Summary
By interpreting the nonlinear σ-model as the Gauss map associated to a certain immersion, we derive an explicit connection between the energy-momentum tensor and the geometry of space-time. We are then able to point out what follows: i) the cosmological term is intimately connected with the mean curvature of the immersion, ii) the possible space-times are restricted by a requirement on the associated immersions, iii) the energy of the solution is given by the square of the mean curvature of the immersion.
Riassunto
Interpretando il modello σ non lineare come mappatura di Gauss associata ad una certa immersione, si deriva una connessione esplicita tra tensore energia-impulso e la geometria dello spazio-tempo. Siamo quindi in grado di porre in evidenza quanto segue: 1) il termine cosmologico è intimamente connesso con la curvatura media dell'immersione, 2) gli spazi-tempo possibili sono ristretti da un'esigenza riguardante le immersioni associate, 3) l'energia della soluzione è data dal quadrato della curvatura media dell'immersione.
Резюме
Интерпретируя нелинейную σ-модель, как Гауссово отображение, связанное с иммерсией, мы выводим явную связь между тензором энергии-имиульса и геометреней пространства-времени. Затем отмечается: 1) космологический член тесно связан с средней кривизной иммерсии; 2) возможные простванства-времена ограничены условием на ассоциированных иммерсиях; 3) энергия решения определяется квадратом средней кривизны иммерсии.
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Nielsen, B. The stress-energy-momentum tensor of the Gauss map in the extended nonlinear σ-model. Nuov Cim B 74, 159–166 (1983). https://doi.org/10.1007/BF02721674
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DOI: https://doi.org/10.1007/BF02721674