# BSSN

#### 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: