# Slicing Conditions

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## Contents

#### Notation

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

$\begin{array}{rcl} D^2\alpha & = & \alpha R \ ,\quad {\rm or} \\ D^2 \alpha & = & \alpha K_{ab}K^{ab} \end{array}$

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

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

The two forms are related by the Hamiltonian constraint.

#### K freezing

Set $\partial_t K = 0$. Then

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

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

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

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:

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