Gödel's universe and induced-matter theory

Abstract

We discuss Gödel's universe in the context of the induced-matter theory. We show that the problem of generating Gödel's metric from an extra dimension is equivalent to finding an embedding of Gödel's universe in a Ricci-flat five-dimensional space. On the other hand, according to the Campbell-Magaard theorem, any spacetime can be locally embedded into a five-dimensional pseudo-Riemannian Ricci-flat manifold. We obtain explicitly a global embedding of Gödel's universe which is Ricci-flat and has a non-Lorentzian signature of type (+ + − − −).

Curvature properties of Gödel metric

“…We note that [51] is the first publication, in which a semi-Riemannian manifold satisfying (8) was named the pseudosymmetric manifold. In [51] pseudosymmetric warped products with one-dimensional base manifold and (n− 1)-dimensional fiber, n ≥ 4, which is not a semi-Riemannian space of constant curvature, were investigated.…”

“…However, the converse statement is not true. For instance, the Schwarzschild spacetime, the Kottler spacetime and the Reissner-Nordström spacetime satisfy (8) with nonzero function L R [55] (see also [56,57]). We also mention that Friedmann-Lemaître-Robertson-Walker spacetimes are pseudosymmetric (cf.…”

“…However, both notions of pseudosymmetry are not equivalent. Throughout the paper we will confine the pseudosymmetry related to (8).…”

“…In 2001 Radojević [7] presented modification of Gödel metric in order to find out some other perfect fluid solutions. Induced matter theory and embedding of Gödel universe in five-dimensional Ricci flat space was studied by Fonseca-Neto et al [8] in 2005. Riemann extension of Gödel metric was considered by Dryuma [2] in 2005, and Dautcourt and Abdel-Megied [9] studied light cone of Gödel universe.…”

“…al. [8] in 2005. Riemann extension of Gödel metric was considered by Dryuma [2] in 2005, and Dautcourt et.…”