Constructing a boosted, spinning black hole in the damped harmonic gauge

The damped harmonic gauge is important for numerical relativity computations based on the generalized harmonic formulation of Einstein's equations and is used to reduce coordinate distortions near binary black hole mergers. However, currently there is no prescription to construct quasiequilibrium binary black hole initial data in this gauge. Instead, initial data are typically constructed using a superposition of two boosted analytic single black hole solutions as free data in the solution of the constraint equations. Then, a smooth time-dependent gauge transformation is done early in the evolution to move into the damped harmonic gauge. Using this strategy to produce initial data in the damped harmonic gauge would require the solution of a single black hole in this gauge, which is not known analytically. In this work we construct a single boosted, spinning, equilibrium black hole in damped harmonic coordinates as a regular time-independent coordinate transformation from Kerr-Schild coordinates. To do this, we derive and solve a set of four coupled, nonlinear, elliptic equations for this transformation, with appropriate boundary conditions. This solution can now be used in the construction of damped harmonic initial data for binary black holes.