Causal solutions of the Gödel type, for which the line element is ds2 = dt2 - 2b e mxdtdv - c e 2mxdv2 - dx2 - dz2 with c = 0, are known to exist for gravitational theories containing a cosmological constant Λ and quadratic higher-derivative terms defined by the Lagrangian L = -(1/2)κ-2(R + 2Λ) + A1R2 + A2RijRij. Here, we show that acausal solutions, for which c < 0, containing closed time-like lines, can be constructed only if A2 = 0. Extension of this analysis to the heterotic superstring theory, including a generic massless scalar field ϕ plus quadratic and quartic gravitational terms [Formula: see text] and [Formula: see text], again yields a causal solution with c = 0, and also Λ = 0, as required for anomaly freedom, while solutions with c < 0 are ruled out. More general rotational space–times appear to be intractable analytically, and therefore it remains a matter of conjecture that the heterotic superstring admits only classical Lorentzian solutions which respect causality. For the energy density ρ(ϕ) of the scalar field is positive semi-definite only when g00 ≥ 0, which is equivalent to the causality condition g11 ≤ 0 or c ≥ 0 in a Lorentzian space–time for which det gij < 0; while ρ(ϕ) is unbounded from below in the presence of closed time-like lines, when g11 > 0, implying instability of ϕ, which will react back on the metric until it becomes causal.
Higher Derivative Theory for Curvature Term Coupling with Scalar Field
