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{{distinguish|ADHM construction}}

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[[File:ArnowittDeserMisner2009 01.jpg|thumb|236px|[[Richard Arnowitt]], [[Stanley Deser]] and [[Charles Misner]] at the ''ADM-50: A Celebration of Current GR Innovation'' conference held in November 2009<ref>[http://adm-50.physics.tamu.edu ADM-50: A Celebration of Current GR Innovation]</ref> to honor the 50th anniversary of their paper.]]

The '''ADM formalism''' (named for its authors [[Richard Arnowitt]], [[Stanley Deser]] and [[Charles W. Misner]]) is a [[Hamiltonian mechanics|Hamiltonian]] formulation of [[general relativity]] that plays an important role in [[canonical quantum gravity]] and [[numerical relativity]]. It was first published in 1959.<ref name="ADM">{{cite journal |doi=10.1103/PhysRev.116.1322 |title=Dynamical Structure and Definition of Energy in General Relativity |year=1959 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=Physical Review |volume=116 |issue=5 |pages=1322–1330 |bibcode=1959PhRv..116.1322A|url=https://authors.library.caltech.edu/72877/1/PhysRev.116.1322.pdf }}</ref>

The comprehensive review of the formalism that the authors published in 1962<ref>Chapter 7 (pp. 227–265) of [[Louis Witten]] (ed.), ''Gravitation: An introduction to current research'', Wiley: New York, 1962.</ref> has been reprinted in the journal ''[[General Relativity and Gravitation]]'',<ref>{{cite journal |doi=10.1007/s10714-008-0661-1 |arxiv=gr-qc/0405109 |title=Republication of: The dynamics of general relativity |year=2008 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=General Relativity and Gravitation |volume=40 |issue=9 |pages=1997–2027 |bibcode=2008GReGr..40.1997A}}</ref> while the original papers can be found in the archives of ''[[Physical Review]]''.<ref name="ADM"/><!--

   Start of reference block:

--><ref> The papers are:
*{{cite journal |doi=10.1103/PhysRev.113.745 |title=Quantum Theory of Gravitation: General Formulation and Linearized Theory |year=1959 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |journal=Physical Review |volume=113 |issue=2 |pages=745–750 |bibcode=1959PhRv..113..745A|url=https://authors.library.caltech.edu/72875/1/PhysRev.113.745.pdf }}
*{{cite journal |doi=10.1103/PhysRev.117.1595 |title=Canonical Variables for General Relativity |year=1960 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=Physical Review |volume=117 |issue=6 |pages=1595–1602 |bibcode=1960PhRv..117.1595A|url=https://authors.library.caltech.edu/88200/1/PhysRev.117.1595.pdf }}
*{{cite journal |doi=10.1103/PhysRevLett.4.375 |title=Finite Self-Energy of Classical Point Particles |year=1960 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=Physical Review Letters |volume=4 |issue=7 |pages=375–377 |bibcode=1960PhRvL...4..375A|url=https://authors.library.caltech.edu/72917/1/PhysRevLett.4.375.pdf }}
*{{cite journal |doi=10.1103/PhysRev.118.1100 |title=Energy and the Criteria for Radiation in General Relativity |year=1960 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=Physical Review |volume=118 |issue=4 |pages=1100–1104 |bibcode=1960PhRv..118.1100A|url=https://authors.library.caltech.edu/88201/1/PhysRev.118.1100.pdf }}
*{{cite journal |doi=10.1103/PhysRev.120.313 |title=Gravitational-Electromagnetic Coupling and the Classical Self-Energy Problem |year=1960 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=Physical Review |volume=120 |issue=1 |pages=313–320 |bibcode=1960PhRv..120..313A|url=https://authors.library.caltech.edu/88202/1/PhysRev.120.313.pdf }}
*{{cite journal |doi=10.1103/PhysRev.120.321 |title=Interior Schwarzschild Solutions and Interpretation of Source Terms |year=1960 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=Physical Review |volume=120 |issue=1 |pages=321–324 |bibcode=1960PhRv..120..321A |url=https://authors.library.caltech.edu/88203/1/PhysRev.120.321.pdf }}
*{{cite journal |doi=10.1103/PhysRev.121.1556 |title=Wave Zone in General Relativity |year=1961 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=Physical Review |volume=121 |issue=5 |pages=1556–1566 |bibcode=1961PhRv..121.1556A|url=https://authors.library.caltech.edu/88204/1/PhysRev.121.1556.pdf }}
*{{cite journal |doi=10.1103/PhysRev.122.997 |title=Coordinate Invariance and Energy Expressions in General Relativity |year=1961 |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. |journal=Physical Review |volume=122 |issue=3 |pages=997–1006 |bibcode=1961PhRv..122..997A|url=https://authors.library.caltech.edu/88205/1/PhysRev.122.997.pdf }}</ref><!--

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==Overview==
The formalism supposes that [[spacetime]] is [[foliation|foliated]] into a family of spacelike surfaces <math>\Sigma_t</math>, labeled by their time coordinate <math>t</math>, and with coordinates on each slice given by <math>x^i</math>.  The dynamic variables of this theory are taken to be the [[metric tensor]] of three dimensional spatial slices <math>\gamma_{ij}(t,x^k)</math> and their [[canonical coordinates|conjugate momenta]] <math>\pi^{ij}(t,x^k)</math>.  Using these variables it is possible to define a [[Hamiltonian mechanics|Hamiltonian]], and thereby write the equations of motion for general relativity in the form of [[Hamilton's equation]]s.

In addition to the twelve variables <math>\gamma_{ij}</math> and <math>\pi^{ij}</math>, there are four [[Lagrange multipliers]]: the [[lapse function]], <math>N</math>, and components of [[shift vector field]], <math>N_i</math>.  These describe how each of the "leaves" <math>\Sigma_t</math> 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 systems|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 <math>g_{ij}</math> and the metric tensor for the full four-dimensional spacetime <math>{^{(4)}}g_{\mu \nu}</math>.

The text here uses [[Einstein notation]] in which summation over repeated indices is assumed.

Two types of derivatives are used:  [[Partial derivative]]s are denoted either by the operator <math>\partial_{i}</math> or by subscripts preceded by a comma.  [[Covariant derivative]]s are denoted either by the operator <math>\nabla_{i}</math> or by subscripts preceded by a semicolon.

The absolute value of the [[determinant]] of the matrix of metric tensor coefficients is represented by <math>g</math> (with no indices).  Other tensor symbols written without indices represent the trace of the corresponding tensor such as <math>\pi = g^{ij}\pi_{ij}</math>.

==Derivation==

===Lagrangian formulation===
The starting point for the ADM formulation is the [[Lagrangian (field theory)|Lagrangian]]
:<math>\mathcal{L} = {^{(4)}R} \sqrt{^{(4)}g},</math>
which is a product of the square root of the [[determinant]] of the four-dimensional [[metric tensor]] for the full spacetime and its [[scalar curvature|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
:<math>g_{ij} = {^{(4)}}g_{ij}</math>
will be the [[generalized coordinates]] for a Hamiltonian formulation. The [[conjugate momenta]] can then be computed as
:<math>\pi^{ij} = \sqrt{^{(4)}g} \left( {^{(4)}}\Gamma^0_{pq} - g_{pq} {^{(4)}}\Gamma^0_{rs}g^{rs} \right) g^{ip}g^{jq},</math>
using standard techniques and definitions. The symbols <math>{^{(4)}}\Gamma^0_{ij}</math> are [[Christoffel symbols]] associated with the metric of the full four-dimensional spacetime. The lapse
:<math>N = \left( -{^{(4)}g^{00}} \right)^{-1/2}</math>
and the shift vector
:<math>N_{i} = {^{(4)}g_{0i}}</math>
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
:<math>\mathcal{L} = -g_{ij} \partial_t \pi^{ij} - NH - N_i P^i - 2 \partial_i \left( \pi^{ij} N_j - \frac{1}{2} \pi N^i + \nabla^i N \sqrt{g} \right)</math>
is conveniently written in terms of the two new quantities
:<math>H = -\sqrt{g} \left[^{(3)}R + g^{-1} \left(\frac{1}{2} \pi^2 - \pi^{ij} \pi_{ij} \right) \right]</math>
and
:<math>P^i = -2 \pi^{ij}{}_{;j},</math>
which are known as the [[Hamiltonian constraint]] and the momentum constraint respectively. Note also that 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]] on three-dimensional spaces embedded in the four-dimensional [[spacetime]], it is possible and desirable to use the usual procedures from [[Lagrangian mechanics]] to derive "equations of motion" that describe the time evolution of both the metric <math>g_{ij}</math> and its conjugate momentum <math>\pi^{ij}</math>.  The result

:<math>\partial_t g_{ij} = \frac{2N}{\sqrt{g}} \left( \pi_{ij} - \tfrac{1}{2} \pi g_{ij} \right) + N_{i;j} + N_{j;i}</math>

and

:<math>\begin{align}
 \partial_t \pi^{ij} = &-N \sqrt{g} \left( R^{ij} - \tfrac{1}{2} R g^{ij} \right) + \frac{N}{2\sqrt{g}} g^{ij} \left( \pi^{mn} \pi_{mn} - \tfrac{1}{2} \pi^2 \right ) - \frac{2N}{\sqrt{g}} \left( \pi^{in} {\pi_n}^j - \tfrac{1}{2} \pi \pi^{ij} \right) \\
 &- \sqrt{g} \left (\nabla^i \nabla^j N - g^{ij} \nabla^n \nabla_n N \right ) + \nabla_n \left (\pi^{ij} N^n \right ) - {N^i}_{;n} \pi^{nj} - {N^j}_{;n} \pi^{ni}
\end{align}</math>

is a [[non-linear]] set of [[partial differential equations]].

Taking variations with respect to the lapse and shift provide constraint equations
:<math>H = 0</math>
and
:<math>P^i = 0,</math>
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===
{{main|Wheeler–DeWitt equation}}

Using the ADM formulation, it is possible to attempt to construct a [[Quantum gravity|quantum theory of gravity]] in the same way that one constructs the [[Schrödinger equation]] corresponding to a given Hamiltonian in [[quantum mechanics]]. That is, replace the canonical momenta <math>\pi^{ij}(t, x^k)</math> and the spatial metric functions by linear functional differential operators
:<math> \hat{g}_{ij}(t, x^k) \mapsto g_{ij}(t, x^k),</math>
:<math> \hat{\pi}^{ij}(t, x^k) \mapsto -i \frac{\delta}{\delta g_{ij}(t, x^k)}. </math>
More precisely, the replacing of classical variables by operators is restricted by [[Canonical commutation relation|commutation relations]]. The hats represents operators in quantum theory. This leads to the [[Wheeler–DeWitt equation]].

===Application to numerical solutions of the Einstein equations===
{{main|Numerical relativity}}

There are relatively few known exact solutions to the [[Einstein field equations]]. 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]] 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 the reduction of order of differential equations must{{clarify|date=May 2016}} be done, if we want to prepare equations for a computer.

==ADM energy and mass==
{{Also|Mass in general relativity}}

ADM energy is a special way to define the [[energy]] in [[general relativity]], which is only applicable to some special geometries of [[spacetime]] that asymptotically approach a well-defined [[metric tensor]] at infinity – for example a spacetime that asymptotically approaches [[Minkowski space]]. 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]]. [[Noether's theorem]] 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]]. [[Cosmic inflation]] in particular is able to produce energy (and mass) from "nothing" because the [[vacuum energy]] density is roughly constant, but the volume of the Universe [[exponential growth|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".<ref>{{Cite journal|arxiv=0908.0679|title=Hamiltonian formulation of f(Riemann) theories of gravity|journal=Progress of Theoretical Physics|volume=123|issue=1|pages=169–185|last1=Deruelle|first1=Nathalie|last2=Sasaki|first2=Misao|last3=Sendouda|first3=Yuuiti|last4=Yamauchi|first4=Daisuke|year=2010|doi=10.1143/PTP.123.169|bibcode = 2010PThPh.123..169D }}</ref>

==See also==
* [[Canonical coordinates]]
* [[Hamilton–Jacobi–Einstein equation]]
* [[Peres metric]]

==Notes==
{{Reflist|2}}

==References==
* {{cite book | author=Kiefer, Claus| title = Quantum Gravity | location= Oxford, New York | publisher= [[Oxford University Press]] | year=2007 |isbn = 978-0-19-921252-1}}

{{Relativity}}

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[[Category:Mathematical methods in general relativity]]
[[Category:Formalism (deductive)]]