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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 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 \Sigma_t, labeled by their time coordinate t, and with coordinates on each slice given by x^i. 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 \gamma_(t,x^k) and their conjugate momenta \pi^(t,x^k). 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 \gamma_ and \pi^, there are four Lagrange multipliers: the lapse function, N, and components of shift vector field, N_i. These describe how each of the "leaves" \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 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 g_ and the metric tensor for the full four-dimensional spacetime g_. 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 \partial_ 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 \nabla_ 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 g (with no indices). Other tensor symbols written without indices represent the trace of the corresponding tensor such as \pi = g^\pi_.


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: : ds^2 = -N^2 dt^2 + g_ (dx^i + N^i dt)(dx^j + N^j dt) where N is the lapse function encoding the proper time evolution, N_i is the shift vector, encoding how spatial coordinates change between hypersurfaces. g_ 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 :\mathcal = \sqrt, 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 :g_ = g_ will be the generalized coordinates for a Hamiltonian formulation. The conjugate momenta can then be computed as :\pi^ = \sqrt \left( \Gamma^0_ - g_ \Gamma^0_g^ \right) g^g^, using standard techniques and definitions. The symbols \Gamma^0_ 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 :N = \left( - \right)^ and the shift vector :N_ = 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 :\mathcal = -g_ \partial_t \pi^ - NH - N_i P^i - 2 \partial_i \left( \pi^ N_j - \frac \pi N^i + \nabla^i N \sqrt \right) is conveniently written in terms of the two new quantities :H = -\sqrt \left R + g^ \left(\frac \pi^2 - \pi^ \pi_ \right) \right/math> and :P^i = -2 \pi^_, which are known as the Hamiltonian constraint and the momentum constraint respectively. The lapse and the shift appear in the Lagrangian as Lagrange multipliers.


Equations of motion

Although the variables in the Lagrangian represent 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 ...
on three-dimensional spaces embedded in the four-dimensional
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 ...
, it is possible and desirable to use the usual procedures from
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
to derive "equations of motion" that describe the time evolution of both the metric g_ and its conjugate momentum \pi^. The result :\partial_t g_ = \frac \left( \pi_ - \tfrac \pi g_ \right) + N_ + N_ and :\begin \partial_t \pi^ = &-N \sqrt \left( R^ - \tfrac R g^ \right) + \frac g^ \left( \pi^ \pi_ - \tfrac \pi^2 \right ) - \frac \left( \pi^ ^j - \tfrac \pi \pi^ \right) \\ &+ \sqrt \left (\nabla^i \nabla^j N - g^ \nabla^n \nabla_n N \right ) + \nabla_n \left (\pi^ N^n \right ) - _ \pi^ - _ \pi^ \end is a
non-linear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
set of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...
. Taking variations with respect to the lapse and shift provide constraint equations :H = 0 and :P^i = 0, and the lapse and shift themselves can be freely specified, reflecting the fact that coordinate systems can be freely specified in both space and time.


Applications


Application to quantum gravity

Using the ADM formulation, it is possible to attempt to construct a quantum theory of gravity in the same way that one constructs the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
corresponding to a given Hamiltonian in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. That is, replace the canonical momenta \pi^(t, x^k) and the spatial metric functions by linear functional differential operators : \hat_(t, x^k) \mapsto g_(t, x^k), : \hat^(t, x^k) \mapsto -i \frac. More precisely, the replacing of classical variables by operators is restricted by commutation relations. The hats represent operators in quantum theory. This leads to the Wheeler–DeWitt equation.


Application to numerical solutions of the Einstein equations

There are relatively few known exact solutions to the
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
. In order to find other solutions, there is an active field of study known as numerical relativity in which supercomputers are used to find approximate solutions to the equations. In order to construct such solutions numerically, most researchers start with a formulation of the Einstein equations closely related to the ADM formulation. The most common approaches start with an
initial value problem In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ...
based on the ADM formalism. In Hamiltonian formulations, the basic point is replacement of set of second order equations by another first order set of equations. We may get this second set of equations by Hamiltonian formulation in an easy way. Of course this is very useful for numerical physics, because reducing the order of differential equations is often convenient if we want to prepare equations for a computer.


ADM energy and mass

ADM energy is a special way to define the
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
in
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 ...
, which is only applicable to some special geometries of
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 ...
that asymptotically approach a well-defined
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 ...
at infinity – for example a spacetime that asymptotically approaches
Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...
. The ADM energy in these cases is defined as a function of the deviation of the metric tensor from its prescribed asymptotic form. In other words, the ADM energy is computed as the strength of the gravitational field at infinity. If the required asymptotic form is time-independent (such as the Minkowski space itself), then it respects the time-translational
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
.
Noether's theorem Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...
then implies that the ADM energy is conserved. According to general relativity, the conservation law for the total energy does not hold in more general, time-dependent backgrounds – for example, it is completely violated in
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...
. Cosmic inflation in particular is able to produce energy (and mass) from "nothing" because the
vacuum energy Vacuum energy is an underlying background energy that exists in space throughout the entire universe. The vacuum energy is a special case of zero-point energy that relates to the quantum vacuum. The effects of vacuum energy can be experiment ...
density is roughly constant, but the volume of the Universe grows exponentially.


Application to modified gravity

By using the ADM decomposition and introducing extra auxiliary fields, in 2009 Deruelle et al. found a method to find the Gibbons–Hawking–York boundary term for modified gravity theories "whose Lagrangian is an arbitrary function of the Riemann tensor".


See also

*
Canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
* Hamilton–Jacobi–Einstein equation *
Peres metric In mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to proble ...
* Shape dynamics * Calculus of moving surfaces


Notes


References

* {{DEFAULTSORT:Adm Formalism Mathematical methods in general relativity Formalism (deductive)