A positive semi-definite action in a Kaluza-Klein theory with compactification onto time-like extra dimensions

Abstract

The (4+ N)-dimensional theory is considered whose lagrangian function is L 4+N = √− g ̂ α R ̂ 2 , where R̂ is the Ricci scalar and α is a positive constant. The metric is g ̂ AB = diag(g ab, φ −1 g ̄ mn) . Dimensional reduction leads to an effective four-dimensional lagrangian of induced-gravity type, if L 4 = √−g (κ 1φ −(N−4) 2 + κ 2φ −(N−2) 2 R + φ −N 2 R 2 , where R is the four-dimensional Ricci scalar and κ 1 and κ 2 are constants. The positive semi-definiteness of L avoids the difficulties, pointed out recently by Horowitz and by Rubakov, which can arise in quantum cosmology when the (euclidean) action becomes negative. The compactification is onto a time-like internal space g ̄ mn , as suggested by Aref'eva and Volovich, giving a four-dimensional de Sitter space-time with φ = constant, which, however, is classically unstable on a time scale ∼ H −1. Decrease of the radius φ −1 2 of the internal space is ultimately halted by quantum effects, via some V( φ), and L 4 then includes the usual Hilbert term and a cosmological constant.