# Shift Conditions

## Contents

#### Notation

For the minimal strain and minimal distortion conditions, $g_{ab}$ denotes the spatial metric. For the Gamma freezing and Gamma driver conditions, $g_{ab}$ denotes the conformal metric. The spatial covariant derivative is $D_a$, the extrinsic curvature is $K_{ab}$, the shift vector is $\beta^a$ and the lapse function is $\alpha$.

#### Minimal Strain

Extremize $\int d^3 x\, \Theta_{ab}\Theta^{ab}$ with respect to the shift, where $\Theta_{ab} \equiv \frac{1}{2} \partial_t g_{ab} = -\alpha K_{ab} + D_{(a}\beta_{b)}$. This gives the condition:

$D^b D_{(a}\beta_{b)} = D^b(\alpha K_{ab})$

#### Minimal Distortion

Extremize $\int d^3 x\, \Sigma_{ab}\Sigma^{ab}$ with respect to the shift, where $\Sigma_{ab} \equiv \Theta_{ab} - g_{ab}\Theta/3$ is the trace-free part of $\Theta_{ab}$. This gives the condition:

$D^b \left[ D_{(a}\beta_{b)} - \frac{1}{3} D_c\beta^c g^{ab} \right] = D^b \left[ \alpha\left(K_{ab} - \frac{1}{3} g_{ab} K \right) \right]$

#### Gamma freezing

The Gamma freezing condition is $\partial_t \Gamma^a = 0$ where $\Gamma^a \equiv -\partial_b g^{ab}$ are the conformal connection functions and $g_{ab}$ is the conformal metric with unit determinant, $g=1$. Explicitly,

$0 = 2\alpha \Gamma^c_{ab}A^{ab} -\frac{4}{3}\alpha g^{ca} \partial_a K + 12\alpha A^{ca} \partial_a\varphi - g^{ab}\Gamma^d_{ab} \partial_d\beta^c + \frac{2}{3} g^{ab} \Gamma^c_{ab} \partial_d\beta^d + \beta^a\partial_a \Gamma^c + g^{ab} \partial_a \partial_b \beta^c + \frac{1}{3} g^{ca}\partial_a\partial_b \beta^b - 2 A^{ca}\partial_a \alpha$

where the physical metric is defined by $e^{4\varphi}g_{ab}$ and the physical extrinsic curvature is defined by $e^{4\varphi} (A_{ab} + K g_{ab}/3)$. (See the BSSN system of evolution equations.)

#### Gamma driver

The Gamma driver condition is

$\begin{array}{rcl} \partial_t \beta^a & = & \beta^c\partial_c \beta^a + \frac{3}{4} B^a \\ \partial_t B^a & = & \beta^c\partial_c B^a + (\partial_t \Gamma^a - \beta^c\partial_c \Gamma^a) - \eta B^a \end{array}$

where $\Gamma^a \equiv -\partial_b g^{ab}$ are the conformal connection functions built from the conformal metric $g_{ab}$. The conformal metric has unit determinant, $g=1$. The numerical parameter $\eta$ is usually set to a value between zero and ${\rm few} \times 0.1$. The term $\partial_t\Gamma^a$ is given by

$\partial_t \Gamma^a = 2\alpha \Gamma^c_{ab}A^{ab} -\frac{4}{3}\alpha g^{ca} \partial_a K + 12\alpha A^{ca} \partial_a\varphi - g^{ab}\Gamma^d_{ab} \partial_d\beta^c + \frac{2}{3} g^{ab} \Gamma^c_{ab} \partial_d\beta^d + \beta^a\partial_a \Gamma^c + g^{ab} \partial_a \partial_b \beta^c + \frac{1}{3} g^{ca}\partial_a\partial_b \beta^b - 2 A^{ca}\partial_a \alpha$

where the physical metric is defined by $e^{4\varphi}g_{ab}$ and the physical extrinsic curvature is defined by $e^{4\varphi} (A_{ab} + K g_{ab}/3)$. (See the BSSN system of evolution equations.)

Variants of the Gamma driver condition are obtained by dropping one or more of the advection terms $\beta^c\partial_c \beta^a$, $\beta^c\partial_c B^a$, or $\beta^c\partial_c \Gamma^a$.