Submanifolds in five-dimensional pseudo-Euclidean spaces and four-dimensional FRW universes

Abstract

Equations for submanifolds, which correspond to embeddings of the four-dimensional FRW universes in five-dimensional pseudo-Euclidean spaces, are presented in convenient form in general case. Several specific examples are considered.It is well known that in general case a four-dimensional pseudo-Riemannian manifold can be represented as a submanifold in a flat ten-dimensional space-time [1]. If the metric possesses additional symmetries the dimensionality of the ambient space-time may be smaller, for example, it is well known that the de Sitter space d S 4 can be represented as a hyperboloid in five-dimensional Minkowski space (for the first time explicit formulas for this embedding were presented in [2]). It is less known that the FRW (Friedmann-Robertson-Walker) space-times can also be embedded in five-dimensional Minkowski space (more generally, in pseudo-Euclidean spaces as we will show below). For the first time this fact was noticed almost 80 years ago in [3], where explicit formulas for the embeddings of the FRW universes were presented (see note D in [3]). In 1965 analogous formulas were presented in [4], where various embeddings of the solutions to equations of General Relativity in pseudo-Euclidean spaces were considered. Nevertheless, in spite of the recent activity in the field of embeddings of the four-dimensional General Relativity in five-dimensional flat or Ricci-flat space-times (see, for example, [5,6] and references therein), it looks as if

Embedding FLRW geometries in pseudo-Euclidean and anti–de Sitter spaces

“…which easily follows from (2.15) above. Note that (2.15) was explicitly derived in [10]. However, its interpretation as a unit-speed parametrized curve was not noted.…”

“…These were derived independently by Rosen, who constructed pseudo-Euclidean embeddings of many other known spacetimes and was unaware of Robertson's work [9]. Interestingly, the same formulae kept being rediscovered in various contexts until fairly recently, sometimes with erroneous or premature conclusions; see [10] for a comprehensive list of references. 2 In any case, the collective summary is as follows: all possible k = 0, 1 FLRW geometries, but only a certain type of the k = −1 (e.g.…”

Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems

“…(There are many papers on isometric embeddings in physics journals where some references may be found in [29], [57], [40]. Unfortunately, most of these papers are virtually non-existent, since they are not freely accessible on the web: they cannot be read and referred to.…”

“…40 That was meant to imply all conceivable versions of Nash's theorem; likely it does, but it is too heavy to be usable in practice. 41 Probabaly, the eventual general theorem, in order to be painlessly applicable to concrete problems, should be even more "abstract" and go significantly beyond the scope of Nash's construction, including, in particular, the variation suggested by Günther.…”

“…The continuity of the operator D −1 implied by Nash's estimates allows regularisation solutions of the equation D(f ) = g by 40 This was needed in [34] for construction of local isometric C ∞ -immersions X n → R s n −1 , where the obviouos inversion of L D comes by solution of ODE that is not a differential operator M .…”