Immersions of Einstein-Riemann spacetimes infive-dimensional flat spaces

Abstract

Gauge-theoretic constructs and Cartan's method of movingcoframe fields are used to obtain immersions of four-dimensionalEinstein-Riemann spacetimes in flat five-dimensional spaces. Ageneralization of Cayley's representation theorem (for matrix elements ofSO(n)) to the groups SO(2,3) and SO(1,4) leads to a choice of thefourimmersion parameters such that all salient geometric quantities areevaluated in terms of rational algebraic functions. After an analysis ofthe general immersion problem, considerations are restricted to classesof immersions for which the matrix of gauge curvature 2-forms is ofmaximal rank. All solutions of these immersion problems are shown to begenerated by vector spaces of 1-form fields, so that there is asuperposition principle at this level. These 1-form fields lead to familiesof associated metric and curvature tensor fields with specificcomposition laws. Several multiple-parameter families of immersions arecalculated explicitly. A spherically symmetric, deflation-inflation modelwith a central black hole is presented.