协变符号变化引起的暴胀：一种几何机制
Inflation from Covariant Signature Change: A Geometric Mechanism
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中文总结 AI 辅助
提出一种协变机制，通过度规从欧几里得到洛伦兹的平滑符号变化驱动有限时段的加速膨胀，并导出暴胀持续时间的模型无关判据。
中文摘要 AI 辅助
我们提出一种协变机制，其中度规符号从欧几里得区域到洛伦兹区域的平滑变化驱动了一个有限时段的加速膨胀。这种转变由沿类时曲线的标量插值函数编码，发生在余维一超曲面上，在该曲面上连续度规退化但曲率不变量保持有限，因此该曲面是曲率正则的。利用这种协变延拓，我们将连续度规的爱因斯坦张量重写为局域化的、依赖于插值函数的有效源，用于过渡后的洛伦兹分支，从而得到一个在穿越附近支持的纯几何应力张量。在洛伦兹区域，我们推导出一个与模型无关的局域加速判据：当插值函数的斜率超过由初始超曲面上的外曲率和空间里奇曲率确定的临界值时，暴胀持续；当该不等式首次饱和时，暴胀结束。标准光滑轮廓（tanh、广义逻辑函数和幂律/反正切）给出了加速阶段固有时长的闭式表达式，表明对于固定的几何数据，轮廓形状控制着这一时长。该构造提供了一条从正则欧几里得起源到早期洛伦兹加速膨胀相的无奇点、无暴胀子路径，且与无边界型边界条件兼容。
英文摘要
We present a covariant mechanism in which a smooth change of metric signature, from a Euclidean to a Lorentzian regime, drives a finite interval of accelerated expansion. The transition, encoded by a scalar interpolator along a timelike congruence, occurs on a codimension-one hypersurface where the continued metric is degenerate but curvature invariants remain finite, so the surface is curvature-regular. Using this covariant continuation, we rewrite the Einstein tensor of the continued metric as a localized, interpolator-dependent effective source for the post-transition Lorentzian branch, yielding a purely geometric stress tensor supported near the crossing. In the Lorentzian regime, we derive a model-independent, local criterion for acceleration: inflation persists while the interpolator's slope exceeds a critical value fixed by the extrinsic curvature and the spatial Ricci curvature on the initial hypersurface, and ends when this inequality is first saturated. Standard smooth profiles (tanh, generalized logistic, and power-law/arctan) admit closed-form expressions for the proper-time duration of the accelerated epoch, showing that, for fixed geometric data, the profile shape controls this duration. The construction provides a non-singular, inflaton-free route from a regular Euclidean origin to an early Lorentzian phase of accelerated expansion, in a manner compatible with no--boundary--type boundary conditions.
