On wormholes in spacetimes of embedding class one

Abstract

An $n$-dimensional Riemannian space is said to be of embedding class $m$ if $n+m$ is the lowest dimension of the flat space in which the given space can be embedded. A spherically symmetric spacetime of class two can be reduced to class one by a suitable transformation of coordinates. Applied to wormholes, given a well-defined shape function $b=b(r)$, the resulting wormhole has an event horizon and is therefore nontraversable. On a macroscopic scale, $b(r)$ can be replaced by $m(r)$, the effective mass of a spherical star of radius $r$ with $m(0)=0$, to yield a valid solution. Spacetimes of embedding class one have been used successfully for modeling compact stellar objects. On a microscopic scale, one can invoke noncommutative geometry to obtain a charged nontraversable wormhole, i.e., an Einstein-Rosen bridge, and hence a model for a charged particle.