DOES THE HETEROTIC SUPERSTRING THEORY CONTAIN CLOSED TIME-LIKE LINES?

Causal solutions of the Godel type, for which the line element is ds(2) = dt(2) - 2be(mx) dtdv-ce(2mx)dv(2) - dx(2) - dz(2) with c = 0, are known to exist for gravitational theories containing a cosmological constant. and quadratic higher-derivative terms defined by the Lagrangian L = -(1/2)kappa(-2)(R + 2 Lambda) + A(1)R(2) + A(2)R(ij)R(ij). Here, we show that acausal solutions, for which c < 0, containing closed time-like lines, can be constructed only if A(2) = 0. Extension of this analysis to the heterotic superstring theory, including a generic massless scalar field phi plus quadratic and quartic gravitational terms R-2 = R-2 - RijRij and R-4, again yields a causal solution with c - 0, and also Lambda - 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 rho(phi) of the scalar field is positive semi-definite only when g(00) >= 0, which is equivalent to the causality condition g(11) <= 0 or c >= 0 in a Lorentzian space-time for which det g(ij) < 0; while rho(phi) is unbounded from below in the presence of closed time-like lines, when g(11) > 0, implying instability of phi, which will react back on the metric until it becomes causal.