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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.^[2]

The comprehensive review of the formalism that the authors published in 1962^[3] has been reprinted in the journal General Relativity and Gravitation,^[4] while the original papers can be found in the archives of Physical Review.^[2][5]

Overview

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.

In addition to the twelve variables ${\displaystyle \gamma _{ij}}$ and ${\displaystyle \pi ^{ij}}$, there are four Lagrange multipliers: the lapse function, ${\displaystyle N}$, and components of shift vector field, ${\displaystyle N_{i}}$. These describe how each of the "leaves" ${\displaystyle \Sigma _{t}}$ 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). 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 ${\displaystyle g_{ij}}$ and the metric tensor for the full four-dimensional spacetime ${\displaystyle {^{(4)}}g_{\mu \nu }}$.

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 ${\displaystyle \partial _{i}}$ or by subscripts preceded by a comma. Covariant derivatives are denoted either by the operator ${\displaystyle \nabla _{i}}$ or by subscripts preceded by a semicolon.

The absolute value of the determinant of the matrix of metric tensor coefficients is represented by ${\displaystyle g}$ (with no indices). Other tensor symbols written without indices represent the trace of the corresponding tensor such as ${\displaystyle \pi =g^{ij}\pi _{ij}}$.

ADM Split

The ADM split denotes the separation of the spacetime metric into three spatial components and one temporal component (foliation). It separates the spacetime metric into its spatial and temporal parts, which facilitates the study of the evolution of gravitational fields. The basic idea is to express the spacetime metric in terms of a lapse function that represents the time evolution between hypersurfaces, and a shift vector that represents spatial coordinate changes between these hypersurfaces) along with a 3D spatial metric. Mathematically, this separation is written as:

${\displaystyle ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)}$

where ${\displaystyle N}$ is the lapse function encoding the proper time evolution, ${\displaystyle N_{i}}$ is the shift vector, encoding how spatial coordinates change between hypersurfaces. ${\displaystyle g_{ij}}$ is the emergent 3D spatial metric on each hypersurface. This decomposition allows for a separation of the spacetime evolution equations into constraints (which relate the initial data on a spatial hypersurface) and evolution equations (which describe how the geometry of spacetime changes from one hypersurface to another).

Derivation of ADM formalism

Lagrangian formulation

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

${\displaystyle {\mathrm{\pi <!--\; \pi \; -->}}^{ij}=\sqrt{{}^{(4)}g}({}^{(4)}{\mathrm{\Gamma <!--\; \Gamma \; -->}}_{pq}^0-<!--\; -\; -->g_{pq}{}^{(4)}{\mathrm{\Gamma <!--\; \Gamma \; -->}}_{rs}^0}$Zdroj:https://en.wikipedia.org?pojem=ADM_mass
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