Some zero rest mass test fields in general relativity

Zero rest mass test fields of pure algebraic type are defined and studied via the newly developed GHP formalism. The field equations are written explicitly and an immediate generalisation of Robinson's theorem is obtained. The form for the general zero rest mass test field of pure type is determined in terms of the tetrad components of the background Weyl spinor and an additional gauge dependent function which can be thought of as representing a test neutrino field.     

The algebraic Rainich conditions

It is well known that the Einstein-Maxwell field eguations are the Euler-Lagrange equations associated with a particular Lagrange density. It is also well known that, in a four-dimensional space, the Einstein-Maxwell field equations give rise to the Rainich conditions (which can be divided into two types, the algebraic and the differential Rainich conditions). In this note it is shown that the algebraic Rainich conditions are inevitably the consequence of every Euler-Lagrange equation associated with each member of a special class of Lagrange densities. However, in general, these Euler-Lagrange equations are not the Einstein-Maxwell field equations, although the Lagrange density associated with the latter is a particular member of this class.     

Neutrino radiation in gravitational fields

A differential equation representing radiation solutions of the general relativistic Weyl equation is derived. Their optical properties and the group of motion of the corresponding energy-momentum tensor are studied. If there exists neutrino radiation the Riemann space must be algebraically special and the propagation of the neutrinos occurs only along one of the principal null directions. Gravitational- and neutrinopp-waves taken together, represent an exact solution of the Weyl-Einstein system of field equations.     

The uniqueness of the Einstein-Maxwell field equations

The most general Lagrange density (which is a concomitant of the metric tensor together with a vector field and its first derivatives) for which the associated Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although it is more general than the Lagrangian which is commonly used, it still has essentially the same energy momentum tensor.     

First order formalism off(R) gravity

The first order formalism is applied to study the field equations of a general Lagrangian density for gravity of the form . These field equations correspond to theories which are a subclass of conformally metric theories in which the derivative of the metric is proportional to the metric by a Weyl vector field. The resulting geometrical structure is unique, except whenf(R)=aR ², in the sense that the Weyl field is identifiable in terms of the trace of the energy-momentum tensor and its derivatives. In the casef(R)=aR ² the metric is only defined up to a conformai factor. We discuss the matter conservation equations which are implied by the invariance of the theories under diffeomorphisms. We apply the results to the case of dust and obtain that in general the dust particles will not follow geodesic Unes. We consider the linearized field equations and apply them to obtain the weak field slow motion limit. It is found that the gravitational potential acquires a new term which depends linearly on the mass density. The importance of these new equations is briefly discussed.     

Extended Einstein–Cartan theory à la Diakonov: The field equations

Diakonov formulated a model of a primordial Dirac spinor field interacting gravitationally within the geometric framework of the Poincaré gauge theory (PGT). Thus, the gravitational field variables are the orthonormal coframe (tetrad) and the Lorentz connection. A simple gravitational gauge Lagrangian is the Einstein–Cartan choice proportional to the curvature scalar plus a cosmological term. In Diakonov?s model the coframe is eliminated by expressing it in terms of the primordial spinor. We derive the corresponding field equations for the first time. We extend the Diakonov model by additionally eliminating the Lorentz connection, but keeping local Lorentz covariance intact. Then, if we drop the Einstein–Cartan term in the Lagrangian, a nonlinear Heisenberg type spinor equation is recovered in the lowest approximation.     

The Hamiltonian formulation of regular rth-order Lagrangian field theories

A Hamiltonian formulation of regular rth-order Lagrangian field theories over an m-dimensional manifold is presented in terms of the Hamilton-Cartan formalism. It is demonstrated that a uniquely determined Cartan m-form may be associated to an rth-order Lagrangian by imposing conditions of congruence modulo a suitably defined system of contact m-forms. A geometric regularity condition is given and it is shown that, for a regular Lagrangian, the momenta defined by the Hamilton-Cartan formalism, together with the coordinates on the (r−1)st-order jet bundle, are a minimal set of local coordinates needed to express the Euler-Lagrange equations. When r is greater than one, the number of variables required is strictly less than the dimension of the (2r−1)st order jet bundle. It is shown that, in these coordinates, the Euler-Lagrange equations take the first-order Hamiltonian form given by de Donder. It is also shown that the geometrically natural generalization of the Hamilton-Jacobi procedure for finding extremals is equivalent to de Donder's Hamilton-Jacobi equation. Research supported by the Natural Sciences and Engineering Research Council.     

A Note on the Equivalence of Post-Newtonian Lagrangian and Hamiltonian Formulations

Recently,it has been generally claimed that a low order post-Newtonian(PN)Lagrangian formulation,whose Euler-Lagrange equations are up to an infinite PN order,can be identical to a PN Hamiltonian formulation at the infinite order from a theoretical point of view.In general,this result is difficult to check because the detailed expressions of the Euler-Lagrange equations and the equivalent Hamiltonian at the infinite order are clearly unknown.However,there is no difficulty in some cases.In fact,this claim is shown analytically by means of a special first-order post-Newtonian(1PN)Lagrangian formulation of relativistic circular restricted three-body problem,where both the Euler-Lagrange equations and the equivalent Hamiltonian are not only expanded to all PN orders,but have converged functions.It is also shown numerically that both the Euler-Lagrange equations of the low order Lagrangian and the Hamiltonian are equivalent only at high enough finite orders.     

The Hamilton-Cartan formalism for higher order conserved currents: 1. Regular first order Lagrangians

The Hamilton–Cartan formalism for regular first order Lagrangian field theories is extended to deal with conserved currents which depend on higher order derivatives of the field variables. These conserved currents are characterized. Exterior differential systems I^{(k + 1)} and I equivalent to the k-th and infinite prolongations of the Euler-Lagrange equations are defined. It is shown that to each conserved current is associated an equivalence class of infinitesimal symmetries of I. Conserved charges are defined and a Poisson bracket is constructed by analogy with the usual definition. The sine-Gordon equation is treated briefly as an application of the formalism.     

Heisenberg Equations of Motion in a Nonabelian Gauge Theory

Heisenberg type equations of motion are established in a nonabelian gauge theory with minimal and nonminimal couplings and various relativistic particle equations of motion are derived from them. These equations for pointlike particles possessing a nonabelian gauge interaction (chosen for definiteness to be of SO(4,1) type) ore obtained in classical limit, ħ → 0, or in a semiclassical limit in which contributions of first order in ħ are retained. As a byproduct of the formalism, which can be applied to an arbitrary gauge group, a simple derivation of the Lorentz equation and the Bargmann-Michel-Telegdi equation from spinor electrodynamics with anomalous (i.e. nonminimal) coupling is given starting from the associated quantum mechanical Heisenberg equations of motion and specializing the gauge group to the electromagnetic U(1) group.     

Spacetimes with killing tensors

For Einstein-Maxwell fields for which the Weyl spinor is of type {2, 2}, and the electromagnetic field spinor is of type {1, 1} with its principal null directions coaligned with those of the Weyl spinor, the integrability conditions for the existence of a certain valence two Killing tensor are shown to reduce to a simple criterion involving the ratio of the amplitude of the Weyl spinor to the amplitude of a certain test solution of the spin two zero restmass field equations. The charged Kerr solution provides an example of a spacetime for which the criterion is satisfied; the chargedC-metric provides an example for which it is not.This piece of work was completed, in part, during the authors' summer 1972 stays at The University of Texas at Dallas, Division of Mathematics and Mathematical Physics, the Max-Planck-Institut für Physik und Astrophysik in München, and the Black Hole session of the Ecole d'été de Physique Théorique in Les Houches; supported, in part, by the National Science Foundation, Grants GP-8868, GP-3463 9X, GP-20023, and GU-1598; the Air Force Office of Scientific Research, Grant 903-67; the National Aeronautics and Space Administration, Grant 44-004-001; the Westinghouse Corporation; the Clark Foundation; and the Rhodes Trust.     

Conservation Laws and Associated Noether Type Vector Fields via Partial Lagrangians and Noether’s Theorem for the Liang Equation

We show how one can construct conservation laws of the Liang equation which is not variational but may be regarded as Euler-Lagrange in part. This first requires the determination of the Noether-type symmetries associated with the partial Lagrangian. The final construction of the conservation laws resort to a formula equivalent to Noether’s theorem. A variety of subclasses are given and, for each, a large number of conserved flows are found—the method is usable for any general choice of the variable speed of sound.     

Exact self-consistent plane-symmetric solutions of the equations of two interacting fields (spinor field and scalar field)

A spinor field interacting with a zero-mass neutral scalar field is considered for the case of the simplest type of direct interaction, where the interaction Lagrangian has the formL _int =1/2 ϕ_αϕ_,α F(S) whereF(S) is an arbitrary function of the spinor field invariantS=ψψ. Exact solutions of the corresponding systems of equations that take into account the natural gravitational field in a plane-symmetric metric are obtained. It is proved that the initial system of equations has regular localized soliton-type solutions only if the energy density of the zero-mass scalar field is negative as it “disengages” from interaction with the spinor field. In two-dimensional space-time the system of field equations we are studying describes the configuration of fields with constant energy densityT ₀₀ , i.e., no soliton-like solutions exist in this case. Russian People’s Friendship University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 69–75, July, 1998.     

The uniqueness of the Einstein-Yang-Mills equations

The most general gauge-invariant Lagrange density (concomitant of the metric tensor together with the gauge potentials of a gauge and its first derivatives) for which the associated Euler-Lagrange equations are precisely Yang-Mills equations is obtained. It is more general than the Lagrangian which is commonly used, but it still has essentially the same energy momentum tensor.     

Integrability of Einstein–Weyl Equations for Spatially Homogeneous Models of Type III by Bianchi

This paper deals with nonisotropic spatially homogeneous models for a self-consistent system of Einstein–Weyl equations with a spinor field. It is shown that for spaces of type III by Bianchi the system of Einstein–Weyl equations is integrable.     

Third-order equation for two-component spinors

We present the properties of a two-component spinor field that obeys a third-order equation. It is separated into a massive part that corresponds closely to a Dirac field, and a massless part that obeys the Weyl equation. We discuss the interaction of such a field with an external electromagnetic field and the (weak) interactions of two such fields. They can be considered both in terms of relativistic quantum mechanics and quantum field theory. We conclude that this formulation has some attractive features, such as a unified treatment of electrons and muons with their neutrinos, a special role of thePC transformation, a more convergent propagator and a new approach to interactions. It also has some serious difficulties, aside from those generally associated with higher-order equations. These are mainly related to inconsistencies in the simultaneous considerations of electromagnetic and weak interactions. The approach also suggests a further unification of the electron and muon fields into a single bispinor field.     

Second-order scalar-tensor field equations in a four-dimensional space

Lagrange scalar densities which are concomitants of a pseudo-Riemannian metric-tensor, a scalar field and their derivatives of arbitrary order are considered. The most general second-order Euler-Lagrange tensors derivable from such a Lagrangian in a four-dimensional space are constructed, and it is shown that these Euler-Lagrange tensors may be obtained from a Lagrangian which is at most of second order in the derivatives of the field functions.