# Generalized Harmonic (second order)

### From GRwiki

Let denote the spacetime metric and denote the spacetime covariant derivative. The Ricci tensor can be written as

where are the usual Christoffel symbols and

Define the covariant vector

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

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

When the generalized harmonic equations are equivalent to the Einstein equations.

## Constraints

The constraints are:

The time evolution of the constraints is