Energy and Momentum Densities Associated with Solutions Exhibiting Directional Type Singularities

TL;DR

This paper calculates energy and momentum densities for a general static axially symmetric vacuum space-time using various energy-momentum complexes, revealing differences and similarities among these definitions, especially in the Curzon metric case.

Contribution

It compares multiple energy-momentum complexes in a Weyl metric space-time, highlighting their differences and conditions for agreement, particularly in the Curzon metric.

Findings

Landau-Lifshitz and Bergmann-Thomson give same momentum density but different energy densities.

In Curzon metric, energy densities agree only as R approaches infinity.

Different complexes yield varying energy densities, emphasizing the complexity of energy localization in GR.

Abstract

We obtain the energy and momentum densities of a general static axially symmetric vacuum space-time described by the Weyl metric, using Landau-Lifshitz and Bergmann-Thomson energy-momentum complexes. These two definitions of the energy-momentum complex do not provide the same energy density for the space-time under consideration, while give the same momentum density. We show that, in the case of Curzon metric which is a particular case of the Weyl metric, these two definitions give the same energy only when $R \to \infty$. Furthermore, we compare these results with those obtained using Einstein, Papapetrou and M{\o}ller energy momentum complexes.