= Time-symmetric conformally flat vacuum initial data
{title2=$K_{ij}=0,\quad\gamma_{ij}=\psi^4\delta_{ij},\quad\Delta\psi=0$}
For a spatial initial slice with vanishing <extrinsic curvature>, the vacuum <momentum constraint> is automatic and the <Hamiltonian constraint> requires zero <Ricci scalar>. In three spatial dimensions, the indicated <conformally flat metric> has $\mathcal R=-8\psi^{-5}\Delta\psi$, with a nonzero <conformal factor>. Thus a <harmonic function> supplies the data on its domain. A positive example is $\psi=1+a/r$ on a punctured or exterior region. There is no nonconstant globally smooth example on all of Euclidean three-space tending to one at infinity, by the <maximum principle for harmonic functions>. Time symmetry of an initial slice does not imply that its entire evolved spacetime is static.
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