Generalized Harmonic (second order)

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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

\Gamma_{cab} & \equiv & g_{cd} \Gamma^d_{ab}  \\
\Gamma_c & \equiv & g^{ab} \Gamma_{cab} 

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.


The constraints are:

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

The time evolution of the constraints is

   n^b\nabla_b C_a & = & 2 M_a +   \\
   n^b\nabla_b M_a & = & 
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