A Conceptual Introduction To Signature Change Through a Natural Extension of Kaluza-Klein Theory
Vincent Moncrief, Nathalie E. Rieger
TL;DR
The paper addresses how signature-changing geometries can emerge naturally within Kaluza-Klein frameworks by allowing the higher-dimensional Lorentzian manifold to develop a Cauchy horizon, which causes the $U(1)$-generated Killing field to transition from spacelike to timelike across the horizon and yields a quotient metric that changes from Lorentzian to Riemannian. It develops explicit Misner-type and generalized Taub-NUT constructions, showing that the base fields $g,A,\,\\Phi$ are singular at the horizon interface but can be transformed to a smooth transverse-type-changing metric $\bar{g}$, with the base equations switching from hyperbolic to elliptic across the horizon. The analysis reveals obstructions to projecting geodesics across the horizon (via the constant of motion $p_5$) and frames the phenomenon as a natural, holographic-like emergence of signature change from a globally smooth higher-dimensional geometry. Finally, it connects these geometric insights to the structure of Killing horizons: analytic compact Cauchy horizons are Killing horizons, and their symmetry properties (e.g., generated by a $U(1)$ leading to $T^{2}$ or $T^{3}$ isometries) underpin the feasibility and constrained nature of such signature-changing KK constructions.
Abstract
We propose an extension of basic Kaluza-Klein theory in which the higher-dimensional Lorentzian manifold develops a Cauchy horizon rather than remaining globally hyperbolic as in the conventional framework. In this setting, the $U(1)$-generating Killing field, assumed to exist in Kaluza-Klein theory, undergoes a transition in its causal character, from spacelike in the globally hyperbolic region to timelike in an acausal extension through a horizon. This yields a (lower-dimensional) quotient manifold whose metric changes signature from Lorentzian to Riemannian. In this way, one observes a singular, signature changing transition emerging rather naturally from the projection of a globally smooth, even analytic, Lorentzian geometry ``up in the bundle''. This reveals a ``signature change without signature change'' scenario -- a phrasing inspired by John Wheeler -- and extends the usual Kaluza-Klein framework in a conceptually natural direction.