Possible consistent extra time dimensions in the early universe

Abstract

Gravity cannot be quantized unless the quantized theory is cast on a manifold whose concomitant number of physical space dimensions and number of physical time dimensions correspond to physical reality, and not simply to the perception of reality. At present, the accepted number of physical time dimensions is dictated more by folklore than by science. In this paper we discuss a model of the early universe in which the number of physical time dimensions is four, and formulate Theorem[\ref{tj}], which underlies an explanation of why the extra time dimensions do not source unphysical effects. In this paper we describe a new model of gravitational inflation that is driven by dark energy and "mediated" by a real massless scalar inflaton field \(\varphi\) whose potential is identically equal to zero. The coupled Einstein gravitational and inflaton field equations are formulated on an eight-dimensional spacetime manifold of \textbf{four space} dimensions and \textbf{four time} dimensions. We find explicit solutions to these field equations that exhibit temporal exponential \textbf{deflation of three of the four time dimensions}, and then study the dynamics of a massive complex scalar field \(\psi\) that propagates on the background ground state Einstein gravitational field to determine whether its quantum fluctuations \(\delta \psi\) are stable or unstable. We compute explicit approximate solutions to the \(\delta \psi\) field equations that are \textbf{stable}, meaning that the quantum fluctuations \(\delta \psi\) of the field \(\psi\) do not grow exponentially with time. \textbf{Instabilities} driven by the momenta associated to the three extra time dimensions do not appear in the physical solutions of the field equations of this model.