Canonical formulation of the embedded theory of gravity equivalent to Einstein’s general relativity

Abstract

We study the approach in which independent variables describing gravity are functions of the space-time embedding into a flat space of higher dimension. We formulate a canonical formalism for such a theory in a form that requires imposing additional constraints, which are a part of Einstein's equations. As a result, we obtain a theory with an eight-parameter gauge symmetry. This theory becomes equivalent to Einstein's general relativity either after partial gauge fixing or after rewriting the metric in the form that is invariant under the additional gauge transformations. We write the action for such a theory.

Gravity as a field theory in flat space-time

“…We use the formula for the second fundamental form on a surface defined by the function y a (x) of embedding into a flat ambient space (an exposition of the embedding theory formalism can be found in [4] and in more detail in [9]…”

“…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.…”

“…That is, in addition to all the solutions of the Einstein equation, these equations also contain other (so-called extra) solutions. These solutions can be eliminated by introducing additional constraints into the theory as previously proposed in [2] and investigated in detail in [3], [4] for N = 10. 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.…”

New dynamical variables in Einstein’s theory of gravity

“…. , 10, are the immersion coordinates (sufficiently complete information about this approach can be found in [7], [8]). But the equations of motion in that theory differ from the HilbertEinstein equations.…”

“…For the authors of [7], this circumstance resulted in disappointment with this approach, while the authors of [8] present substantial arguments in favor of it. We liked the main formulas of the immersion theory very much, which is why we decided to modify them by regarding the f A μ not as the gradients of scalars but as independent vector fields.…”