Shift Conditions

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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.

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