I point out a simple expression for the "Hubble" parameter $\mathscr{H}$, defined by Borde, Guth and Vilenkin (BGV) in their proof of past incompleteness of inflationary spacetimes. I show that the parameter $\mathscr{H}$ which an observer $O$ with four-velocity $\bf v$ will associate with a congruence $\bf u$ is equal to the fractional rate of change of the magnitude $\xi$ of the Jacobi field associated with $\bf u$, measured along the points of intersection of $O$ with $\bf u$, with its direction determined by $\bf v$. I then analyse the time dependence of $\mathscr{H}$ and $\xi$ using the geodesic deviation equation, computing these exactly for some simple spacetimes, and perturbatively for spacetimes close to maximally symmetric ones. The perturbative solutions are used to characterise the rms fluctuations in these quantities arising due to possible fluctuations in the curvature tensor.

中文翻译：

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BGV 定理、测地线偏差和量子涨落

我指出了“哈勃”参数 $\mathscr{H}$ 的一个简单表达式，该表达式由 Borde、Guth 和 Vilenkin (BGV) 在他们证明过去膨胀时空的不完整性中定义。我证明了参数 $\mathscr{H}$ 与四速度 $\bf v$ 的观察者 $O$ 将与同余 $\bf u$ 关联的参数 $\bf u$ 等于幅度 $\与 $\bf u$ 相关的 Jacobi 场的 xi$，沿 $O$ 与 $\bf u$ 的交点测量，其方向由 $\bf v$ 确定。然后，我使用测地线偏差方程分析 $\mathscr{H}$ 和 $\xi$ 的时间相关性，为一些简单的时空精确计算这些，并为接近最大对称时空的时空进行微扰。