Ngô Quốc Anh

December 7, 2012

Derivation of the Einstein constraint equations: The Hamitonian constraint

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 16:41

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 \Lambda, that is

\displaystyle \text{Ric}_{\overline g}-\frac{1}{2}\overline g \text{Scal}_{\overline g}+\Lambda\overline g = \kappa T

where \kappa is a constant. The above equation is understood over the Lorentzian manifold (V,\overline g) of the dimension n+1. We shall use \langle\cdot, \cdot\rangle to denote \overline g(\cdot, \cdot).

Let us take (M,g) a submanifold of V of the dimension n. It is well-known that the Levi-Civita connection \nabla verifies the following decomposition

\displaystyle\overline\nabla_X Y=\nabla_X Y + \mathrm{I\!I}(X,Y)

for any smooth vector fields X and Y tangent to M and \mathrm{I\!I} is the second fundamental form.

We now mention the so-called Gauss equation. Before we do that, let us recall the Riemann curvature tensor R given by

\displaystyle R(X,Y)Z = \nabla_X\nabla_Y Z-\nabla_Y\nabla_X Z - \nabla_{[X,Y]}Z.

Furthermore, there holds

\displaystyle\nabla_X Y = \top(\overline\nabla_X Y),\quad \mathrm{I\!I}(X,Y) = \bot(\overline\nabla_X Y).

Then the Gauss equation is given by

\displaystyle\left\langle {\overline R (X,Y)Z,W} \right\rangle = \left\langle {R\left( {X,Y)Z} \right),W} \right\rangle + \left\langle {\mathrm{I\!I}(X,Z)} , {\mathrm{I\!I}(Y,W)} \right\rangle - \left\langle {\mathrm{I\!I}(Y,Z)} , {\mathrm{I\!I}(X,W)} \right\rangle.

We now let e_1,...,e_n be a local orthonormal frame field for M. Using the Gauss equation, we arrive at

\displaystyle\left\langle {\overline R \left( {{e_i},{e_j}} \right){e_i},{e_j}} \right\rangle = \left\langle {R({e_i},{e_j}){e_i},{e_j}} \right\rangle + \left\langle {\mathrm{I\!I}({e_i},{e_i})} ,{\mathrm{I\!I}({e_j},{e_j})} \right\rangle - \left\langle {\mathrm{I\!I}({e_i},{e_j})}, {\mathrm{I\!I}({e_i},{e_j})} \right\rangle,

(more…)

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