The problem of dynamical stability of spontaneous compactification in Kaluza–Klein models with vacuum corrections

A semiclassical model of a nonminimally coupled scalar field in a multidimensional space with spherically compactified additional dimensions is considered. It is noted that for the self-consistent description of time-dependent perturbations of the radius of the internal space one needs at least a complete adiabatic expansion of the vacuum energy-momentum tensor, including all higher derivatives of the metric. The proposed technique makes it possible to obtain such expansions linearized around an arbitrary (quasi)static solution. It is found that the frequency Fourier components of the energy-momentum tensor converge absolutely only in a finite disk of complex frequencies, and unique analytic continuation to the remainder of the complex plane is impossible. This means that rapid oscillations are nonlocal and can be investigated only nonperturbatively. Nevertheless, within the disk of absolute convergence there exist in general eigenfrequencies, and if these include complex frequencies, then local perturbation theory gives a proof of instability. As an illustration, the energy-momentum tensor for a six-dimensional spacetime is calculated.