
The Arnowitt–Deser–Misner (ADM) formalism (named for its authors
Richard Arnowitt,
Stanley Deser and
Charles W. Misner) is a
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
formulation of
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
that plays an important role in
canonical quantum gravity
In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by ...
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
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
is
foliated into a family of spacelike surfaces
, labeled by their time coordinate
, and with coordinates on each slice given by
. The dynamic variables of this theory are taken to be the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
of three-dimensional spatial slices
and their
conjugate momenta . Using these variables it is possible to define a
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, and thereby write the equations of motion for general relativity in the form of
Hamilton's equations.
In addition to the twelve variables
and
, there are four
Lagrange multipliers: the
lapse function,
, and components of
shift vector field,
. These describe how each of the "leaves"
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 geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
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
and the metric tensor for the full four-dimensional spacetime
.
The text here uses
Einstein notation
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
in which summation over repeated indices is assumed.
Two types of derivatives are used:
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s are denoted either by the operator
or by subscripts preceded by a comma.
Covariant derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
s are denoted either by the operator
or by subscripts preceded by a semicolon.
The absolute value of the
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of the matrix of metric tensor coefficients is represented by
(with no indices). Other tensor symbols written without indices represent the trace of the corresponding tensor such as
.
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:
:
where
is the lapse function encoding the proper time evolution,
is the shift vector, encoding how spatial coordinates change between hypersurfaces.
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
:
which is a product of the square root of the
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of the four-dimensional
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
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
:
will be the
generalized coordinates for a Hamiltonian formulation. The
conjugate momenta can then be computed as
:
using standard techniques and definitions. The symbols
are
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metri ...
associated with the metric of the full four-dimensional spacetime. The lapse
:
and the shift vector
:
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
:
is conveniently written in terms of the two new quantities
: