Uniqueness theorems for the Lanczos BF formulation

Establish explicit uniqueness theorems for the Lanczos BF formulation of vacuum General Relativity, using its equations of motion to determine whether solutions are uniquely specified under appropriate initial or boundary conditions.

Background

The paper reformulates vacuum General Relativity as a BF theory in which a gauge vector field derived from the Lanczos potential encodes the nontrivial gravitational degrees of freedom. Its dynamics are governed by the BF equations of motion and, in Lorentz gauge, a source-free wave equation for Ricci-flat spacetimes.

Unlike standard four-dimensional General Relativity, for which uniqueness theorems are established in vacuum and in certain matter-coupled settings, the proposed formulation has no explicit uniqueness results. The authors identify proving such theorems as a natural next step, making the uniqueness question an unresolved problem for this formulation.

References

Unlike classical 4D General Relativity, which possesses well-established uniqueness theorems in vacuum and for certain matter content, our formulation currently lacks explicit uniqueness results. However, the simplicity of the equations of motion makes proving such theorems a natural next step.

— The Lanczos potential in the Plebanski paradigm of gravity  (2609.29434 - Espín et al., 24 Sep 2026) in Discussion