ADM formalism

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The ADM Formalism developed by Arnowitt, Deser and Misner is a Hamiltonian formulation of general relativity.

The formalism supposes that spacetime is foliated into a family of spacelike surfaces $Σ t$ , labeled by their time coordinate $t$ , and with coordinates on each slice given by $x i$ . The dynamic variables of this theory are taken to be the metric of three dimensional spatial slices $γ i j (t, x k)$ and their conjugate momenta $π i j (t, x k)$ . Using these variables it is possible to define a Hamiltonian, and thereby write the equations of motion for general relativity in the form of Hamilton's equations.

In addition to the two variables $γ i j$ and $π i j$ , there are two Lagrange multipliers called the lapse $N$ and shift $N i$ . These describe how each of the "leaves" $Σ t$ of the foliation of spacetime are welded together. These variables are nondynamical, and their "equations of motion" are constraint equations that enforce invariance under time reparameterizations and coordinate changes on the spatial slices, respectively.

Using the ADM formulation, it is possible to attempt to construct a quantum theory of gravity, in the same way that one constructs the Schrödinger equation corresponding to a given Hamiltonian in quantum mechanics. That is, replace the canonical momenta $π i j (t, x k)$ by functional differential operators

$\pi^{ij}(t,x^k) \to -i \frac{\delta}{\delta \gamma_{ij}(t,x^k)}$