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 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 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. 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). In the derivation here, a superscript (4) is prepended to quantities that typically have both a three-dimensional and a four-dimensional version, such as the metric tensor for three-dimensional slices g i j {displaystyle g_{ij}} and the metric tensor for the full four-dimensional spacetime ( 4 ) g μ ν {displaystyle {^{(4)}}g_{mu u }} . The text here uses Einstein notation in which summation over repeated indices is assumed. Two types of derivatives are used: Partial derivatives are denoted either by the operator ∂ i {displaystyle partial _{i}} or by subscripts preceded by a comma. Covariant derivatives are denoted either by the operator ∇ i {displaystyle abla _{i}} or by subscripts preceded by a semicolon. The absolute value of the determinant of the matrix of metric tensor coefficients is represented by g {displaystyle g} (with no indices). Other tensor symbols written without indices represent the trace of the corresponding tensor such as π = g i j π i j {displaystyle pi =g^{ij}pi _{ij}} . The starting point for the ADM formulation is the Lagrangian