  We study quantum aspects of the Einstein gravity with one time-like and one
space-like Killing vector commuting with each other. The theory is formulated
as a $\coset$ nonlinear $\sigma$-model coupled to gravity. The quantum analysis
of the nonlinear $\sigma$-model part, which includes all the dynamical degrees
of freedom, can be carried out in a parallel way to ordinary nonlinear
$\sigma$-models in spite of the existence of an unusual coupling. This means
that we can investigate consistently the quantum properties of the Einstein
gravity, though we are limited to the fluctuations depending only on two
coordinates. We find the forms of the beta functions to all orders up to
numerical coefficients. Finally we consider the quantum effects of the
renormalization on the Kerr black hole as an example. It turns out that the
asymptotically flat region remains intact and stable, while, in a certain
approximation, it is shown that the inner geometry changes considerably however
small the quantum effects may be.
