ADM formalism

The ADM formalism, named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner, is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first published in 1959.

The comprehensive review of the formalism that the authors published in 1962 has been reprinted in the journal General Relativity and Gravitation, while the original papers can be found in the archives of Physical Review.

The formalism supposes that spacetime is foliated into a family of spacelike surfaces ${\displaystyle \Sigma _{t}}$, labeled by their time coordinate ${\displaystyle t}$, and with coordinates on each slice given by ${\displaystyle x^{i}}$. The dynamic variables of this theory are taken to be the metric tensor of three dimensional spatial slices ${\displaystyle \gamma _{ij}(t,x^{k})}$ and their conjugate momenta ${\displaystyle \pi ^{ij}(t,x^{k})}$. 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.

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