The purpose of this article is to study the geometry of static space–times. We show that the energy density and pressure vanish on the boundary of a perfect fluid space–time, provided that it satisfies a suitable condition. Moreover, we provide an upper bound volume growth for geodesic balls of the base of static vacuum space similar to a classical result due to Bishop. In addition, we derive a weak version of the maximum principle of Omori–Yau at infinity for such spaces.

中文翻译：

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3流形上静态真空空间的测地线球的体积增长

本文的目的是研究静态时空的几何。我们证明，只要满足合适的条件，能量密度和压力就会在理想的流体时空边界上消失。此外，我们为静态真空空间底部的测地线球提供了上限体积增长，这与Bishop的经典结果类似。此外，我们推导了此类空间在无穷远处的大森-丘最大原理的一个弱形式。