PRAMANA c Indian Academy of Sciences Vol. 77, No. 3 — journal of September 2011 physics pp. 415–428 Isometric embeddings in cosmology and astrophysics GARETH AMERY∗, JOTHI MOODLEY and JAMES PAUL LONDAL Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa ∗Corresponding author. E-mail:
[email protected] Abstract. Recent interest in higher-dimensional cosmological models has prompted some signifi- cant work on the mathematical technicalities of how one goes about embedding spacetimes into some higher-dimensional space. We survey results in the literature (existence theorems and simple explicit embeddings); briefly outline our work on global embeddings as well as explicit results for more com- plex geometries; and provide some examples. These results are contextualized physically, so as to provide a foundation for a detailed commentary on several key issues in the field such as: the mean- ing of ‘Ricci equivalent’ embeddings; the uniqueness of local (or global) embeddings; symmetry inheritance properties; and astrophysical constraints. Keywords. Embedding theory; cosmology; astrophysics. PACS Nos 04.20.Jb, 04.20.Ex, 04.50.−h 1. Introduction Extensions to general relativity often involve extra dimensions. Inspired by string theory [1], D-brane theory [2] and Horava–Witten theory [3], there has been a great deal of inter- est in extra-dimensional effective theories, evidenced by phenomenological models such as the Randall–Sundrum [4], Arkani–Hamed–Dimopoulos–Dvali [5] and Dvali–Gabadadze– Porrati [6] scenarios, which attempt to tackle problems such as the mass-hierarchy problem, the origins of the primordial spectrum, the inflationary field and dark energy.