tensors via generalized Christoffel symbols and their derivatives in the Colombeau framework.
• Proves a generalized version of the Fundamental Lemma of (pseudo-)Riemannian geometry, ensuring consistency with classical geometric identities.
• Applies the framework to impulsive pp-wave spacetimes by modeling the metric as a generalized line element involving generalized delta functions, enabling solution of the geodesic equations in the Colombeau setting.

Experimental results

Research questions

• RQ1 How can a consistent pseudo-Riemannian geometry be formulated for singular, distributional metrics using nonlinear generalized functions?
• RQ2 What conditions ensure invertibility of generalized functions in Colombeau algebras, and how does this relate to defining a generalized metric tensor?
• RQ3 How can geodesics be rigorously defined and solved in spacetimes with impulsive gravitational waves using generalized functions?
• RQ4 To what extent does the generalized curvature and connection structure recover known results from linear distribution theory and smooth geometry?
• RQ5 Can the framework consistently describe physical phenomena such as geodesic refraction in impulsive pp-wave spacetimes?

Key findings

• The paper establishes a generalized version of the Fundamental Lemma of (pseudo-)Riemannian geometry, ensuring that key geometric identities hold in the Colombeau setting.
• Geodesics in impulsive pp-wave spacetimes are shown to be uniquely solvable in the algebra of generalized functions ${\mathcal{G}}(\mathbb{R})^3$ for given initial conditions.
• The distributional shadow of the generalized geodesics exhibits physically meaningful behavior: straight-line motion is refracted at the null hypersurface $u=0$, with position and velocity discontinuities proportional to the metric's jump.
• The solution for $v(u)$ includes a Heaviside function term $H(u)$ and a term proportional to $u_+$, reflecting the impulse's effect on the geodesic path.
• The framework provides a consistent, comprehensive interpretation of prior heuristic calculations in general relativity involving singular metrics, such as cosmic strings and impulsive waves.
• The generalized curvature of the metric in the impulsive pp-wave case vanishes away from $u=0$, consistent with the physical expectation of curvature concentrated on a null hypersurface.