User:Maschen/Hamilton-Jacobi-Einstein equation

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

Selected papers

 * (Equation A.3 in appendix).