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The case of asymptotic Minkowskian space-times is considered. A special classof asymptotic rectilinear coordinates at the spatial infinity, related to aspecific system of free falling observers, is chosen. This choice is applied inparticular to the Schwarzschild metric, obtaining a vanishing energy for thisspace-time. This result is compared with the result of some known theorems onthe uniqueness of the energy of any asymptotic Minkowskian space, showing thatthere is no contradiction between both results, the differences becoming fromthe use of coordinates with different operational meanings. The suitability ofGauss coordinates when defining an {\em intrinsic} energy is considered and itis finally concluded that a Schwarzschild metric is a particular case ofspace-times with vanishing {\em intrinsic} 4-momenta.

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