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We study quantum aspects of the Einstein gravity with one time-like and onespace-like Killing vector commuting with each other. The theory is formulatedas a $\coset$ nonlinear $\sigma$-model coupled to gravity. The quantum analysisof the nonlinear $\sigma$-model part, which includes all the dynamical degreesof freedom, can be carried out in a parallel way to ordinary nonlinear$\sigma$-models in spite of the existence of an unusual coupling. This meansthat we can investigate consistently the quantum properties of the Einsteingravity, though we are limited to the fluctuations depending only on twocoordinates. We find the forms of the beta functions to all orders up tonumerical coefficients. Finally we consider the quantum effects of therenormalization on the Kerr black hole as an example. It turns out that theasymptotically flat region remains intact and stable, while, in a certainapproximation, it is shown that the inner geometry changes considerably howeversmall the quantum effects may be.Comment: 16 pages, LaTeX. The hep-th number added on the cover, and minor  typos correcte

Field theoretical quantum effects on the Kerr geometry