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A new exact time-dependent Kerr-like black hole solution is found on a Randall–Sundrum brane world spacetime. The solution is also valid on the Wick-rotated Euclidean counterpart space, so represents equally well a gravitational instanton, i.e. a bump in spacetime. Since the r-dependent part is determined by a first-order differential equation, one conjectures that the solution is a self-dual solution comparable with the Eguchi–Hanson solution. The zeroes and poles of the spacetime are determined by a quintic polynomial. To describe the Hawking particles, one uses the antipodal boundary condition on a complex Klein-bottle hypersurface ${ℂ}^1\times {ℂ}^1$. We used the Hopf fibration to get $S^2$ as the black hole horizon, where the centrix is not in a torus but in the Klein bottle. The twist fits very well with the antipodal identification of the point on the horizon. No “cut-and-paste” is necessary, so the Hawing particles remain pure without instantaneous information transport. A local observer passing the horizon will not notice a central singularity in suitable coordinates. The black hole paradoxes are also revisited in our new black hole model. A connection is made with the geometric quantization of ${ℂ}^1\times {ℂ}^1\sim S^3$, by considering the symplectic 2-form. The model can be easily extended to the non-vacuum situation by including a scalar field. Both the dilaton and the scalar field can be treated as quantum fields when approaching the Planck area.