# Slicing Conditions

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## Contents |

#### Notation

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 slicing

Harmonic time slices are defined by where is the spacetime covariant derivative. In adapted coordinates,

#### 1+log slicing

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".

#### Bona-Masso family

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

#### Maximal slicing

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.

#### K freezing

Set . Then

Alternatively, set . Then

In each case, the two forms are related by the Hamiltonian constraint.

#### K driver

First order form:

where , , and are constants. Second order form: