As we have already seen in this post, the Einstein equations are essentially hyperbolic if one assumes that the harmonic condition holds, i.e. . This type of condition was first introduced by De Donder in 1921 and have played an important role in theoretical developments, notably in the Choquet-Bruhat work of the well-posedness of the Cauchy problem for Einstein equations.
The harmonic slicing is defined by requiring that the harmonic condition holds only for the coordinate , i.e. , leaving freedom to choose any coordinate , in each hypersurface .
Using the formula for the d’Alembertian operator, which is the Laplace-Beltrami operator in the Minkowski space, we obtain
that is to say
Thanks to the identity , one can see that
we find that
By expanding, we obtain
In view of the divergence for vector field, there holds
Thanks to the evolution equation of the spatial metric, i.e.
we find that
Since , we obtain
By collecting all above formulas, we find that
Using the formula for the Lie derivative for scalar functions, we get that
Hence, we have got an evolution equation for the lapse function . Thanks to
the above equation is nothing but
Thus, for the scalar density depending on the shift vector , the lapse function is
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- The evolution equations for the constraint equations
- decomposition of the spacetime metric