Hamiltonian (ADM)

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Notation:

\kappa \equiv 8\pi G where G is Newton's constant. g_{ab} is the spatial metric, \alpha is the scalar lapse function, \beta^a is the shift vector, and D_a is the spatial covariant derivative. P^{ab} is the conjugate momentum, related to the extrinsic curvature by


\begin{array}{rcl}
 P^{ab} & = & \frac{1}{2\kappa} \sqrt{g} (K g^{ab}-K^{ab}) \\
 K_{ab} & = & \frac{\kappa}{\sqrt{g}} (Pg_{ab}-2P_{ab}) 
\end{array}

Hamiltonian

Time evolution is defined via Poisson brackets with the Hamiltonian

H = \int d^3x ( \alpha {\mathcal H} + \beta^a {\mathcal M}_a )

The Hamitonian and momentum constraints are


\begin{array}{rcl}
{\mathcal H} & \equiv & \frac{\kappa}{\sqrt{g}} ( 2 P^{ab} P_{ab} - P^2 ) - \frac{1}{2\kappa} \sqrt{g} R = 0  \\
{\mathcal M}_a & \equiv & -2D_b P^b_a = 0  
\end{array}

The fundamental Poisson brackets relations are  [g_{ij}(\vec x),P^{k\ell}(\vec y) ] = \delta_{(i}^k \delta_{j)}^\ell \delta^3(\vec x - \vec y) where  \delta^3(\vec x - \vec y) is the three--dimensional Dirac delta function.

Equations of motion:

In terms of the time derivative operator \partial_\perp \equiv \partial_t - {\mathcal L}_\beta, the ADM (Hamiltonian) equations are


\begin{array}{rcl}
\partial_\perp g_{ab} & = &  2\kappa\frac{\alpha}{\sqrt{g}} (2 P_{ab} - P g_{ab})   \\
\partial_\perp P^{ab} & = &  \kappa\frac{\alpha}{\sqrt{g}} (P^{cd}P_{cd} - P^2/2 ) g^{ab} 
     - 2\kappa\frac{\alpha}{\sqrt{g}} (2 P^{ac}P_c^b - P P^{ab}) - \frac{\alpha\sqrt{g}}{2\kappa} G^{ab} 
     + \frac{\sqrt{g}}{2\kappa} (D^a D^b \alpha - g^{ab} D_c D^c\alpha )  
\end{array}

where G^{ab} is the spatial Einstein tensor. The constraint evolution system is:


\begin{array}{rcl}
\partial_\perp{\mathcal H} & = &  \alpha D_a{\mathcal M}^a + 2 {\mathcal M}^a D_a\alpha  \\
\partial_\perp{\mathcal M}_a & = & {\mathcal H} D_a\alpha 
\end{array}

Other relations:

The time derivative of the extrinsic curvature is

\partial_\perp K_{ab} = \alpha K K_{ab} - 2\alpha K_{ac}K^c_b + \alpha R_{ab} - D_a D_b\alpha 
     - \frac{\alpha}{4} ( K^2 - K^{cd}K_{cd} + R) g_{ab}

where R_{ab} and R are the spatial Ricci tensor and spatial curvature scalar. The zero density constraints defined by York (see the gdot-Kdot system) are {\mathcal H}^{\rm Y} \equiv K^2 - K_{ab}K^{ab} + R = 0 and {\mathcal M}_a^{\rm Y} \equiv D_b K^b_a - D_a K  = 0 . They are related to the ADM constraints by {\mathcal H} = -\sqrt{g} {\mathcal H}^{\rm Y}/(2\kappa) and {\mathcal M}_a = \sqrt{g} {\mathcal M}_a^{\rm Y}/\kappa. The York form of the constraints evolved with the Hamiltonian (ADM equations) are


\begin{array}{rcl}
\partial_\perp{\mathcal H}^{\rm Y} & = &  \alpha K {\mathcal H}^{\rm Y} -4{\mathcal M}_a^{\rm Y} D^a\alpha - 2\alpha D^a {\mathcal M}_a^{\rm Y}  \\
\partial_\perp{\mathcal M}_a^{\rm Y} & = &  - \frac{1}{2}{\mathcal H}^{\rm Y} D_a\alpha + \alpha K {\mathcal M}_a^{\rm Y} 
\end{array}
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