[Paper Review] The definition of Finsler spacetime

E. Minguzzi|arXiv (Cornell University)|Dec 13, 2014

Advanced Differential Geometry Research45 references3 citations

TL;DR

This paper establishes the essential uniqueness of Finsler spacetime theories by proving that the widely used 'conic' Finsler Lagrangian—defined only over a conic subbundle—is equivalent to John Beem's classical Finsler Lagrangian defined over the entire slit tangent bundle. By showing that causality theory depends only on the future cone and that the conic case is a restriction of a Beem-type Lagrangian, the work confirms that modified Finsler spacetime models do not extend the theory but remain within its foundational framework.

ABSTRACT

In recent works the author developed the local and global causality theory of Finsler spacetimes. Subsequently other authors restated these results for a modified theory in which the Finsler Lagrangian is defined only over a conic subbundle, thus apparently departing from John Beem's classical definition of Finsler spacetime for which the Lagrangian is defined over the whole slit tangent bundle. It is shown here that this modified theory is no more general, since the `conic' Finsler Lagrangian is the restriction of a Beem's Lagrangian. Since causality theory depends on curves defined through the future cone, this work establishes the essential uniqueness of Finsler spacetime theories and Finsler causality.

Motivation & Objective

• To resolve confusion in the literature about whether conic Finsler spacetimes generalize Beem's classical Finsler spacetime definition.
• To clarify whether causality theory in Finsler spacetimes depends on the domain of the Lagrangian.
• To establish that the conic Finsler Lagrangian is not a new theory but a restriction of Beem's original formulation.
• To confirm the essential uniqueness of Finsler spacetime theories by showing causality is preserved under this restriction.

Proposed method

• Analyzing the geometric structure of Finsler Lagrangians defined on conic subbundles of the slit tangent bundle.
• Demonstrating that any conic Finsler Lagrangian can be extended to a globally defined Finsler Lagrangian on the full slit tangent bundle.
• Using the causal structure derived from the future cone to compare causality in conic and Beem-type Finsler spacetimes.
• Applying results from prior causality theory in Finsler spacetimes to show equivalence in causal properties.
• Proving that the restriction of a Beem-type Lagrangian to a conic subbundle yields the same causal curves as the conic model.
• Establishing that causality theory is invariant under this restriction, thus confirming theoretical equivalence.

Experimental results

Research questions

• RQ1 Is the conic Finsler Lagrangian a genuine generalization of Beem's classical Finsler spacetime definition?
• RQ2 Can causality in Finsler spacetimes be consistently defined using only the future cone, independent of the Lagrangian's domain?
• RQ3 Is the conic Finsler model fundamentally different from Beem's original theory, or is it merely a restricted case?
• RQ4 Does the restriction of a Beem-type Lagrangian to a conic subbundle preserve the causal structure of the spacetime?
• RQ5 What is the theoretical relationship between the conic Finsler model and Beem's classical Finsler spacetime in terms of causality?

Key findings

• The conic Finsler Lagrangian is not a new theory but the restriction of a Beem-type Finsler Lagrangian defined on the full slit tangent bundle.
• Causality in Finsler spacetimes depends only on the future cone, which remains invariant under the restriction to a conic subbundle.
• The modified conic Finsler theory does not extend the scope of Beem's classical Finsler spacetime theory.
• The causality theory developed for Beem's Finsler spacetimes applies directly to the conic model, confirming theoretical equivalence.
• The essential uniqueness of Finsler spacetime theories is established, as all such models are equivalent to Beem's original formulation.