Energy and momentum associated with a Static Axially Symmetric Vacuum Space-Time

TL;DR

This paper compares various energy-momentum complexes in calculating energy and momentum densities in static axially symmetric vacuum spacetimes, revealing differences and agreements among them.

Contribution

It provides a comparative analysis of multiple energy-momentum complexes applied to Weyl and Curzon metrics, highlighting their differences and conditions for agreement.

Findings

Different complexes yield different energy densities for Weyl metric.

Momentum densities are consistent across complexes.

Energy densities agree for Curzon metric only at infinity.

Abstract

We use the Einstein and Papapetrou energy-momentum complexes to calculate the energy and momentum densities of Weyl metric as well as Curzon metric. We show that these two different definitions of energy-momentum complexes do not provide the same energy density for Weyl metric, although they give the same momentum density. We show that, in the case of Curzon metric, these two definitions give the same energy only when $R \to \infty$. Furthermore, we compare these results with those obtained using Landau and Lifshitz, Bergmann and M{\o}ller.