Dynamical perturbations of compactified space in a multidimensional model with nonlocal vacuum corrections

Time-dependent isotropic perturbations of the internal space $S^2$ are studied for a six-dimensional model with matter represented by a quantized scalar field. In the framework of “partial summation” of local vacuum corrections, an exact equation is obtained for the eigenfrequencies of the multidimensional universe. The solvability of this equation is proved numerically. Some general properties of the spectrum and details relating to the nonlocality of the vacuum are discussed. It is found that spontaneous compactification is unstable irrespective of the values of the constant of the nonminimal coupling. Direct calculations confirm the invalidity of the previously used approximation of weak nonstationarity, so we still do not yet possess a single example of semiclassically stable compactification.