Abstract
Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be required to have the same symmetry as the metric. On the other hand, it is possible to require the embedding to be unfolded so that the surface in the ambient space would occupy the subspace of the maximum possible dimension. In the weak gravitational field limit, such a requirement together with a large enough dimension of the ambient space makes embedding gravity equivalent to general relativity, while at lower dimensions it guarantees the linearizability of the equations of motion. We discuss symmetric embeddings for the metrics of flat Euclidean three-dimensional space and Minkowski space. We propose the method of sequential surface deformations for the construction of unfolded embeddings. We use it to construct such embeddings of flat Euclidean three-dimensional space and Minkowski space, which can be used to analyze the equations of motion of embedding gravity.
Highlights
According to the Janet–Cartan–Friedman (JCF) theorem [1], an arbitrary n-dimensional pseudo-Riemannian space can be locally isometrically embedded into the ambient flat space of dimension N n(n + 1)/2 with suitable signature
In this case, the unfolded embedding of the flat metric is chosen as the background in the decomposition (5), within the framework of the perturbation theory bμaν (4) can be removed from the RT equations (3) as a non-singular matrix that is a factor in a homogeneous equation
Isometric embeddings of flat metrics can be nontrivial, i.e., different from a plane in the ambient space. Such nontrivial embeddings are of interest from the point of view of describing gravity within the framework of the embedding theory
Summary
According to the Janet–Cartan–Friedman (JCF) theorem [1], an arbitrary n-dimensional pseudo-Riemannian space can be locally isometrically embedded into the ambient flat space of dimension N n(n + 1)/2 with suitable signature. After such a substitution the theory might change (additional solutions appear, see [11] for a discussion of gravity modifications resulting from differential transformations of field variables) It happens even if the number of new variables ya(x), which is equal to the number of ambient space dimensions, corresponds to the JCF theorem value N = 10 and does not differ from the number of the old metric variables gμν(x). This string-inspired approach was first proposed in [12] and was subsequently studied in a number of works [13,14,15,16,17,18,19,20] under the names like embedding theory, geodetic brane gravity and embedding gravity.
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