User:Maschen/Hamilton-Jacobi-Einstein equation

Further information: ADM formalism

In general relativity, the Hamilton-Jacobi-Einstein equation (EHJE) is an equation in the Hamiltonian formulation of geometrodynamics using superspace, cast in the "geometrodynamics era" around the 1960s, by A. Peres in 1962 and others. 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 equations (EFEs). It is a modification of the Hamilton-Jacobi equation (HJE) from classical mechanics.

Background

The HJE is:

$-{\frac {\partial S}{\partial t}}=H$

where S is action S and H the Hamiltonian, a function of the configuration (position vectors) and motion (momenta vectors) of the system, and time t.

It is of interest in the various alternative formulations of classical mechanics (see analytical mechanics), because it provides a very direct analogue to the general Schrödinger equation (SE):

$i\hbar {\frac {\partial \Psi }{\partial t}}={\hat {H}}\Psi$

where Ψ is the wavefunction describing the entire quantum system, Ĥ is the Hamiltonian operator, and ħ is Planck's reduced constant. The phase of Ψ can be interpreted as the action:

$\Psi ={\sqrt {\rho }}e^{iS/\hbar }$

which yields the HJE in the limit ħ → 0.

Equation

In general relativity, the metric tensor is an essential object, since proper time, arc length, geodesic motion in curved spacetime, 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 spatial coordinates (x, y, z) without the coordinate time t.

The equation can be written:

$\left[{\frac {1}{2}}g_{pq}g_{rs}-g_{pr}g_{qs}\right]{\frac {\delta S}{\delta g_{pq}}}{\frac {\delta S}{\delta g_{rs}}}+gR=0$

where g is the determinant of the metric tensor and R the Ricci scalar curvature, and the "δ" instead of "d" denotes the variational derivative rather than the ordinary derivative. These derivatives correspond to the field momenta:

$\pi ^{pq}={\frac {\delta S}{\delta g_{pq}}}$

the rate of change of action per unit coordinate distance.

Description

The equation describes how wavefronts of constant action propagate in superspace - where three-dimensional space is understood to be "dynamic" and changing with time (not four-dimensional spacetime dynamic in all four dimensions), as waves of particles, i.e. matter waves, unfold in space.

The formalism incorperates the quantum mechanical concept of a wavefunction.

Applications

The equation takes various complicated forms in:

• Quantum gravity
• Quantum cosmology

See also

• Wheeler–DeWitt equation
• Quantum geometry
• Calculus of variations
• Principle of least action

References

Further reading

Books

• J.L. Lopes (1977). Quantum mechanics, a half century later: Papers of a Colloquium on Fifty Years of Quantum Mechanics. Strasbourg, France: Springer, Kluwer Academic Publishers. ISBN 9-789-0277-07840.
• C. Rovelli (2004). Quantum Gravity. Cambridge University Press. ISBN 0-521-83733-2.
• C. Kiefer (2012). Quantum Gravity (3rd ed.). Oxford University Press. ISBN 0-199-58520-2.

Selected papers

• T. Banks (1984). "TCP, Quantum Gravity, The Cosmological Constant and all that..." (PDF). Stanford, USA. (Equation A.3 in appendix).
• B. K. Darian (1997). "Solving the Hamilton-Jacobi equation for gravitationally interacting electromagnetic and scalar fields" (PDF). Canada, USA. arXiv:gr-qc/9707046v2.
• J. R. Bond, D. S. Salopek (1990). "Nonlinear evolution of long-wavelength metric fluctuations in inflationary models". Canada (USA), Illinois (USA). {{cite news}}: Unknown parameter |jounal= ignored (help)
• Sang Pyo Kim (1996). "Classical spacetime from quantum gravity". Phys. Rev. D. Kunsan, Korea: IoP. doi:10.1088/0264-9381/13/6/011.
• S.R. Berbena, A.V. Berrocal, J. Socorro, L.O. Pimentel (2006). "The Einstein-Hamilton-Jacobi equation: Searching the classical solution for barotropic FRW" (PDF). Guanajuato and Autónoma Metropolitana (Mexico). arXiv:gr-qc/0607123.{{cite news}}: CS1 maint: multiple names: authors list (link)