Spontaneous compactification and the higher-dimensional vacuum Einstein equations

The vacuum Einstein equations in (4+m+n) dimensions are displayed for a class of metrics obtained by the Kaluza-Klein geometrisation of the (4+m)-dimensional Einstein-Maxwell equations. Consistency problems are solved by requiring that the metric be one of two types. In both cases the field equations split into spacetime and internal field equations, and the full metric includes components which can be interpreted as internal mutually orthogonal electromagnetic fields. The authors show that spontaneous compactification in vacuum occurs only if observable spacetime has a huge cosmological constant, or if the internal spaces have no global isometries, or if the number of internal dimensions is very large. (author)