ADM formalism

The starting point for the ADM formulation is the Lagrangian

${\displaystyle {\mathcal {L}}={^{(4)}R}{\sqrt {^{(4)}g}},}$

which is a product of the square root of the determinant of the four-dimensional metric tensor for the full spacetime and its Ricci scalar. This is the Lagrangian from the Einstein–Hilbert action.

The desired outcome of the derivation is to define an embedding of three-dimensional spatial slices in the four-dimensional spacetime. The metric of the three-dimensional slices

${\displaystyle g_{ij}={^{(4)}}g_{ij}}$

will be the generalized coordinates for a Hamiltonian formulation. The conjugate momenta can then be computed as

${\displaystyle \pi ^{ij}={\sqrt {^{(4)}g}}\left({^{(4)}}\Gamma _{pq}^{0}-g_{pq}{^{(4)}}\Gamma _{rs}^{0}g^{rs}\right)g^{ip}g^{jq},}$

using standard techniques and definitions. The symbols ${\displaystyle {^{(4)}}\Gamma _{ij}^{0}}$ are Christoffel symbols associated with the metric of the full four-dimensional spacetime. The lapse

${\displaystyle N=\left(-{^{(4)}g^{00}}\right)^{-1/2}}$

and the shift vector

${\displaystyle N_{i}={^{(4)}g_{0i}}}$

are the remaining elements of the four-metric tensor.

Having identified the quantities for the formulation, the next step is to rewrite the Lagrangian in terms of these variables. The new expression for the Lagrangian

${\displaystyle {\mathcal {L}}=-g_{ij}\partial _{t}\pi ^{ij}-NH-N_{i}P^{i}-2\partial _{i}\left(\pi ^{ij}N_{j}-{\frac {1}{2}}\pi N^{i}+\nabla ^{i}N{\sqrt {g}}\right)}$

is conveniently written in terms of the two new quantities

${\displaystyle H=-{\sqrt {g}}\left[^{(3)}R+g^{-1}\left({\frac {1}{2}}\pi ^{2}-\pi ^{ij}\pi _{ij}\right)\right]}$

and

${\displaystyle P^{i}=-2\pi ^{ij}{}_{;j},}$

which are known as the Hamiltonian constraint and the momentum constraint respectively. The lapse and the shift appear in the Lagrangian as Lagrange multipliers.

Although the variables in the Lagrangian represent the metric tensor on three-dimensional spaces embedded in the four-dimensional spacetime, it is possible and desirable to use the usual procedures from Lagrangian mechanics to derive "equations of motion" that describe the time evolution of both the metric ${\displaystyle g_{ij}}$ and its conjugate momentum ${\displaystyle \pi ^{ij}}$. The result

${\displaystyle \partial _{t}g_{ij}={\frac {2N}{\sqrt {g}}}\left(\pi _{ij}-{\tfrac {1}{2}}\pi g_{ij}\right)+N_{i;j}+N_{j;i}}$

and

${\displaystyle {\begin{aligned}\partial _{t}\pi ^{ij}=&-N{\sqrt {g}}\left(R^{ij}-{\tfrac {1}{2}}Rg^{ij}\right)+{\frac {N}{2{\sqrt {g}}}}g^{ij}\left(\pi ^{mn}\pi _{mn}-{\tfrac {1}{2}}\pi ^{2}\right)-{\frac {2N}{\sqrt {g}}}\left(\pi ^{in}{\pi _{n}}^{j}-{\tfrac {1}{2}}\pi \pi ^{ij}\right)\\&+{\sqrt {g}}\left(\nabla ^{i}\nabla ^{j}N-g^{ij}\nabla ^{n}\nabla _{n}N\right)+\nabla _{n}\left(\pi ^{ij}N^{n}\right)-{N^{i}}_{;n}\pi ^{nj}-{N^{j}}_{;n}\pi ^{ni}\end{aligned}}}$

is a non-linear set of partial differential equations.

Taking variations with respect to the lapse and shift provide constraint equations

${\displaystyle H=0}$

and

${\displaystyle P^{i}=0,}$

and the lapse and shift themselves can be freely specified, reflecting the fact that coordinate systems can be freely specified in both space and time.

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 ${\displaystyle \pi ^{ij}(t,x^{k})}$ and the spatial metric functions by linear functional differential operators

${\displaystyle {\hat {g}}_{ij}(t,x^{k})\mapsto g_{ij}(t,x^{k}),}$

${\displaystyle {\hat {\pi }}^{ij}(t,x^{k})\mapsto -i{\frac {\delta }{\delta g_{ij}(t,x^{k})}}.}$

More precisely, the replacing of classical variables by operators is restricted by commutation relations. The hats represent operators in quantum theory. This leads to the Wheeler–DeWitt equation.

There are relatively few known exact solutions to the Einstein field equations. In order to find other solutions, there is an active field of study known as numerical relativity in which supercomputers are used to find approximate solutions to the equations. In order to construct such solutions numerically, most researchers start with a formulation of the Einstein equations closely related to the ADM formulation. The most common approaches start with an initial value problem based on the ADM formalism.

In Hamiltonian formulations, the basic point is replacement of set of second order equations by another first order set of equations. We may get this second set of equations by Hamiltonian formulation in an easy way. Of course this is very useful for numerical physics, because reducing the order of differential equations is often convenient if we want to prepare equations for a computer.