MathSciDoc: An Archive for Mathematician ∫

In a recent paper we presented a model for a unified quantization of gravity with other fundamental sources of nature. After quantization we had to solve a Wheeler-DeWitt equation which was a hyperbolic equation in a fiber bundle which was equipped with a Lorentzian metric with a time function $t$ ranging from $0$ to infinity. The Lorentzian metric had a big bang singularity in $t=0$ and the coefficients of the hyperbolic operator also inherited this singularity. The solutions of the Wheeler-DeWitt equation were products of spatial and temporal eigenfunctions and in order to prove that these solutions also experience a big bang singularity in $t=0$ the temporal eigenfunctions $w(t)$ had to become unbounded if $t$ tends to $0$. In our former paper this was only proved in special cases where $w$ could be expressed with the help of Bessel functions but not in general. In the present paper we prove that also in the general case the temporal solutions become unbounded near $t=0$, or more precisely: $\lim_{t\ra 0} (\abs w^2+t^2 \abs{\dot w}^2)=\un$ and $\limsup_{t\ra 0}\abs w^2=\un$.

quantization of gravity, quantum gravity, big bang, asymptotic behaviour near a singularity, solutions of the Wheeler-DeWitt equation, Prüfer substitution