Equilibrium initial data for moving puncture simulations: The stationary 1+log slicing

We propose and explore a "stationary 1+log" slicing condition for the construction of solutions to Einstein's constraint equations. For stationary spacetimes, these initial data will give a stationary foliation when evolved with "moving puncture" gauge conditions that are often used in black hole evolutions. The resulting slicing is time-independent and agrees with the slicing generated by being dragged along a time-like Killing vector of the spacetime. When these initial data are evolved with moving puncture gauge conditions, numerical errors arising from coordinate evolution are minimized. In the construction of initial data for binary black holes it is often assumed that there exists an approximate helical Killing vector that generates the binary's orbit. We show that, unfortunately, 1+log slices that are stationary with respect to such a helical Killing vector cannot be asymptotically flat, unless the spacetime possesses an additional axial Killing vector.