A couple of days ago, we showed that the Einstein equations are essentially hyperbolic under the harmonic gauge. To be precise, the solvability of the equations
is equivalent to solving the following hyperbolic system
provided for all . Later on, when we consider the Cauchy problem for the Einstein equations on some appropriate framework, the first and second fundamental forms need to verify some constraint equations, [here and here].
By tracing the above equation, we obtain
which helps us to write
In the vacuum case, the above equation is nothing but
The spacetime we choose will be the product . What we really need is to construct a metric on such that the Einstein equations fulfill. In fact, thanks to the hyperbolic system showed above, we aim to find a suitable initial condition such that
- The gauge condition can propagate in time;
- The hyperbolic system is solvable for small time .
While the latter is standard (we shall touch this issue later), the former needs to study and this is the main point of this notes.
In this entry, given a solution of the constraint equations on a manifold of the dimension , we shall construct an appropriate spacetime of the dimension . Recall that
Initially, we set
As a consequence of the choice above, we also have for non-zero . To fully construct the initial condition, we still need to find .
First, in view of the gauge condition, , we find that
Therefore, at , we get that
Hence, in order to guarantee that at for all , we first select such that at . Once this task is done, we can determine at such that at .
In other words, the initial data that preserves the gauge condition can be found in this way. Within a small time, the hyperbolic system mentioned above always admits a solution . That is to say that the reduced Einstein equations have solution, i.e.,
where the -tensor is given by
However, it is not necessary to have that solves the Einstein equations unless the gauge condition remains valid within the small time. We shall prove this affirmatively.
We now consider how could the gauge condition propagate in time so long as the metric solves the reduced equations. As we shall see, all satisfy a homogeneous linear wave equation which is a consequence of the Bianchi identities with vanishing initial time derivatives which come from the constraint equations.
First, since the metric solves
and thanks to the Einstein equation for the curvature, we immediately have
Let us write
Using this, we have
where is the trace of . Since the Einstein tensor is divergence free, we obtain
Keep in mind that . A simple calculation shows that
we have proved that
The above long calculation also shows that
at . Hence, at , we get
By using , i.e. the Hamiltonian constraint, we obtain . We still need to prove that for all . Indeed, thanks to for any , we know that
Since this is true for all and the matrix is invertible, we have that all vanish at for all . Since satisfy a homogeneous linear hyperbolic system with vanishing initial data, vanish identically by the uniqueness for solutions of the Cauchy problem for the hyperbolic evolutions.