Simplification of the Hamilton-Jacobi functional of general relativity

In follow-up of an earlier paper by Komar it is shown that the Lagrangian of general relativity can be chosen so thatS=p ^mng_mnd³x. This result holds without the requirement of special boundary conditions.Supported in part by the U.S. Air Force under Grant No. AF-AFOSR 68-1524 to Yeshiva University and Contract No. ARL F 33615-70-C-1110 to Syracuse University.     

The Ernst equation in general relativity

Stationary axisymmetric gravitational fields are governed by the Ernst equation. Its internalSU(1, 1) symmetry gives rise to a linear problem and to Bäcklund transformations which map known solutions into new ones. The main features of the solution generating techniques are outlined, the generalization to Einstein-Maxwell fields is discussed and some applications are given.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.This paper summarizes some common work with G. Neugebauer. I would like to thank him for many stimulating discussions.     

Kinetic equation in the general theory of relativity

An attempt is made to discover the physical content of the general-relativistic theory of gases. Under an invariant interpretation of the collision term, this theory does not satisfy the correspondence principle for classical theory. The collision term in the classical Boltzmann form is meaningful only in an isolated reference frame realized by a locally inertial frame with the origin at the collision point. With this formulation of the kinetic equations, the class of equilibrium states in GR expands considerably and also covers nonstationary distributions. A specific example of such a distribution that is locally Maxwellian is given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 72–76, February, 1979.In conclusion, the author considers it his pleasant duty to thank Prof. N. A.Chernikov for useful discussion of certain problems touched on this paper, and A. V. Zakharov, who made this result of his article koown to the author.     

On ellipsoidal solutions of Liouville's equation in general relativity

A relativistic, collisionless gas of gravitating particles all having the same proper mass (possibly equal to zero) is studied under the assumption that the oneparticle distribution function is locally ellipsoidal in momentum space with respect to some timelike vector field (observer). Liouville's equation implies that the distribution function depends only on a quadratic form in the 4- momenta, whose coefficients are a Killing tensor in the case of non- vanishing proper mass, and a conformal Killing tensor in the case of vanishing rest mass of the particles. It is suggested that cosmological models of Bianchi-type I can be described in terms of ellipsoidal momentum distribution functions whose ellipsoidal tensor is built out of the Killing vectors associated with the spatial homogeneity.     

The Hamilton-Jacobi equation revisited

A new analysis of the nature of the solutions of the Hamilton-Jacobi equation of classical dynamics is presented based on Caratheodory’s theorem concerning canonical transformations. The special role of a principal set of solutions is stressed, and the existence of analogous results in quantum mechanics is outlined.     

On the Hamilton-Jacobi equation for second-class constrained systems

We discuss a general procedure for arriving at the Hamilton-Jacobi equation of second-class constrained systems, and illustrate it in terms of a number of examples by explicitely obtaining the respective Hamilton principal function, and verifying that it leads to the correct solution to the Euler-Lagrange equations.     

Mayer expansions and the Hamilton-Jacobi equation

We review the derivation of Wilson's differential equation in (infinitely) many variables, which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal integration out is performed. We show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. The differential equation approach gives a remarkable identity for connected parts and precise estimates which include criteria for convergence of iterated Mayer expansions. Applications include the Yukawa gas in two dimensions past the=4 threshold and another derivation of some earlier results of Göpfert and Mack.     

An approximate Lorentz-invariant Hamilton-Jacobi equation

The transformation properties of the nonrelativistic and quasirelativistic Hamilton-Jacobi equations (HJE) are investigated as far as approximate Lorentz invariance is concerned. It is pointed out that these properties depend on whether or not the potential rest energy is included in the particle energy. An approximate Lorentz invariance of the many-particle nonrelativistic HJE is established by means of the multiple time Dirac-Fock-Podolsky formalism.     

Coordinate-dependent 3 + 1 formulation of the general relativity equation of motion

This paper gives a coordinate-dependent 3 + 1 decomposition of the geodesic law of motion. The equation is similar in form to the equation of motion of electrodynamics except there is an extra term proportional to the rate of change of kinetic energy and the vector potential. The equation is solved for the acceleration explicitly.     

General relativity in two dimensions: A Hamilton-Jacobi analysis

We analyzed the constraint structure of the Einstein-Hilbert first-order action in two dimensions using the Hamilton-Jacobi approach. We were able to find a set of involutive, as well as a set of non-involutive constraints. Using generalized brackets we showed how to assure integrability of the theory, to eliminate the set of non-involutive constraints and how to build the field equations.     

Improved Hamiltonian for general relativity

A Hamiltonian formalism for asymptotically flat spaces in general relativity which is manifestly covariant under Poincaré transformations at infinity is proposed and some of its implications are briefly discussed.     

Non-local equations for general relativity

The field equations for two non-local variables, equivalent to the Einstein vacuum equations, are presented. These variables are the holonomy operator associated with special paths and the light cone cut function.

Starting from these equations, one shows via a perturbation argument that a single, fourth-order equation for the cut function can be derived. This single equation encodes the entire conformal structure of a vacuum space—time. The same perturbation technique yields, via quadratures, solutions to the vacuum Einstein equations to any order.     

Heisenberg equations of motion for spin-1/2 wave equation in general relativity

The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.     

Heisenberg equations of motion for spin-1/2 wave equation in general relativity

The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.     

Momenta and reduction for general relativity

A slice theorem for the action of Diff on the space of solutions of the Einstein equations in the asymptotically flat case is proved.     

相对论哈密顿-雅可比方程及其应用

利用相对论哈密顿-雅可比方法求出了电子在激光场中的相对论性运动方程的解析解.并且在电子与激光脉冲散射的实验室参照系、电子初始静止参照系、电子平均静止系中,对于给定的任意椭圆偏振的激光场,得到了解析表达式.     

Geometric algebra techniques for general relativity

Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor.     

Solving the Bethe-Salpeter equation for positronium

We propose a Coulomb-like kernel for the relativistic two-fermion Bethe-Salpeter equation, to be used as the lowest-order approximation in systematic perturbative calculation of bound-state energy levels in QED. The kernel is symmetric in the two fermions and for the exchange of in and out momenta. The resulting equation is exactly soluble, unlike previously considered unperturbed kernels. We give explicitly the Green function and eigen-functions. We also discuss the problem of the behaviour of the wave functions at zero relative coordinate in connection with the contribution to energy levels from the one-photon annihilation channel in QED.