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

which yields

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

Therefore,

For the remaining term, we observe that

In other words, we have shown that

By definition of the Ricci curvature, we have

since . Therefore,

Since

we obtain

Thus, we have proved that

known as the Hamitonian constraint equation.

See also:

- 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.

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Comment by doanchi — December 7, 2012 @ 17:10

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Comment by Ngô Quốc Anh — December 7, 2012 @ 17:12

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Comment by doanchi — December 10, 2012 @ 12:17