In general relativity, the Hamilton–Jacobi–Einstein equation (HJEE) or Einstein–Hamilton–Jacobi equation (EHJE) is an equation in the Hamiltonian formulation of geometrodynamics in superspace, cast in the "geometrodynamics era" around the 1960s, by Asher Peres in 1962 and others. It is an attempt to reformulate general relativity in such a way that it resembles quantum theory within a 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 equations (EFEs). 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 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-valued probability amplitude Ψ (more formally as a quantum state ket |Ψ⟩ – an element of a Hilbert space). Using the polar form of the wave function, so making a Madelung transformation:
Ψ = ρ e i S / ℏ {\displaystyle \Psi ={\sqrt {\rho }}e^{iS/\hbar }}
the phase of Ψ is interpreted as the action, and the modulus √ρ = √Ψ*Ψ = |Ψ| is interpreted according to the Copenhagen interpretation as the probability density function. The reduced Planck constant ħ is the quantum of angular momentum. Substitution of this into the quantum general Schrödinger equation (SE):
i ℏ ∂ Ψ ∂ t = H ^ Ψ , {\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}={\hat {H}}\Psi \,,}
and taking the limit ħ → 0 yields the classical HJE:
− ∂ S ∂ t = H , {\displaystyle -{\frac {\partial S}{\partial t}}=H\,,}
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 in these theories. In non-relativistic QM, space and time are not on equal footing; time is a parameter while position is an operator. In RQM and QFT, position returns to the usual 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. 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 x and momentum p, and energy E and time t, are pairwise subject to the uncertainty principles
Δ x Δ p ≥ ℏ 2 , Δ E Δ t ≥ ℏ 2 , {\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}},\quad \Delta E\Delta t\geq {\frac {\hbar }{2}}\,,}
which imply that small intervals in space and time mean large fluctuations in energy and momentum are possible. Since in GR mass–energy and 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. 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. 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. In any case, a four-dimensional curved spacetime 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 (see Metric space for details). The metric tensor in general relativity 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 is only a function of the 3d spatial coordinates r, (for example r = (x, y, z) in Cartesian coordinates) without the coordinate time t:
g i j = g i j ( r ) . {\displaystyle g_{ij}=g_{ij}(\mathbf {r} )\,.}
In this context gij is referred to as the "metric field" or simply "field".
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