Nonexistence of Conformally Flat Slices in Kerr and Other Stationary Spacetimes

It is proved that stationary solutions to the vacuum Einstein field equations with a nonvanishing angular momentum have no Cauchy slice that is maximal, conformally flat, and nonboosted. The proof is based on results coming from a certain type of asymptotic expansion near null and spatial infinity—which also show that the development of Bowen-York–type data cannot have a development admitting a smooth null infinity—and from the fact that stationary solutions do admit a smooth null infinity.

Figure 1

The region of the unphysical spacetime near spatial and null infinities. The conformal factor leading to this representation is given by $\Theta =\omega {\vartheta }^{-2}(1-{\tau }^2)$ with $\omega =\vartheta /|d\vartheta |$.