On the uniqueness of the space-time energy in General Relativity: the illuminating case of the Schwarzschild metric
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On the uniqueness of the space-time energy in General Relativity: the illuminating case of the Schwarzschild metric Ramon Lapiedra · Juan Antonio Morales-Lladosa
Received: 6 November 2012 / Accepted: 22 February 2013 © Springer Science+Business Media New York 2013
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 intrinsic energy is considered and it is finally concluded that a Schwarzschild metric is a particular case of space-times with vanishing intrinsic 4-momenta. Keywords Energy and asymptotic flatness · Schwarzschild metric · Weinberg complex
1 Introduction It has been largely discussed and carefully established that there is a sound definition of energy (and also of linear 3-momentum and angular 4-momentum) of any space-time which is asymptotically Minkowskian, once we have selected a symmetric complex, as for example, the one of Weinberg [1] or the other one of Landau and Lifshitz [2], or some other rather different, but likely equivalent, prescriptions (cf. [3–6] for instance).
R. Lapiedra · J. A. Morales-Lladosa (B) Departament d’Astronomia i Astrofísica, Universitat de València, Burjassot, València 46100, Spain e-mail: [email protected] R. Lapiedra e-mail: [email protected]
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R. Lapiedra, J. A. Morales-Lladosa
Whatever it be, it is generally assumed (see [1] for example) that in order to obtain a sound definition of this energy we must rely on some coordinate system which, fast enough, becomes a rectilinear one in the spatial infinity and then, to simplify the calculation, use the Gauss theorem to write the energy 3-volume integral as a 2-surface integral at this infinity (see [7,8] for a clear account on this and related topics). Nevertheless, this theorem can only be applied if the first derivatives of the field in the 3-volume integrand are continuous, though in some situations we can overcome these conditions by redefining, in the sense of distributions, non smooth enough derivatives, as it is for instance the case for an elementary charge in electrostatics. We will take the fact of these conditions into account along the present paper. Furthermore, is this sound definition of energy unique? According to some well known theorems [9–13], the answer is “yes”, provided that the metric approaches asymptotically the Minkowski metric fast enough, but not too fast (see, to begin with, the readable considerations presented in [1] and our “Appendix A” for some comments on it). We wil
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