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About Signature-Change Metrics on Manifolds

Javier Lafuente-López

TL;DR

The article develops a one-parameter family of Lorentz-Riemann signature-change metrics on manifolds, extending Kossowski's transverse type-change framework to Σ^{\alpha}-spaces. It defines α-transversality, radical-geodesic data, and Σ-Adapted structures, establishing that normality is equivalent to the existence of a differentiable radical geodesic distribution crossing the singular hypersurface, yielding local representations $g = \Sigma g_{ij}dx_i dx_j - \operatorname{sign}(x_m)|x_m|^{1/\alpha}dx_m^2$ near $\Sigma$ (with α=1 recovering the classical case). A key analytic component shows that a generalized function $F(\lambda,t)$ is differentiable for all $r>-1$, proven via Baldomero's Theorem and an inductive scheme involving auxiliary sequences, ensuring the required smoothness for the local geometry. The work discusses limitations (not all Σ^{\alpha}-spaces admit radical geodesic distributions) and outlines extensions to negative α (polar) regimes, highlighting potential implications for cosmology and signature-change phenomena in geometry.

Abstract

We provide a one-parameter family of Lorentz-Riemann signature-change models of metric manifolds. This family generalizes the Kossowski's signature type-changihg stablished in [9]. Simple local expressions are sought around the hypersurface of change.

About Signature-Change Metrics on Manifolds

TL;DR

The article develops a one-parameter family of Lorentz-Riemann signature-change metrics on manifolds, extending Kossowski's transverse type-change framework to Σ^{\alpha}-spaces. It defines α-transversality, radical-geodesic data, and Σ-Adapted structures, establishing that normality is equivalent to the existence of a differentiable radical geodesic distribution crossing the singular hypersurface, yielding local representations near (with α=1 recovering the classical case). A key analytic component shows that a generalized function is differentiable for all , proven via Baldomero's Theorem and an inductive scheme involving auxiliary sequences, ensuring the required smoothness for the local geometry. The work discusses limitations (not all Σ^{\alpha}-spaces admit radical geodesic distributions) and outlines extensions to negative α (polar) regimes, highlighting potential implications for cosmology and signature-change phenomena in geometry.

Abstract

We provide a one-parameter family of Lorentz-Riemann signature-change models of metric manifolds. This family generalizes the Kossowski's signature type-changihg stablished in [9]. Simple local expressions are sought around the hypersurface of change.
Paper Structure (15 sections, 12 theorems, 101 equations)

This paper contains 15 sections, 12 theorems, 101 equations.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. $\Sigma^{\alpha}$-spaces.
  4. Conclussion.
  5. Discussion.
  6. APPENDIX: Baldomero's Theorem.
  7. Preliminaries.
  8. Notations.
  9. Calculation Rules.
  10. Rule C1
  11. Rule H.
  12. Rule P.
  13. Proof of Theorem \ref{['Baldo gen']}
  14. End of the Proof of Theorem \ref{['Baldo gen']}
  15. Proof of Lemma \ref{['L1']}

Key Result

Proposition 6

If $\Sigma:_{s}\sigma=0$ and $\varphi:M_{0}\rightarrow\mathbb{R}$ is another equation around $p$ for $\Sigma$ (i.e. $\Sigma:\varphi=0$), then there exists a differentiable $h:M_{0}\rightarrow\mathbb{R}$ such that $\varphi=h\sigma$; furthermore, $\Sigma:_{s}\varphi=0 \Leftrightarrow h$ is non-zero at

Theorems & Definitions (25)

  • Definition 1
  • Remark 2: Notational convention
  • Definition 3
  • Remark 4
  • Definition 5
  • Proposition 6
  • Remark 7
  • Remark 8
  • Definition 9
  • Theorem 10
  • ...and 15 more