Generalized BSSN

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Definitions and Notation:

The Generalized 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 variables g_{ab}, \varphi, K, and A_{ab} are related to 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}/g}\right)^{-2/3} g_{ab}^{\rm phys} \\
\varphi & = & \frac{1}{6} \ln \left(\sqrt{g_{\rm phys}/g}\right) \\
K & = & g^{ab}_{\rm phys} K_{ab}^{\rm phys} \\
A_{ab} & = & \left(\sqrt{g_{\rm phys}/g}\right)^{-2/3} \left(K_{ab}^{\rm phys} - \frac{1}{3} g_{ab}^{\rm phys} K \right)
\end{array}

where g is the determinant of the conformal metric and g_{\rm phys} is the determinant of the physical metric. Note that g_{ab} is a type 0-2 tensor, \varphi is a scalar, K is a scalar, and A_{ab} is a type 0-2 tensor.

The conformal connection functions \Gamma^a 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} = -\frac{1}{\sqrt{g}}\partial_a\left(\sqrt{g} g^{ac} \right)

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

Matter is defined by the mass density \rho = T_{\mu\nu} u^\mu u^\nu, the momentum density j_\mu = -\perp_\mu^\sigma T_{\sigma\nu} u^\nu, and the spatial stress s_{\mu\nu} = \perp_\mu^\sigma \perp_\nu^\rho T_{\sigma\rho} . Here, T_{\mu\nu} is the stress-energy-momentum tensor, u^\mu is the future pointing unit normal to the spacelike hypersurfaces, and \perp^\nu_\mu = \delta^\nu_\mu - u_\mu u^\nu is the projection tensor for spacelike hypersurfaces.

Choose units with 8\pi G = 1, where G is Newton's constant.

Equations of motion:

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


\begin{array}{rcl}
    \partial_\perp  {g}_{ab}  & = & \frac{1}{3} g_{ab}\, \partial_\perp\ln g -2{\alpha} {A}_{ab} \\
    \partial_\perp {A}_{ab} & = & \frac{1}{3} {A}_{ab}\, \partial_\perp \ln g 
        -2{\alpha} {A}_{ac}{A}^c_{b} 
       + {\alpha} {A}_{ab} K   \\
       & &  + e^{-4{\varphi}}  \left[ -2{\alpha} D_a D_b {\varphi} 
         + 4{\alpha} D_a{\varphi} D_b{\varphi} + 4 D_{(a}{\alpha} D_{b)} {\varphi} 
         - D_a D_b{\alpha} + {\alpha} R_{ab} - {\alpha} s_{ab}  \right]^{\rm TF} \\
    \partial_\perp  {\varphi} 
        & = &  -\frac{1}{12} \, \partial_\perp\ln g - \frac{1}{6} {\alpha} K \\
    \partial_\perp  K  & = &  \frac{1}{3} {\alpha} K^2 + {\alpha} {A}_{ab} {A}^{ab} 
        - e^{-4{\varphi}} \left( D^2{\alpha} + 2 D^a{\alpha} D_a{\varphi} 
           \right) + \frac{1}{2}{\alpha} (s + \rho)  \\
       \partial_\perp \Gamma^a  & = & -\frac{1}{3} g^{bc}\Gamma^a_{bc}\,\partial_\perp\ln g 
        - \frac{1}{6} g^{ab} \partial_b \partial_\perp\ln g  - 2 {A}^{ab} \partial_b{\alpha} 
        + 2{\alpha} \left[ 6 {A}^{ab}\partial_b{\varphi} + \Gamma^a_{bc} {A}^{bc} 
          - \frac{2}{3} g^{ab}\partial_b K - g^{ab}j_b \right] 
\end{array}

Indices are raised and lowered with the conformal metric. The superscript TF denotes the trace-free part. The Ricci tensor is


  R_{ab} = -\frac{1}{2} g^{cd} \partial_c \partial_d g_{ab} + g_{c(a}\partial_{b)}\Gamma^c 
      + g^{cd} \left(\Gamma^e_{cd} \Gamma_{(ab)e}+ 2 \Gamma^e_{c(a} \Gamma_{b)ed} + \Gamma^e_{ac} \Gamma_{ebd} \right)

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} 
      + 2g_{c(a} \partial_{b)}\beta^c \\
   {\mathcal L}_\beta \varphi & = & \beta^c\partial_c\varphi \\
   {\mathcal L}_\beta K & = &  \beta^c\partial_c K \\
   {\mathcal L}_\beta A_{ab} & = & \beta^c\partial_c A_{ab} 
      + 2A_{c(a} \partial_{b)}\beta^c \\
   {\mathcal L}_\beta \Gamma^a & = & \beta^c\partial_c \Gamma^a - g^{bc}\Gamma^d_{bc} \partial_d\beta^a + g^{bc} \partial_b \partial_c \beta^a
\end{array}

The Lie derivative of \Gamma^a is computed using the fact that Lie derivatives and partial derivatives commute.

Conformal Invariance

The equations of motion are invariant under conformal transformations:


\begin{array}{rcl}
   g_{ab} & \to & \xi^4 g_{ab} \\
   \varphi & \to & \varphi - \ln\xi \\
   A_{ab} & \to & \xi^4 A_{ab} \\
   K & \to & K \\
   \Gamma^a & \to & \xi^{-4} \Gamma^a - 2\xi^{-5} g^{ab} \partial_b \xi
\end{array}

The ambiguity is removed from the equations of motion by specifying the evolution of g. There are two natural choices. The Eulerian case is \partial_\perp g = 0. The Lagrangian case is \partial_t g  = 0, which is equivalent to  \partial_\perp g = -2D_a\beta^a.

Constraints:

The GBSSN constraints in vacuum are:


\begin{array}{rcl}
        {\mathcal H} & = & \frac{2}{3} K^2 - A_{ab} A^{ab} + e^{-4\varphi} 
                \left( R - 8 D^2\varphi - 8 D_c\varphi D^c\varphi \right) \\
        {\mathcal M}_a & = & D_b A^b_a + 6A_a^b D_b\varphi - \frac{2}{3} D_a K \\
        {\mathcal G}^a & = & \Gamma^a +  \partial_b \left(
                \sqrt{g} g^{ab} \right) /\sqrt{g} 
\end{array}

The spatial curvature R is built from the conformal metric only, so \Gamma^a does not appear anywhere in {\mathcal H}. The constraint evolution system is


\begin{array}{rcl}
        \partial_t {\mathcal H} & = & \beta^c\partial_c {\mathcal H} + \frac{2}{3} \alpha K {\mathcal H} - 4 e^{-4\varphi} \left( 
                D^a\alpha + \alpha D^a\varphi \right) {\mathcal M}_a 
                - 2\alpha e^{-4\varphi} D^a{\mathcal M}_a - 2\alpha e^{-4\varphi} A^a_b 
                \partial_a {\mathcal G}^b \\
        \partial_t {\mathcal M}_a & = &  \beta^c\partial_c {\mathcal M}_a + {\mathcal M}_c\partial_a \beta^c
            - \frac{1}{3} {\mathcal H} D_a\alpha
                + \frac{\alpha}{6} D_a {\mathcal H} + \alpha K {\mathcal M}_a 
                + e^{-6\varphi} D_c \left[ \alpha e^{2\varphi} ( g^{cb} g_{d(a} 
                \partial_{b)}{\mathcal G}^d )^{\rm TF} \right] \\
        \partial_t {\mathcal G}^a & = & \beta^c\partial_c {\mathcal G}^a + 2\alpha g^{ab}{\mathcal M}_b 
\end{array}

for any choice of \partial_\perp g.

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