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In the early 1980s, Hartle and Hawking proposed that signature changes might be conceptually significant, paving the way for the “no boundary” proposal for the universe’s initial conditions. These singularity-free universes have no beginning, but they do have an origin of time. Mathematically, this involves signature-type changing manifolds where a Riemannian region is smoothly joined to a Lorentzian region at the transition surface where time begins. We present a transformation procedure to convert an arbitrary Lorentzian manifold into a singular signature-type changing manifold. Subsequently, we establish the local version of the Transformation Theorem, asserting that, under certain conditions, such a metric \((M,\tilde {g})\) can be derived from some Lorentz metric g through this transformation procedure.

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From Lorentzian Manifolds to Signature-Type Change with Singular Transverse Metrics

  • 会议论文

  • 发表日期：

    2025年

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  • Wolfgang Hasse,
  • Nathalie E. Rieger

摘要

In the early 1980s, Hartle and Hawking proposed that signature changes might be conceptually significant, paving the way for the “no boundary” proposal for the universe’s initial conditions. These singularity-free universes have no beginning, but they do have an origin of time. Mathematically, this involves signature-type changing manifolds where a Riemannian region is smoothly joined to a Lorentzian region at the transition surface where time begins. We present a transformation procedure to convert an arbitrary Lorentzian manifold into a singular signature-type changing manifold. Subsequently, we establish the local version of the Transformation Theorem, asserting that, under certain conditions, such a metric \((M,\tilde {g})\) can be derived from some Lorentz metric g through this transformation procedure.