# Generalized BSSN

#### 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 $\partial_\perp g$.