Hamiltonian Formulation of General Relativity

1. The ADM Picture: Variables and Dynamics

Let's start by visually summarizing the 3+1 decomposition we've developed.

Our task is to find the equations that govern these variables. These equations split into two groups: six evolution equations and four constraint equations. Today, we focus on the constraints.

2. Derivation I: Projection of the Einstein Field Equations

The most direct way to find the constraints is to take the full 4D Einstein Field Equations (EFE) and project them onto the directions normal and tangent to our spatial slice ${\Sigma }_t$.

Let's write the EFE as:

where we use the superscript ${}^{(4)}$ to emphasize 4D quantities. It's often easier to work with the trace-reversed form:

where $T=g^{\mu \nu }{T}_{\mu \nu }$.

We also need to decompose the stress-energy tensor $T_{\mu \nu }$ relative to an observer moving with 4-velocity $n^{\mu }$ (our normal observer):

• Energy Density: $E\equiv T_{\mu \nu }{n}^{\mu }{n}^{\nu }$
• Momentum Density: $p_{\alpha }\equiv -T_{\mu \nu }{n}^{\mu }{\gamma }_{\alpha }^{\nu }$
• Spatial Stress Tensor: $S_{\alpha \beta }\equiv T_{\mu \nu }{\gamma }_{\alpha }^{\mu }{\gamma }_{\beta }^{\nu }$

With these definitions, we can project the EFE.

Let's analyze these two results.

The Hamiltonian Constraint

Projecting the EFE twice along the normal vector $n^{\mu }$ (i.e., calculating $n^{\mu }{n}^{\nu }{G}_{\mu \nu }=8\pi n^{\mu }{n}^{\nu }{T}_{\mu \nu }$) and using the Gauss-Codazzi relations from the previous lesson yields:

This is the Hamiltonian constraint (or energy constraint). Let's interpret its terms:

• ${}^{(3)}R$: The intrinsic Ricci scalar curvature of the 3D spatial slice.
• $K^2-K_{ij}{K}^{ij}$: Terms quadratic in the extrinsic curvature, representing the "kinetic energy" of the gravitational field (since $K_{ij}$ is related to ${\dot{\gamma }}_{ij}$).
• $16\pi E$: The energy density of matter and radiation on the slice.

This equation is a powerful statement about the allowed geometry of space. It dictates that the spatial curvature, its rate of change, and the energy density it contains are not independent but must satisfy this specific relationship. It is the GR equivalent of Gauss's law in electromagnetism, but for gravity.

The Momentum Constraint

Projecting the EFE once along the normal and once along the slice (i.e., calculating ${\gamma }_{\alpha }^{\mu }{n}^{\nu }{G}_{\mu \nu }=8\pi {\gamma }_{\alpha }^{\mu }{n}^{\nu }{T}_{\mu \nu }$) gives:

This is the momentum constraint (or diffeomorphism constraint). Here, $D_j$ is the 3D covariant derivative compatible with ${\gamma }_{ij}$. This equation relates the spatial derivatives of the extrinsic curvature to the momentum density of matter $p_i$. It ensures that momentum is conserved on the slice.

The full derivations of these equations involve careful application of the Gauss-Codazzi relations. You can see the details in Appendix D of the recommended text if you're interested in the algebraic steps.

3. Derivation II: The ADM Hamiltonian Formalism

While direct projection is insightful, a more profound derivation comes from the action principle. This approach, central to modern physics, reveals the deep structure of GR as a constrained Hamiltonian system, directly analogous to systems you've studied in classical mechanics.

Our goal is to rewrite the Einstein-Hilbert action in 3+1 variables, identify the canonical variables, and construct the Hamiltonian.

Let's summarize the key results of this derivation.

1. The ADM Action: After substituting the 3+1 decomposition of ${}^{(4)}R$ into the Einstein-Hilbert action $S_{EH}=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}{}^{(4)}R$, and dropping boundary terms, one arrives at the ADM action:

   $$S_{ADM}=\frac{1}{16\pi G}\int dt{\int }_{{\Sigma }_t}{d}^{3}xN\sqrt{\gamma }\left({}^{(3)}R+K_{ij}{K}^{ij}-K^2\right)$$

   The crucial observation is that the Lagrangian density depends on $N$ and $N^i$, but not on their time derivatives $\dot{N}$ and ${\dot{N}}^i$. In the language of constrained systems, this means $N$ and $N^i$ are not dynamical variables but Lagrange multipliers. Their role is to enforce constraints.

2. Canonical Momentum: The dynamical variable is ${\gamma }_{ij}$. Its conjugate momentum ${\pi }^{ij}$ is defined by the functional derivative of the Lagrangian with respect to ${\dot{\gamma }}_{ij}$. The calculation yields:

   $${\pi }^{ij}=\frac{\delta L_{ADM}}{\delta {\dot{\gamma }}_{ij}}=\frac{\sqrt{\gamma }}{16\pi G}(K^{ij}-{\gamma }^{ij}K)$$

   This confirms the relationship we anticipated: the canonical momentum ${\pi }^{ij}$ is algebraically related to the extrinsic curvature $K_{ij}$.

3. The Hamiltonian: Performing the Legendre transformation, $H_{ADM}=\int ({\pi }^{ij}{\dot{\gamma }}_{ij}-{\mathcal{L}}_{ADM})d^{3}x$, and expressing everything in terms of the canonical variables $({\gamma }_{ij},{\pi }^{ij})$ leads to the total ADM Hamiltonian:

   $$H_{ADM}={\int }_{{\Sigma }_t}{d}^{3}x\left(N\mathcal{H}+N_i{\mathcal{H}}^i\right)$$

   Here, $\mathcal{H}$ and ${\mathcal{H}}^i$ are functions of ${\gamma }_{ij}$, ${\pi }^{ij}$, and matter fields.

4. The Constraint Equations: Since $N$ and $N_i$ are Lagrange multipliers, varying the action with respect to them must yield zero. This leads to the equations of motion for $N$ and $N_i$:

   $$\frac{\delta S_{ADM}}{\delta N}=0⟹\mathcal{H}=0$$ $$\frac{\delta S_{ADM}}{\delta N_i}=0⟹{\mathcal{H}}^i=0$$

   These are precisely the constraint equations! The functions $\mathcal{H}$ and ${\mathcal{H}}^i$ are the Hamiltonian constraint density and momentum constraint density. When written out explicitly (for vacuum, $T_{\mu \nu }=0$) they are:

   • Hamiltonian Constraint:
     $\mathcal{H}=\frac{16\pi G}{\sqrt{\gamma }}\left({\pi }_{ij}{\pi }^{ij}-\frac{1}{2}{\pi }^2\right)-\frac{\sqrt{\gamma }}{16\pi G}{}^{(3)}R=0$
   • Momentum Constraint:
     ${\mathcal{H}}^i=-2D_j{\pi }^{ji}=0$
     (where $\pi ={\gamma }_{ij}{\pi }^{ij}$). You can verify that substituting the definition of ${\pi }^{ij}$ in terms of $K_{ij}$ recovers the geometric forms we derived in the first section.

Conclusion: The Structure of Spacetime Dynamics

In this lesson, we have derived the four crucial constraint equations of General Relativity from two different perspectives.

Key Takeaways:

• The ten Einstein Field Equations split into four constraint equations and six evolution equations in the 3+1 formalism.
• The Hamiltonian constraint ($\mathcal{H}=0$) relates the spatial curvature, its rate of change, and the matter energy density on a slice. It can be seen as an equation for the total "energy" of a closed gravitational system, which must be zero.
• The three momentum constraints (${\mathcal{H}}^i=0$) relate the spatial derivatives of the extrinsic curvature to the matter momentum density, enforcing momentum conservation.
• From the Hamiltonian viewpoint, these constraints arise because the lapse $N$ and shift $N^i$ are not true dynamical variables but Lagrange multipliers. Their "equations of motion" enforce the constraints on the physical states $({\gamma }_{ij},{\pi }^{ij})$.
• This constrained structure is a direct consequence of the diffeomorphism invariance (general covariance) of General Relativity. The freedom to choose our time-slicing and spatial coordinates manifests as these constraints on the initial data.

Preview of the Next Lesson:

The Hamiltonian formulation has given us the classical constraint equation $\mathcal{H}=0$. This is the perfect launchpad for canonical quantization. In our next lesson, we will "promote" the classical variables to quantum operators (${\gamma }_{ij}\to {\hat{\gamma }}_{ij}$, ${\pi }^{ij}\to -i\hslash \frac{\delta }{\delta {\gamma }_{ij}}$) and impose the Hamiltonian constraint as an operator equation on a "wavefunction of the universe" $\Psi [{\gamma }_{ij}]$. This will lead us to one of the most famous and enigmatic equations in theoretical physics: the Wheeler-DeWitt equation, $\hat{\mathcal{H}}\Psi =0$.