Slicing Conditions

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The lapse function is \alpha and K is the trace of the extrinsic curvature. The time derivative operator \partial_\perp \equiv \partial_t - {\mathcal L}_\beta is used, where {\mathcal L}_\beta is the Lie derivative along the shift vector. The spatial covariant derivative is denoted  D_a.

Harmonic slicing

Harmonic time slices t = {\rm const} are defined by \nabla^\mu \nabla_\mu t = 0 where \nabla_\mu is the spacetime covariant derivative. In adapted coordinates,

\partial_\perp \alpha = -\alpha^2 K

1+log slicing

1+log slicing is defined by

 \partial_\perp \alpha = -2\alpha K

The name "1+log" is also used for the slicing condition  \partial_t \alpha = -2\alpha K, obtained by dropping the advection term above. The origin of the term "1+log" appears to come from yet another variant,  \partial_t \alpha = -2\alpha K + 2 D_a \beta^a. This latter form can be re-expressed as \partial_t\alpha = \partial_t(\ln g) and integrated to \alpha = f(x) + \ln g, where f(x) is a function of space. If f(x) equals unity, the lapse has the form "1+log".

Bona-Masso family

The Bona-Masso family of slicings is

 \partial_\perp \alpha = -\alpha^2 f(\alpha) K

where f(\alpha) is a free function of the lapse. The choice f = 2/\alpha gives 1+log slicing, and f = 1 gives harmonic slicing.

Specified lapse antidensity

Let the lapse antidensity  N \equiv \alpha/\sqrt{g} be a given function of spacetime, where g is the determinant of the spatial metric. Then

 \partial_\perp \alpha + \alpha^2 K = \alpha \partial_\perp(\ln N)

Maximal slicing

Let K = \partial_t K = 0. This gives

D^2\alpha & = & \alpha R  \ ,\quad  {\rm or} \\
D^2 \alpha & = & \alpha K_{ab}K^{ab}

where R is the spatial curvature scalar and K_{ab} is the extrinsic curvature. The two forms for maximal slicing are related by the Hamiltonian constraint.

Constant mean curvature

Choose K and \partial_t K to be constant in space. Then

   D^2 \alpha & = & \alpha(R + K^2) - \partial_t K \ ,\quad {\rm or} \\
   D^2 \alpha & = & \alpha K_{ab}K^{ab} - \partial_t K

The two forms are related by the Hamiltonian constraint.

K freezing

Set  \partial_t K = 0. Then

   D^2 \alpha & = & \alpha (R + K^2) + \beta^a D_a K \ ,\quad {\rm or}\\
   D^2 \alpha & = & \alpha K_{ab}K^{ab} + \beta^a D_a K 

Alternatively, set  \partial_\perp K = 0. Then

   D^2 \alpha & = & \alpha (R + K^2)  \ ,\quad {\rm or}\\
   D^2 \alpha & = & \alpha K_{ab}K^{ab} 

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

K driver

First order form:

  \partial_\perp \alpha + C_1 \alpha^2 + C_2 \alpha^2 (K - K_0) = 0

where C_1, C_2, and K_0 are constants. Second order form:

   \partial_t^2\alpha + C_1 \alpha \partial_t\alpha   = C_2 \alpha^2 \left[ 
      D^2 \alpha - \alpha K_{ab}K^{ab} - \beta^c D_c K\right]
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