This entry is devoted to a derivation of the Einstein constraint equations. In fact, this is an expanded version of the previous entry. For the sake of simplicity, here we only derive the Hamiltonian constant equation. The momentum constaint equation will be considered in the coming entry.
Before we start, let us recall the following form of the Einstein equation with the cosmological constant , that is
where is a constant. The above equation is understood over the Lorentzian manifold of the dimension . We shall use to denote .
Let us take a submanifold of of the dimension . It is well-known that the Levi-Civita connection verifies the following decomposition
for any smooth vector fields and tangent to and is the second fundamental form.
We now mention the so-called Gauss equation. Before we do that, let us recall the Riemann curvature tensor given by
Furthermore, there holds
Then the Gauss equation is given by
We now let be a local orthonormal frame field for . Using the Gauss equation, we arrive at
By definition of the scalar curvature, we have
Besides, by the definition of the (scalar-valued) second fundamental form with respect to the unit normal vector , i.e.,
The respective mean curvature which is nothing but the trace of the above second fundamental form is
In the case of a hypersurface, there holds
For the remaining term, we observe that
In other words, we have shown that
By definition of the Ricci curvature, we have
since . Therefore,
Thus, we have proved that
known as the Hamitonian constraint equation.
- Why do the Einsteins equations describe the propagation of wavelike phenomena?
- Derivation of the Einstein constraint equations: The momentum constraint
Reference: Justin Corvino, Introduction to General Relativity and the Einstein Constraint Equations, Lecture notes.