  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.
