Generalized harmonic spatial coordinates and hyperbolic shift conditions

We propose a generalization of the condition for harmonic spatial coordinates analogous to the generalization of the harmonic time slices introduced by Bona et al., and closely related to dynamic shift conditions recently proposed by Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic spatial coordinates imply a condition for the shift vector that has the form of an evolution equation for the shift components. We find that in order to decouple the slicing condition from the evolution equation for the shift it is necessary to use a rescaled shift vector. The initial form of the generalized harmonic shift condition is not spatially covariant, but we propose a simple way to make it fully covariant so that it can be used in coordinate systems other than Cartesian. We also analyze the effect of the shift condition proposed here on the hyperbolicity of the evolution equations of general relativity in 1+1 dimensions and 3+1 spherical symmetry, and study the possible development of blow-ups. Finally, we perform a series of numerical experiments to illustrate the behavior of this shift condition.