Constraint Algebra for Regge-Teitelboim Formulation of Gravity

Abstract

We consider the formulation of the gravity theory first suggested by Regge and Teitelboim where the space-time is a four-dimensional surface in a flat ten-dimensional space. We investigate a canonical formalism for this theory following the approach suggested by Regge and Teitelboim. Under constructing the canonical formalism we impose additional constraints agreed with the equations of motion. We obtain the exact form of the first-class constraint algebra. We show that this algebra contains four constraints which form a subalgebra (the ideal), and if these constraints are fulfilled, the algebra becomes the constraint algebra of the Arnowitt-Deser-Misner formalism of Einstein's gravity. The reasons for the existence of additional first-class constraints in the canonical formalism are discussed.

The Ambiguity in the Definition and Behavior of the Gravitational and Cosmological ‘Coupling Constants’ in the Theory of Induced Gravity

“…From (53), taking into account (54), follows µ r1 = 0. These conditions are obtained from the requirement of existence and matching solutions Z = const of Equations (45) and (55).…”

“…We consider this model as an intermediate stage before formulating a consistent field theory that 250 takes into account the approaches of local isometric embedding's methods. These methods were considered in [44][45][46][47]. From these researches it follows that each manifold requires a separate study on "embedding" , which makes it difficult to develop a general theory used for comparison with the observed data.…”

Gravity as a field theory in flat space-time

“…We note that this value of N is distinguished because in accordance with the Janet and Cartan theorem [5], [6] (see, e.g., Remark 18 in [7]), an arbitrary four-dimensional Riemannian space can be locally isometrically embedded in just the ten-dimensional space. The correct form of the algebra of first-class constraints, which appears in the canonical formalism for the embedding theory when imposing additional constraints eliminating extra solutions, was found and investigated in [4], [8]. In what follows, we can use an analogous approach for eliminating extra solutions in the theory proposed here because this theory also admits extra solutions.…”

Гравитация Как Теория Поля В Плоском Пространстве-Времени

“…Отметим, что такое значение N выделено тем, что согласно теореме Жане и Картана [5], [6] (см., например, [7], примечание 18) произвольное четырехмерное риманово пространство может быть локально изомет-рически вложено именно в десятимерное пространство. Правильный вид алгебры связей первого рода, которая возникает в каноническом формализме для теории вложения при использовании дополнительных связей, исключающих лишние реше-ния, был найден и исследован в работах [4], [8]. В дальнейшем аналогичный подход к исключению лишних решений может быть использован и для теории, предлага-емой в настоящей работе, поскольку в ней тоже возникают лишние решения.…”

“…Ве-личины R и L m инвариантны (см. замечание после формулы (8)), но детерминант w неинвариантен. Из формул (3), (4) легко увидеть, что при преобразовании (2) под знаком интеграла в (26) появляется множитель в виде модуля якобиана этого пре-образования.…”