[Paper Review] Metrical Multi-Time Lagrange Geometry of Physical Fields

Mircea Neagu|arXiv (Cornell University)|Sep 12, 2000

Advanced Differential Geometry Research8 references3 citations

TL;DR

This paper introduces a metrical multi-time Lagrange geometry framework for unifying gravitational and electromagnetic fields using a Kronecker $h$-regular Lagrangian on the 1-jet fibre bundle $J^1(T,M)$. It derives generalized Einstein and Maxwell equations through Cartan canonical connections and metrical d-tensors, showing that conservation laws and field equations reduce to classical forms under specific conditions, establishing a parametrized, geometrically consistent field theory.

ABSTRACT

The author exposes the metrical multi-time Lagrange geometry of physical fields which naturally generalizes the classical Lagrangian developped by Miron and Anastasiei. In other words, one constructs a natural theory of physical fields on the 1-jet fibre bundle, attached to a Kronecker h-regular multi-time Lagrangian with partial derivatives of order one.

Motivation & Objective

• To develop a geometric framework for physical fields using metrical multi-time Lagrange spaces on the 1-jet bundle $J^1(T,M)$.
• To unify gravitational and electromagnetic field theories within a single geometric structure based on a Kronecker $h$-regular Lagrangian.
• To derive generalized Einstein and Maxwell equations using Cartan canonical connections and metrical d-tensors.
• To establish conservation laws for gravitational and electromagnetic fields in the context of multi-time Lagrangian geometry.
• To demonstrate that the theory reduces to classical field equations under specific conditions, validating its physical consistency.

Proposed method

• Utilizes the metrical multi-time Lagrange space $ML^n_p = (J^1(T,M), L)$ with a Kronecker $h$-regular Lagrangian $L: J^1(T,M) \to \mathbb{R}$.
• Constructs the fundamental metrical d-tensor $g_{ij}(x^k, y^k) = \frac{1}{2} \frac{\partial^2 L}{\partial y^i \partial y^j}$ to model a unified gravitational field.
• Introduces the Cartan canonical connection $C\Gamma = (L^i_{jk}, C^i_{jk})$ via metrical compatibility conditions $g_{ij|k} = 0$ and $g_{ij}|_k = 0$.
• Derives the Sasakian-like metric $G = g_{ij} dx^i \otimes dx^j + g_{ij} \delta y^i \otimes \delta y^j$ on $TM$.
• Applies the nonlinear connection $\Gamma = (N^i_j(x,y))$ to define adapted bases $\{ \delta/\delta x^i, \partial/\partial y^i \}$ and dual forms.
• Derives field equations by imposing Einstein-type curvature conditions on the Ricci tensors $R_{ij}, S_{ij}$ and energy-momentum tensors $\mathcal{T}^{H,V}$.

Experimental results

Research questions

• RQ1 How can a unified geometric framework be constructed for both gravitational and electromagnetic fields using multi-time Lagrangian geometry?
• RQ2 What are the generalized Einstein and Maxwell equations in the context of metrical multi-time Lagrange spaces?
• RQ3 How do conservation laws for gravitational and electromagnetic fields emerge from the Cartan canonical connection and metrical structure?
• RQ4 Under what conditions does the multi-time Lagrangian theory reduce to classical Einstein and Maxwell equations?
• RQ5 What is the role of the gauge group and parametrization in the geometric formulation of physical fields?

Key findings

• The generalized Einstein equations for the metrical multi-time Lagrange space $ML^n_p$ are given by $R_{ij} - \frac{1}{2}R g_{ij} = \mathcal{K} \mathcal{T}^{H}_{ij}$ and $S_{ij} - \frac{1}{2}S g_{ij} = \mathcal{K} \mathcal{T}^{V}_{ij}$, with $\mathcal{K}$ the Einstein constant.
• The Maxwell equations for the electromagnetic field are derived as $F_{ij|k} + F_{jk|i} + F_{ki|j} = -\sum_{\{i,j,k\}} C_{imr} R^r_{jk} y^m$, where $F_{ij}$ are horizontal components and $f_{ij} = 0$.
• For $p=1$, the conservation laws are $\left[\frac{R+S}{2}\right]_{/1} = R^m_{1|m} - P^{(m)}_{(1)1}|^{(1)}_{(m)}$, and similar expressions for other components, ensuring energy-momentum conservation.
• For $p \geq 2$, the conservation laws simplify to $\tilde{\mathcal{T}}^\mu_{\beta/\mu} = 0$ and $\tilde{\mathcal{T}}^m_{j|m} = 0$, indicating natural conservation in the reduced form.
• In the autonomous case ($g_{ij} = g_{ij}(x^k)$), the conservation laws reduce to $\left[H^\mu_\beta - \frac{H+r}{2}\delta^\mu_\beta\right]_{/\mu} = 0$ and $\left[r^m_j - \frac{H+r}{2}\delta^m_j\right]_{|m} = 0$.
• The theory is shown to be a 'metrical-parametrized' field theory in the sense of Marsden, with invariance under the gauge group of the $J^1(T,M)$ bundle, confirming its geometric naturality.