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

Abstract

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 $\mu_K$ 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 ${\mu}_K$, 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 $\mu_K=1/2$ 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 $\mu_K$ 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 $\mu_K$ from zero to $\infty$ deforms the planar sections of the spacetime into ever deepening funnels, eventually degenerating into cylindrical topology.