Summary
An analysis is given of the fundamental question raised by Wheeler as to whether the Schwarzschild field, in the absence of a, finite interior solution, may be regarded as describing an everywhere empty space-time manifold. A model is constructed with a finite source tensor (the « island model »), and it is shown that one obtains a deltafunction point mass source tensor for the Schwarzschild field as a limiting case. It is also shown using Einstein’s pseudotensor that there is a contribution to the total energy of the Schwarzschild field that is not due to gravitational energy. In the course of the analysis, a subtraction formalism for handling gravitational energy divergences in polar co-ordinates is also developed. It is concluded that a source-free interpretation of the Schwarzschild field is not possible. A critique of the point mass is also given, and also an interpretation of the divergences as « gauge-jumps ».
Riassunto
Si analizza la questione della sorgente del campo di Schwarzschild in assenza d’una sorgente interna finita. Storicamente, questo campo è considerato come la soluzione delle equazioni di Einstein di una massa puntiforme, ma nell’interpretazione di Wheeler, c’è la soluzione dello spazio-tempo vuoto. Si esaminano le interpretazioni in contrasto dei due metodi. Primo, si costruisce un modello in oui si deriva la soluzione di Schwarzschild come limite di una soluzione con sorgente di massa distribuita. Secondo, si dimostra per mezzo del pseudotensore di Einstein che si trova un contributo all’energia totale non gravitazionale. Si fà questo calcolo nelle coordinate quasi rettangolari, e con un nuovo formalismo di sottrazione, nelle coordinate polari sferiche e nelle coordinate sotermiche non statiche. Si dimostra che è possibile eliminare in modo coerente le divergenze nelle coordinate polari. Si conclude che non è possibile considerare il campo di Schwarzschild come lo spazio-tempo vuoto. Si espone uno studio critico della massa puntiforme e anche un’interpretazione delle divergenze come « salti di gauge ».
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Tangherlini, F.R. Source of the Schwarzschild field. Nuovo Cim 38, 153–174 (1965). https://doi.org/10.1007/BF02750446
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DOI: https://doi.org/10.1007/BF02750446