arXiv 2210.011522022-10-03gr-qc

Levi-Civita时空的Komar质量密度的几何和物理解释

Geometrical and physical interpretation of the Levi-Civita spacetime in terms of the Komar mass density

Bence Racskó, László Á. Gergely

备注：: 16 pages, 8 figures

摘要：

我们重新解释的圆柱对称，静态真空Levi-Civita度规，已知的外尔，爱因斯坦-罗森，或卡斯纳类坐标。尽管无限轴源，我们通过紧化和紧化半径的随后的爆破推导出它的科马尔质量密度。我们表明，在爱因斯坦-罗森框架中计算的Komar质量密度$μ_K$，作为度量参数时，具有许多优点。它消除了参数空间的双重覆盖，在平坦时空中消失，当它很小时，它对应于无限弦的质量密度。在全面分析了局部和全局几何之后，我们继续对列维-奇维塔时空进行物理解释。首先，我们证明了牛顿引力是吸引力，其大小随着所有正的$μ_K$单调增加，在“径向”方向上渐近到适当距离的倒数。其次，我们揭示了附近测地线之间的潮汐力（因此爱因斯坦意义上的引力）在$μ_K=1/2$达到最大值，然后渐近地减小到零。因此，从物理学的观点来看，列维-奇维塔时空的科马尔质量密度包含两个贡献：牛顿引力和加速度效应。$μ_K$的增加加强了牛顿引力，但也使磁力线越来越平行，最终通过等效原理将牛顿引力转化为纯加速度场，将列维-奇维塔时空转化为平坦的林德勒时空。在几何图像中，$μ_K$从零增加到$\infty$，时空的平面部分变形为越来越深的漏斗，最终退化为圆柱拓扑。

英文摘要：

We revisit the interpretation of the cylindrically symmetric, static vacuum Levi-Civita metric, known in either Weyl, Einstein-Rosen, or Kasner-like coordinates. Despite the infinite axis source, we derive its Komar mass density through a compactification and subsequent blowing up of the compactification radius. We show that, the Komar mass density $μ_K$ calculated in the Einstein-Rosen frame, when employed as the metric parameter, has a number of advantages. It eliminates double coverages of the parameter space, vanishes in flat spacetime and when small, it corresponds to the mass density of an infinite string. After a comprehensive analysis of the local and global geometry, we proceed with the physical interpretation of the Levi-Civita spacetime. First we show that the Newtonian gravitational force is attractive and its magnitude increases monotonically with all positive $μ_K$, asymptoting to the inverse of the the proper distance in the "radial" direction. Second, we reveal that the tidal force between nearby geodesics (hence gravity in the Einsteinian sense) attains a maximum at $μ_K=1/2$ and then decreases asymptotically to zero. Hence, from a physical point of view the Komar mass density of the Levi-Civita spacetime encompasses two contributions: Newtonian gravity and acceleration effects. An increase in $μ_K$ strengthens Newtonian gravity but also drags the field lines increasingly parallel, eventually transforming Newtonian gravity through the equivalence principle into a pure acceleration field and the Levi-Civita spacetime into a flat Rindler-like spacetime. In a geometric picture the increase of $μ_K$ from zero to $\infty$ deforms the planar sections of the spacetime into ever deepening funnels, eventually degenerating into cylindrical topology.

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