ADM formalism

The Arnowitt–Deser–Misner (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.

Overview

The formalism supposes that spacetime is foliated into a family of spacelike surfaces ⁠

Σ

t

{\displaystyle \Sigma _{t}}

⁠, labeled by their time coordinate ⁠

t

{\displaystyle t}

⁠, and with coordinates on each slice given by ⁠

x

i

{\displaystyle x^{i}}

⁠. The dynamic variables of this theory are taken to be the metric tensor of three-dimensional spatial slices ⁠

γ

i

j

(

t

,

x

k

)

{\displaystyle \gamma _{ij}(t,x^{k})}

⁠ and their conjugate momenta ⁠

π

i

j

(

t

,

x

k

)

{\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.

In addition to the twelve variables ⁠

γ

i

j

{\displaystyle \gamma _{ij}}

⁠ and ⁠

π

i

j

{\displaystyle \pi ^{ij}}

⁠, there are four Lagrange multipliers: the lapse function, ⁠

N

{\displaystyle N}

⁠, and components of shift vector field, ⁠

N

i

{\displaystyle N_{i}}

⁠. These describe how each of the "leaves" ⁠

Σ

t

{\displaystyle \Sigma _{t}}

⁠ of the foliation of spacetime are welded together. The equations of motion for these variables can be freely specified; this freedom corresponds to the freedom to specify how to lay out the coordinate system in space and time.

Notation

Most references adopt notation in which four dimensional tensors are written in abstract index notation, and that Greek indices are spacetime indices taking values (0, 1, 2, 3) and Latin indices are spatial indices taking values (1, 2, 3).