# Generalized Harmonic (second order)

Let $g_{ab}$ denote the spacetime metric and $\nabla_a$ denote the spacetime covariant derivative. The Ricci tensor can be written as

$R_{ab} \equiv -\frac{1}{2} g^{cd} \partial_c \partial_d g_{ab} + \partial_{(a} \Gamma_{b)} - \Gamma^c_{ab} \Gamma_c + g^{cd} g^{ef} \left[ \partial_c g_{ae} \partial_d g_{bf} - \Gamma_{ace}\Gamma_{bdf} \right]$

where $\Gamma^c_{ab}$ are the usual Christoffel symbols and

$\begin{array}{rcl} \Gamma_{cab} & \equiv & g_{cd} \Gamma^d_{ab} \\ \Gamma_c & \equiv & g^{ab} \Gamma_{cab} \end{array}$

Define the covariant vector

$C_a \equiv H_a + \Gamma_a$

where $H_a$ are given functions of the spacetime coordinates and metric. In the vacuum case, the generalized harmonic equations are

$R_{ab} - \nabla_{(a} C_{b)} + \gamma \left[ n_{(a}C_{b)} - \frac{1}{2} g_{ab} n^c C_c \right] = 0$

where $n_a$ is the unit normal to a spacelike foliation of spacetime and $\gamma$ is a constant parameter. These are a set of wave equations for the components of the spacetime metric:

$g^{cd} \partial_c \partial_d g_{ab} = -2 [\partial_{(a} H_{b)} - \Gamma^c_{ab} H_c ] + 2g^{cd} g^{ef} \left[ \partial_c g_{ae} \partial_d g_{bf} - \Gamma_{acd}\Gamma_{bdf} \right] + 2 \gamma \left[ n_{(a}C_{b)} - \frac{1}{2} g_{ab} n^c C_c \right]$

When $C_a = 0$ the generalized harmonic equations are equivalent to the Einstein equations.

## Constraints

The constraints are:

$\begin{array}{rcl} 0 & = & C_a \ \equiv\ H_a + \Gamma_a \\ 0 & = & M_a \ \equiv\ n^b \left[ R_{ab} - \frac{1}{2} g_{ab} R \right] \end{array}$

The time evolution of the constraints is

$\begin{array}{rcl} n^b\nabla_b C_a & = & 2 M_a + \\ n^b\nabla_b M_a & = & \end{array}$