Embedding FLRW Geometries in Pseudo-Euclidean and Anti-de Sitter Spaces

Abstract

Contrary to the general consensus in the literature that Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) geometries are of embedding class one (i.e.,\ embeddable in one higher dimensional pseudo-Euclidean spaces), we show that the most general $k=0$ and $k=-1$ FLRW geometries are of embedding class two, and their corresponding pseudo-Euclidean spaces have strictly one and two negative eigenvalues, respectively. These are particular results that follow from the new perspective on FLRW embedding that we develop in this paper, namely that these embeddings are equivalent to unit-speed parametrized curves in two or three dimensions. A careful analysis of appropriate tensor fields then gives identical results and further explains why the class-two geometries remained hidden. However, the signatures of the embedding spaces, as well as the explicit embedding formulae, follow only from the curve picture. This also streamlines the comparatively difficult $k=0$ class and provides new explicit embedding formulae for it and reproduces known embedding formulae for the $k=1,-1$ classes. Embedding into anti-de Sitter space in one higher dimension can likewise be done by constructing associated curves. In particular, we find that all $k=1$ and mildly restricted subclasses of $k=0, -1$ geometries are embeddable in anti-de Sitter space in one higher dimension.