Reduction of BTZ spacetime to hypersurfaces of foliation

We reduce the BTZ spacetime to two kinds of hypersurfaces of foliation: one having a fixed radial coordinate and the other a fixed angular coordinate. The radial reduction leads to a Liouville type theory, and confirms, from the first principle, the expectation laid out in the literature. In the other endeavor, the angular reduction of the 3D gravity is carried out in two different ways; the first again yields a Liouville type theory (different from that of the radial reduction) and the second yields a 2D interacting quantum field theory with quartic potential. Finally we discuss potential implications of our result for the Equivalence Principle and Purity of Hawking radiation.