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Autoparallels and the Inverse Problem of the Calculus of Variations

Lavinia Heisenberg

Abstract

We prove that autoparallel curves associated with a torsion-free but not necessarily metric-compatible affine connection can be derived from an action principle. We explicitly construct the action functional and show by standard variational techniques that it produces the desired equations. Our analysis is based on systematically solving the inverse problem of the calculus of variation and the associated Helmholtz conditions. This demonstrates that the dynamics of autoparallels admit a consistent variational formulation even in the presence of non-metricity. Our results provide a variational framework for particle motion in metric-affine geometries and thereby contribute to the mathematical foundations of the geodesic principle in relativistic gravity.

Autoparallels and the Inverse Problem of the Calculus of Variations

Abstract

We prove that autoparallel curves associated with a torsion-free but not necessarily metric-compatible affine connection can be derived from an action principle. We explicitly construct the action functional and show by standard variational techniques that it produces the desired equations. Our analysis is based on systematically solving the inverse problem of the calculus of variation and the associated Helmholtz conditions. This demonstrates that the dynamics of autoparallels admit a consistent variational formulation even in the presence of non-metricity. Our results provide a variational framework for particle motion in metric-affine geometries and thereby contribute to the mathematical foundations of the geodesic principle in relativistic gravity.
Paper Structure (6 sections, 7 theorems, 52 equations)

This paper contains 6 sections, 7 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Geometric Preliminaries
  3. Autoparallels and the Inverse Problem of the Calculus of Variations
  4. Existence of an action functional
  5. The action and its Euler-Lagrange equations
  6. Discussion

Key Result

Proposition 2.1

Let $(\mathcal{M},g,\nabla)$ be a metric-affine manifold. Then the connection admits the decomposition where $L^{a}{}_{bc}$ and $K^{a}{}_{bc}$ are the components of the disformation and contortion tensors, respectively, which are defined as

Theorems & Definitions (24)

  • Definition 2.1: Affine connection
  • Definition 2.2: Parallel transport
  • Definition 2.3: Autoparallel curves
  • Definition 2.4: Metric-affine geometry
  • Definition 2.5: Torsion tensor
  • Definition 2.6: Curvature tensor
  • Definition 2.7
  • Definition 2.8: Pseudo-Riemannian manifold
  • Proposition 2.1: Decomposition of the connection
  • Corollary 2.1: Autoparallel equation, local form
  • ...and 14 more