The boundary of G？del's spacetime and the chronology protection conjecture

We present a homogenous anisotropic conformal spacetime manifold that provide an example of Hawking's chronology protection conjecture in three-dimensional gravity theory. The solution is based upon the fact that the seven-dimensional group of the automorphism of the Heisenberg motion group H1{\times}U(1), modulo discrete sub-group \Gamma, is the symmetry group of the sub-Riemannian (SR)- manifold, boundary of the Cauchy-Riemann (CR)-manifold, allowing the existence of positive mass, momentum, angular-momentum and timelike-translation. It is shown that many mirror symmetric self-similar G\"odel's surfaces are hidden behind a Cauchy spacelike surface so that causality violation is not visible from outside.