BSSN

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Definitions and Notation:

The BSSN variables are the conformal metric g_{ab}, the conformal factor \varphi, the trace of the extrinsic curvature K, the trace-free part of the extrinsic curvature A_{ab}, and the conformal connection functions \Gamma^a. The conformal metric has unit determinant, g = 1. The variables g_{ab}, \varphi, K, and A_{ab} are defined in terms of the physical spatial metric g_{ab}^{\rm phys} and physical extrinsic curvature K_{ab}^{\rm phys} by


\begin{array}{rcl}
g_{ab}^{\rm phys} & = &  e^{4\varphi} g_{ab} \\
K_{ab}^{\rm phys} & = &  e^{4\varphi} \left( A_{ab} + \frac{1}{3} g_{ab} K \right) 
\end{array}

Equivalently,


\begin{array}{rcl}
g_{ab} & = & \left(\sqrt{g_{\rm phys}}\right)^{-2/3} g_{ab}^{\rm phys} \\
\varphi & = & \frac{1}{6} \ln \left(\sqrt{g_{\rm phys}}\right) \\
K & = & g^{ab}_{\rm phys} K_{ab}^{\rm phys} \\
A_{ab} & = & \left(\sqrt{g_{\rm phys}}\right)^{-2/3} \left(K_{ab}^{\rm phys} - \frac{1}{3} g_{ab}^{\rm phys}\right)
\end{array}

where g_{\rm phys} is the determinant of the physical metric. Note that g_{ab} is a type 0-2 tensor density of weight -2/3, \varphi is the logarithm of a scalar density of weight 1/6, K is a scalar, and A_{ab} is a type 0-2 tensor density of weight -2/3.

The conformal connection functions are defined in terms of the "conformal Christoffel symbols" \Gamma^c_{ab} \equiv \frac{1}{2} g^{cd} (\partial_a g_{db} + \partial_b g_{ad} - \partial_d g_{ab}) by


\Gamma^c \equiv g^{ab}\Gamma^c_{ab} = -\partial_a g^{ac}

The lapse function is \alpha and the shift vector is \beta^a.

Equations of motion:

In terms of the time derivative operator \partial_\perp \equiv \partial_t - {\mathcal L}_\beta, the BSSN equations are


\begin{array}{rcl}
  \partial_\perp g_{ab} & = & -2\alpha A_{ab} \\
  \partial_\perp \varphi & = & -\frac{1}{6}\alpha K \\
  \partial_\perp K & = &  \frac{1}{3} \alpha K^2 + \alpha A_{ab} A^{ab} 
      - \nabla^a\nabla_a \alpha \\
  \partial_\perp A_{ab} & = & -2\alpha A_{ac}A^c_b + \alpha K A_{ab} + e^{-4\varphi} \left[
        - \nabla_a \nabla_b \alpha + \alpha R_{ab} \right]^{\rm TF} \\ 
  \partial_t\Gamma^c & = & 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
\end{array}

where \nabla_a and R_{ab} are the covariant derivative and Ricci tensor built from the physical metric. The indices on A_{ab} are raised with the conformal metric. Indices on \nabla_a are raised with the physical metric. The superscript TF denotes the trace-free part.

In terms of BSSN variables, the physical Ricci tensor is


\begin{array}{rcl}
   R_{ab} & = &  -\frac{1}{2} g^{cd} \partial_c\partial_d g_{ab} + g_{c(a}\partial_{b)}\Gamma^c 
     + g^{cd}\Gamma_{cd}^e \Gamma_{(ab)e} + g^{cd} \left( 2\Gamma^e_{c(a} \Gamma_{b)ed} 
     + \Gamma^e_{ac}\Gamma_{edb}\right)  \\
    & & -2(\partial_a\partial_b \varphi - \Gamma_{ab}^c\partial_c\varphi) 
       - 2 g_{ab} g^{cd} (\partial_c \partial_d \varphi - \Gamma_{cd}^e\partial_e\varphi) 
      + 4\partial_a\varphi \partial_b\varphi - 4g_{ab}g^{cd} \partial_c\varphi \partial_d\varphi
\end{array}

where \Gamma_{cab} \equiv g_{cd}\Gamma^d_{ab}.

The Lie derivatives contained in the operators \partial_\perp on the left-hand sides of the equations of motion are:


\begin{array}{rcl}
   {\mathcal L}_\beta g_{ab} & = &  \beta^c\partial_c g_{ab} - \frac{2}{3}g_{ab}\partial_c\beta^c 
      + 2g_{c(a} \partial_{b)}\beta^c \\
   {\mathcal L}_\beta \varphi & = & \beta^c\partial_c\varphi + \frac{1}{6}\partial_c\beta^c \\
   {\mathcal L}_\beta K & = &  \beta^c\partial_c K \\
   {\mathcal L}_\beta A_{ab} & = & \beta^c\partial_c A_{ab} - \frac{2}{3}A_{ab}\partial_c\beta^c 
      + 2A_{c(a} \partial_{b)}\beta^c
\end{array}

Constraints:

The BSSN constraints are:

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