On the uniqueness of the space-time energy in General Relativity. The illuminating case of the Schwarzschild metric

TL;DR

This paper investigates the concept of energy in asymptotic Minkowskian space-times within General Relativity, demonstrating that the Schwarzschild metric can have zero intrinsic energy depending on the coordinate system used.

Contribution

It shows that the Schwarzschild metric can have vanishing intrinsic energy when using specific asymptotic coordinates, clarifying the coordinate dependence of energy definitions in General Relativity.

Findings

Schwarzschild metric can have zero energy in certain coordinates.

Different coordinate choices lead to different energy interpretations.

Intrinsic energy can vanish for Schwarzschild space-time.

Abstract

The case of asymptotic Minkowskian space-times is considered. A special class of asymptotic rectilinear coordinates at the spatial infinity, related to a specific system of free falling observers, is chosen. This choice is applied in particular to the Schwarzschild metric, obtaining a vanishing energy for this space-time. This result is compared with the result of some known theorems on the uniqueness of the energy of any asymptotic Minkowskian space, showing that there is no contradiction between both results, the differences becoming from the use of coordinates with different operational meanings. The suitability of Gauss coordinates when defining an {\em intrinsic} energy is considered and it is finally concluded that a Schwarzschild metric is a particular case of space-times with vanishing {\em intrinsic} 4-momenta.