195.187.72.238 at 14:45, 29 October 2013

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{{General relativity|cTopic=Equations}}
{{further|ADM formalism}}

In [[general relativity]], the '''Hamilton–Jacobi–Einstein equation''' (HJEE) or '''Einstein–Hamilton–Jacobi equation''' (EHJE) is an equation in the [[Hamiltonian mechanics|Hamiltonian formulation]] of [[geometrodynamics]] in [[superspace]], cast in the "geometrodynamics era" around the 1960s, by [[Asher Peres|A. Peres]]<ref>{{cite news|title=On Cauchy’s problem in general relativity - II|journal=Nuovo Cimento|author=A. Peres|publisher=Springer|pages=53–62 |volume=26|issue=1|year=1962|isbn=|url = http://link.springer.com/article/10.1007%2FBF02754342?LI=true}}</ref> in 1962 and others. It is an attempt to reformulate general relativity in such a way that it resembles quantum theory within a [[semiclassical physics|semiclassical]] approximation, much like the correspondence between [[quantum mechanics]] and [[classical mechanics]].

It is named for [[Albert Einstein]], [[Carl Gustav Jacob Jacobi]], and [[William Rowan Hamilton]]. The EHJE contains as much information as all ten [[Einstein field equation]]s (EFEs).<ref>{{cite news | author = U.H. Gerlach | year = 1968 | location = Princeton, USA | publisher = | journal = Physical Review | issue = 5 | volume = 177 | pages = 1929–1941 | title = Derivation of the Ten Einstein Field Equations from the Semiclassical Approximation to Quantum Geometrodynamics | arxiv = | url = http://prola.aps.org/abstract/PR/v177/i5/p1929_1 | doi = 10.1103/PhysRev.177.1929 }}</ref> It is a modification of the [[Hamilton–Jacobi equation]] (HJE) from [[classical mechanics]], and can be derived from the [[Einstein–Hilbert action]] using the [[principle of least action]] in the [[ADM formalism]].

==Background and motivation==

===Correspondence between classical and quantum physics===

In classical [[analytical mechanics]], the dynamics of the system is summarized by the [[action (physics)|action]] {{math|''S''}}. In quantum theory, namely non-relativistic [[quantum mechanics]] (QM), [[relativistic quantum mechanics]] (RQM), as well as [[quantum field theory]] (QFT), with varying interpretations and mathematical formalisms in these theories, the behavior of a system is completely contained in a [[complex number|complex]]-valued [[probability amplitude]] {{math|Ψ}} (more formally as a [[quantum state]] [[Bra–ket notation|ket]] {{math|{{ket|Ψ}}}} - an element of a [[Hilbert space]]). In the [[Semiclassical physics|semiclassical]] [[Eikonal approximation]]:

:<math>\Psi = \sqrt{\rho}e^{iS/\hbar}</math>

the [[wave phase|phase]] of {{math|Ψ}} is interpreted as the action, and the modulus {{math|{{sqrt|''ρ''}} {{=}} {{sqrt|Ψ*Ψ}} {{=}} {{!}}Ψ{{!}}}} is interpreted according to the [[Copenhagen interpretation]] as the [[probability density function]]. The [[Planck's constant|reduced Planck constant]] {{math|''ħ''}} is the ''quantum of action''. Substitution of this into the quantum general [[Schrödinger equation]] (SE):

:<math>i\hbar\frac{\partial \Psi}{\partial t} = \hat{H}\Psi\,,</math>

and taking the limit {{math|''ħ'' → 0}} yields the classical HJE:

:<math>-\frac{\partial S}{\partial t} = H\,,</math>

which is one aspect of the [[correspondence principle]].

===Shortcomings of four-dimensional spacetime===

On the other hand, the transition between quantum theory and ''general'' relativity (GR) is difficult to make; one reason is the treatment of space and time these theories. In non-relativistic QM, space and time are not on equal footing; time is a parameter while [[position operator|position is an operator]]. In RQM and QFT, position returns to the usual [[Coordinate system|spatial coordinates]] alongside the time coordinate, although these theories are consistent only with SR in four-dimensional ''flat'' [[Minkowski space]], and not [[curved space]] nor GR. It is possible to formulate [[quantum field theory in curved spacetime]], yet even this still cannot incorporate GR because gravity is not [[renormalizable]] in QFT.<ref>{{cite news
| author = A. Shomer
| year = 2007
| location = California, USA
| publisher =
| title = A pedagogical explanation for the non-renormalizability of gravity
| arxiv = 0709.3555v2
| url = http://arxiv.org/abs/0709.3555v2
}}</ref> Additionally, in GR particles move through curved spacetime with a deterministically known position and momentum at every instant, while in quantum theory, the position and momentum of a particle cannot be exactly known simultaneously; space {{math|''x''}} and momentum {{math|''p''}}, and energy {{math|''E''}} and time {{math|''t''}}, are pairwise subject to the [[uncertainty principle]]s

:<math>\Delta x \Delta p \geq \frac{\hbar}{2}, \quad \Delta E \Delta t \geq \frac{\hbar}{2}\,,</math>

which imply that small intervals in space and time mean large fluctuations in energy and momentum are possible. Since in GR [[Mass–energy equivalence|mass–energy]] and [[energy–momentum relation|momentum–energy]] is the source of [[spacetime curvature]], large fluctuations in energy and momentum mean the spacetime "fabric" could potentially become so distorted that it breaks up at sufficiently small scales.<ref name="Lerner Trigg p 1285">{{cite book|pages=1285| author=R.G. Lerner, G.L. Trigg| title=Encyclopaedia of Physics| publisher=VHC Publishers|edition=2nd| year=1991| isbn=0-89573-752-3}}</ref> There is theoretical and experimental evidence from QFT that vacuum does have energy since the motion of electrons in atoms is fluctuated, this is related to the [[Lamb shift]].<ref>{{cite book|title=[[Gravitation (book)|Gravitation]]|author=[[John Archibald Wheeler|J.A. Wheeler]], [[Charles Misner|C. Misner]], [[Kip Thorne|K.S. Thorne]]|publisher=W.H. Freeman & Co|page=1190|year=1973|isbn=0-7167-0344-0}}</ref> For these reasons and others, at increasingly small scales, space and time are thought to be dynamical up to the [[Planck length]] and [[Planck time]] scales.<ref name="Lerner Trigg p 1285"/>

In any case, a four-dimensional [[curved space]]time continuum is a well-defined and central feature of general relativity, but not in quantum mechanics.

==Equation==

One attempt to find an equation governing the dynamics of a system, in as close a way as possible to QM and GR, is to reformulate the HJE in ''[[three-dimensional]] curved space'' understood to be "dynamic" (changing with time), and ''not'' [[four-dimensional]] spacetime dynamic in all four dimensions, as the EFEs are. The space has a [[metric tensor|metric]] (see [[metric space]] for details).

The [[metric tensor (general relativity)|metric tensor in general relativity]] is an essential object, since [[proper time]], [[arc length]], [[Equations of motion#Geodesic equation of motion|geodesic motion]] in [[curved space]]time, and other things, all depend on the metric. The HJE above is modified to include the metric, although it's only a function of the 3d spatial coordinates {{math|'''r'''}}, (for example {{math|'''r''' {{=}} (''x'', ''y'', ''z'')}} in [[Cartesian coordinates]]) without the [[coordinate time]] {{math|''t''}}:

:<math>g_{ij} = g_{ij}(\mathbf{r})\,.</math>

In this context {{math|''gij''}} is referred to as the "metric field" or simply "field".

===General equation (free curved space)===

For a free particle in curved "[[Vacuum|empty space]]" or "free space", i.e. in the absence of [[matter]] other than the particle itself, the equation can be written:<ref>{{cite book|title=[[Gravitation (book)|Gravitation]]|author=[[John Archibald Wheeler|J.A. Wheeler]], [[Charles Misner|C. Misner]], [[Kip Thorne|K.S. Thorne]]|publisher=W.H. Freeman & Co|page=1188|year=1973|isbn=0-7167-0344-0}}</ref><ref>{{cite book|title=The Physicist's Conception of Nature|author=J. Mehra|publisher=Springer|page=224|year=1973|isbn=9-02770-3450|url=http://books.google.co.uk/books?id=lSoRzxFye-4C&pg=PA224&dq=hamilton-jacobi-einstein+equation&hl=en&sa=X&ei=eO6EUbmFBsOGONzfgIgI&ved=0CDUQ6AEwAA#v=onepage&q=hamilton-jacobi-einstein%20equation&f=false}}</ref><ref>{{cite book|title=Physical Origins of Time Asymmetry
|author=J.J. Halliwell, J. Pérez-Mercader, W.H. Zurek|publisher=Cambridge University Press|page=429|year=1996|isbn=0-52156-8374|url=http://books.google.co.uk/books?id=kv-evuvt5c4C&pg=PA429&dq=Hamilton%E2%80%93Jacobi%E2%80%93Einstein+equation+in+superspace&hl=en&sa=X&ei=AjuFUf7BAoHrPKKcgMAD&ved=0CEQQ6AEwAw#v=onepage&q=Hamilton%E2%80%93Jacobi%E2%80%93Einstein%20equation%20in%20superspace&f=false
}}</ref>

{{Equation box 1
|title=
|indent =:
|equation =
<math>\frac{1}{\sqrt{g}}\left(\frac{1}{2}g_{pq}g_{rs}-g_{pr}g_{qs}\right)\frac{\delta S}{\delta g_{pq}}\frac{\delta S}{\delta g_{rs}} + \sqrt{g}R=0</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

where {{math|''g''}} is the [[determinant]] of the metric tensor and {{math|''R''}} the [[Ricci scalar curvature]] of the 3d geometry (not including time), and the "{{math|δ}}" instead of "{{math|d}}" denotes the [[variational derivative]] rather than the [[derivative|ordinary derivative]]. These derivatives correspond to the field momenta "conjugate to the metric field":

:<math>\pi^{ij}(\mathbf{r})=\pi^{ij}=\frac{\delta S}{\delta g_{ij}}\,,</math>

the rate of change of action with respect to the field coordinates {{math|''gij''('''r''')}}. The {{math|''g''}} and {{math|''π''}} here are analogous to {{math|''q''}} and {{math|''p'' {{=}} ∂''S''/∂''q''}}, respectively, in classical [[Hamiltonian mechanics]]. See [[canonical coordinates]] for more background.

The equation describes how [[wavefront]]s of constant action propagate in superspace - as the dynamics of [[matter wave]]s of a free particle unfolds in curved space. Additional source terms are needed to account for the presence of extra influences on the particle, which include the presence of other particles or distributions of matter (which contribute to space curvature), and sources of electromagnetic fields affecting particles with [[electric charge]] or [[spin (physics)|spin]]. Like the Einstein field equations, it is [[Non-linear differential equation|non-linear]] in the metric because of the products of the metric components, and like the HJE it is non-linear in the action due to the product of variational derivatives in the action.

The quantum mechanical concept, that action is the phase of the wavefunction, can be interpreted from this equation as follows. The phase has to satisfy the principle of least action; it must be [[stationary point|stationary]] for a small change in the configuration of the system, in other words for a slight change in the position of the particle, which corresponds to a slight change in the metric components;
:<math>g_{ij} \rightarrow g_{ij} + \delta g_{ij} \,,</math>
the slight change in phase is zero:
:<math>\delta S = \int \frac{\delta S }{\delta g_{ij}(\mathbf{r})}\delta g_{ij}(\mathbf{r}) \mathrm{d}^3 \mathbf{r} = 0\,,</math>
(where {{math|d3'''r'''}} is the [[volume element]] of the [[volume integral]]). So the constructive interference of the matter waves is a maximum.  This can be expressed by the [[superposition principle]]; applied to many non-localized wavefunctions spread throughout the curved space to form a localized wavefunction:

:<math>\Psi = \sum_n c_n\psi_n \,,</math>

for some coefficients {{math|''cn''}}, and additionally the action (phase) {{math|''Sn''}} for each {{math|ψ''n''}} must satisfy:

:<math>\delta S = S_{n+1} - S_n = 0 \,,</math>

for all ''n'', or equivalently,

:<math>S_1 = S_2 = \cdots = S_n = \cdots \,.</math>

Regions where {{math|Ψ}} is maximal or minimal occur at points where there is a probability of finding the particle there, and where the action (phase) change is zero. So in the EHJE above, each wavefront of constant action is where the particle ''could'' be found.

This equation still does not "unify" quantum mechanics and general relativity, because the semiclassical Eikonal approximation in the context of quantum theory and general relativity has been applied, to provide a transition between these theories.

==Applications==

The equation takes various complicated forms in:

*[[Quantum gravity]]
*[[Quantum cosmology]]

==See also==

*[[Foliation]]
*[[Quantum geometry]]
*[[Quantum spacetime]]
*[[Calculus of variations]]
*The equation is also related to the [[Wheeler–DeWitt equation]].
*[[Peres metric]]

==References==

===Notes===

{{reflist}}

===Further reading===

====Books====

* {{cite book|title=Quantum mechanics, a half century later: Papers of a Colloquium on Fifty Years of Quantum Mechanics|location=Strasbourg, France|author=J.L. Lopes|publisher=Springer, Kluwer Academic Publishers|year=1977|isbn=9-789-0277-07840|url =
http://books.google.co.uk/books?hl=en&lr=&id=LW2-riA7d6oC&oi=fnd&pg=PA1&dq=Einstein-Hamilton-Jacobi+equation+observer+wavefunction+wheeler&ots=gx7ykKCEsv&sig=L16F3xkB5U9HAXd-YKNP4QwxvpU#v=onepage&q=Einstein-Hamilton-Jacobi%20equation%20observer%20wavefunction%20wheeler&f=false}}
* {{cite book|title=Quantum Gravity|author=C. Rovelli|publisher=Cambridge University Press|year=2004 |isbn=0-521-83733-2|url = http://books.google.co.uk/books?id=HrAzTmXdssQC&pg=PA446&dq=peres+(1962)+geometrodynamics&hl=en&sa=X&ei=mNDhUMqfAoi20QXt-4HYCw&ved=0CEQQ6AEwAg#v=onepage&q=peres%20(1962)%20geometrodynamics&f=false}}
* {{cite book|title=Quantum Gravity|edition=3rd|author=C. Kiefer|publisher=Oxford University Press|year=2012|isbn=0-199-58520-2|url = http://books.google.co.uk/books?id=ftiyh9e3Ac4C&pg=PA155&dq=Peres+(1962)+Einstein-Hamilton-Jacobi+equation&hl=en&sa=X&ei=scviULucHee20QWHj4CACw&ved=0CDMQ6AEwAA#v=onepage&q=Peres%20(1962)%20Einstein-Hamilton-Jacobi%20equation&f=false}}
* {{cite book|title=Towards Quantum Gravity: Proceedings of the XXXV International Winter School on Theoretical Physics|author=J.K. Glikman|publisher=Springer|page=224|location=Polanica, Poland|year=1999|isbn=3-540-669-108|url=http://books.google.co.uk/books?id=wegUE6L7RH4C&pg=PA177&dq=hamilton-jacobi-einstein+equation&hl=en&sa=X&ei=2u-EUZCMC4rEPMz9gOgC&ved=0CDoQ6AEwAQ#v=onepage&q=hamilton-jacobi-einstein%20equation&f=false}}
* {{cite book|title=Quantum cosmology|author=L.Z. Fang, R. Ruffini|publisher=World Scientific|page=|year=1987|isbn=997-1503-123|volume=3|series=Advanced Series in Astrophysics and Cosmology|url=http://books.google.co.uk/books?id=wegUE6L7RH4C&pg=PA177&dq=hamilton-jacobi-einstein+equation&hl=en&sa=X&ei=2u-EUZCMC4rEPMz9gOgC&ved=0CDoQ6AEwAQ#v=onepage&q=hamilton-jacobi-einstein%20equation&f=false}}

====Selected papers====

*{{cite news
| author = T. Banks
| year = 1984
| location = Stanford, USA
| publisher =
| title = TCP, Quantum Gravity, The Cosmological Constant and all that...
| arxiv =
| url = http://www.slac.stanford.edu/pubs/slacpubs/3250/slac-pub-3376.pdf
}} (''Equation A.3 in the appendix'').
*{{cite news
| author = B. K. Darian
| year = 1997
| location = Canada, USA
| publisher =
| title = Solving the Hamilton-Jacobi equation for gravitationally interacting electromagnetic and scalar fields
| arxiv = gr-qc/9707046v2
| url = http://arxiv.org/pdf/gr-qc/9707046.pdf
}}
*{{cite news
| author = J. R. Bond, D. S. Salopek
| year = 1990
| location = Canada (USA), Illinois (USA)
| journal = Phys. Rev. D
| publisher =
| title = Nonlinear evolution of long-wavelength metric fluctuations in inflationary models
| arxiv =
| url = http://prd.aps.org/abstract/PRD/v42/i12/p3936_1
}}
*{{cite news
| author = Sang Pyo Kim
| year = 1996
| location = Kunsan, Korea
| journal = Phys. Rev. D
| publisher = [[Institute of Physics|IoP]]
| title = Classical spacetime from quantum gravity
| arxiv =
| doi = 10.1088/0264-9381/13/6/011
| url = http://iopscience.iop.org/0264-9381/13/6/011
}}
*{{cite news
| author = S.R. Berbena, A.V. Berrocal, J. Socorro, L.O. Pimentel
| year = 2006
| location = Guanajuato and Autónoma Metropolitana (Mexico)
| publisher =
| title = The Einstein-Hamilton-Jacobi equation: Searching the classical solution for barotropic FRW
| arxiv = gr-qc/0607123
| url = http://arxiv.org/pdf/gr-qc/0607123v1.pdf
}}

{{Relativity}}

{{DEFAULTSORT:Hamilton-Jacobi-Einstein equation}}
[[Category:Quantum mechanics]]
[[Category:Quantum field theory]]
[[Category:General relativity]]
[[Category:Hamiltonian mechanics]]

Gowers norm - Revision history

195.187.72.238 at 14:45, 29 October 2013