Il Nuovo Cimento B (1971-1996)

, Volume 32, Issue 2, pp 381–388 | Cite as

Structure of the Einstein tensor for class-1 embedded space-time

  • J. Krause


Continuing previous work, some features of the flat embedding theory of class-1 curved space-time are further discussed. In the two-metric formalism provided by the embedding approach, the Gauss tensor obtains as the flat-covariant gradient of a fundamental vector potential. The Einstein tensor is then examined in terms of the Gauss tensor. It is proved that the Einstein tensor is divergence free in flat space-time,i.e. a true Lorentz-covariant conservation law for the Einstein tensor is shown to hold. The form of the Einstein tensor in flat space-time also appears as a canonical energy-momentum tensor of the vector potential. The corresponding Lagrangian density, however, does not provide us with a set of field equations for the fundamental vector potential; indeed, the Euler-Lagrange «equations» collapse to a useless identity, while the Lagrangian density has the form of a flat divergence.

Структура тензора Эйнштейна для включенного пространства-времени класса 1


Продолжая предыдущую работу, проводится дальнейщий анализ некоторых особенностей теории плоского включения для искривленного пространства-времени класса 1. В формализме, обеспечивающем подход включения, получается тензор Гаусса, как плоский ковариантный градиент основного векторного потенциала. Затем исследуется тензор Эйнштейна в терминах тензора Гаусса. Доказывается, что тензор Эйнштейна свободен от расходимостей в плоском пространстве-времени, т.е. показывается, что для тензора Эйнштейна справедлив истинный Лорентц-ковариантный закон сохранения. Форма тензора Эйнштейна в плоском пространстве-времени выступает как канонический тензор энергии-импульса векторного потенциала. Однако соответствующая плотность Лагранжиана не дает нам систему уравнений поля для основного векторного потенциала. Действительно, «уравнения» Эйлера-Лагранжа коллапсируют в бесполезное тождество, тогда как плотность Лагранжиана имеет вид плоской дивергенции.


Continuando lavori precedenti, si discutono ulteriormente alcune caratteristiche della teoria dell’incastonamento piano dello spazio-tempo curvo di classe 1. Nel formalismo di due metriche dato dall’approccio dell’incastonamento, si ottiene il tensore di Gauss come il gradiente piano covariante di un potenziale vettoriale fondamentale. Si esamina quindi il tensore di Einstein in termini del tensore di Gauss. Si dimostra poi che il tensore di Einstein è privo di divergenze nello spazio-tempo piano, cioè si dimostra che vale una vera legge di conservazione covariante secondo Lorentz per il tensore di Einstein. La forma del tensore di Einstein nello spazio-tempo piano appare anche come un tensore canonico dell’energia-impulso del potenziale vettoriale. La densità lagrangiana corrispondente, però, non ci fornisce un sistema di equazioni di campo per il potenziale vettoriale fondamentale; infatti le «equazioni» di Eulero-Lagrange collassano in un–inutile identità, mentre la densità lagrangiana ha la forma di una divergenza piana.


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Copyright information

© Società Italiana di Fisica 1976

Authors and Affiliations

  • J. Krause
    • 1
  1. 1.Departamento de Física Aplicada, Facultad de IngenieríaUniversidad Central de VenezuelaCaracasVenezuela

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