The lapse function is and is the trace of the extrinsic curvature. The time derivative operator is used, where is the Lie derivative along the shift vector. The spatial covariant derivative is denoted .
Harmonic time slices are defined by where is the spacetime covariant derivative. In adapted coordinates,
1+log slicing is defined by
The name "1+log" is also used for the slicing condition , obtained by dropping the advection term above. The origin of the term "1+log" appears to come from yet another variant, . This latter form can be re-expressed as and integrated to , where is a function of space. If equals unity, the lapse has the form "1+log".
The Bona-Masso family of slicings is
where is a free function of the lapse. The choice gives 1+log slicing, and gives harmonic slicing.
Specified lapse antidensity
Let the lapse antidensity be a given function of spacetime, where is the determinant of the spatial metric. Then
Let . This gives
where is the spatial curvature scalar and is the extrinsic curvature. The two forms for maximal slicing are related by the Hamiltonian constraint.
Constant mean curvature
Choose and to be constant in space. Then
The two forms are related by the Hamiltonian constraint.
Set . Then
Alternatively, set . Then
In each case, the two forms are related by the Hamiltonian constraint.
First order form:
where , , and are constants. Second order form: