Axisymmetric gravitational fields

In this review paper Einstein equations for axisymmetric vacuum fields are introduced in the form given by Lewis. Following Ernst they are reduced to one complex potential equation. Weyl-type, Schwarzschild, Kerr, Tomimatsu-Sato solutions, and their NUT-like generalizations are then discussed.     

Soliton geometry and the vacuum gravitational field equations

A soliton geometry is introduced on manifolds with arbitrary dimensions. The usual soliton connection 1-form defined by Crampin et al. is recovered when the soldering form is a 0-form. It is shown that Einstein's vacuum field equations admit a soliton connection and a soldering 1-form. An associated linear equation with a spectral parameter of Einstein's vacuum field equations are found and some properties of this equation are explored. An example of a Bäcklund transformation is also given.     

Stationary gravitational fields from static fields

In this note it is shown that the method of Misra and Pandey for generating stationary axially symmetric fields from static ones can only give rise to a class of solutions already known.     

Unified gravitational and Yang-Mills fields

We unify the gravitational and Yang-Mills fields by extending the diffeomorphisms in (N=4+n)-dimensional space-time to a larger group, called the conservation group. This is the largest group of coordinate transformations under which conservation laws are covariant statements. We present two theories that are invariant under the conservation group. Both theories have field equations that imply the validity of Einstein's equations for general relativity with the stress-energy tensor of a non-Abelian Yang-Mills field (with massive quanta) and associated currents. Both provide a geometrical foundation for string theory and admit solutions that describe the direct product of a compactn-dimensional space and flat four-dimensional space-time. One of the theories requires that the cosmological constant shall vanish. The conservation group symmetry is so large that there is reason to believe the theories are finite or renormalizable.     

Mass-energy in static gravitational fields

The mass-energy equation in static gravitational fields is shown to beE =g ₄₄ mc ², which agrees with the expressionE =m ₀ c ²(dx/ds)ξ_μ for the energy in a gravitational field possessing a timelike Killing vector ξ. For the Schwarzschild field this leads toE _s ?m ₀ c ² + 1/2m ₀ v ² ?km ₀ M/r. For the Reissner-Nordström field an additional term describing the interaction between the mass and the charge is found to be 2πkm ₀ Q ²/c ² r ². In the Kerr-Newman case more terms are found due to the central rotating gravitating mass.     

Singularities in Weyl gravitational fields

The peculiarities of the scalarS ≡R _ijkl R ^ijkl are exhibited for two axially-symmetric static (Weyl) gravitational fields. By examiningS along curved families of trajectories to the Weyl singularities, examples are found which contradict previous claims by Gautreau and Anderson regarding ‘directional singularities’. Proper circumferences about the Bach and Weyl line-mass singularity are also examined. There is no apparent correlation between the source structure and the behaviour ofS from this analysis.     

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.     

Linear spin-zero quantum fields in external gravitational and scalar fields

The quantum theory of both linear, and interacting fields on curved space-times is discussed. It is argued that generic curved space-time situations force the adoption of the algebraic approach to quantum field theory: and a suitable formalism is presented for handling an arbitrary quasi-free state in an arbitrary globally hyperbolic space-time.For the interacting case, these quasi-free states are taken as suitable starting points, in terms of which expectation values of field operator products may be calculated to arbitrary order in perturbation theory. The formal treatment of interacting fields in perturbation theory is reduced to a treatment of free quantum fields interacting with external sources.Central to the approach is the so-called two-current operator, which characterises the effect of external sources in terms of purely algebraic (i.e. representation free) properties of the source-free theory.The paper ends with a set of Feynman rules which seems particularly appropriate to curved space-times in that it takes care of those aspects of the problem which are specific to curved space-times (and independent of interaction). Heuristically, the scheme calculates in-in rather than in-out matrix elements. Renormalization problems are discussed but not treated.Work partly supported by the Schweizerische Nationalfonds     

Linear spin-zero quantum fields in external gravitational and scalar fields

We give mathematically rigorous results on the quantization of the covariant Klein Gordon field with an external stationary scalar interaction in a stationary curved space-time. We show how, following Segal, Weinless etc., the problem reduces to finding a “one particle structure” for the corresponding classical system. Our main result is an existence theorem for such a one-particle structure for a precisely specified class of stationary space-times. Byproducts of our approach are:

1. A discussion of when a given “equal-time” surface in a given stationary space-time is Cauchy.
2. A modification and extension of the methods of Chernoff [3] for proving the essential self-adjointness of certain partial differential operators.

     

On the models of gravitational fields

Modelling of the gravitational field is discussed within the framework of Einstein's theory. Modelling is defined as the mapping whereby the geodesic lines (the path of test bodies) go into the family of curves given by the equation of a test body motion in a model. Different possible ways of modelling are singled out — namely, the complete, selective, and approximate-selective ones. The theory is applied to the solar gravitational field.Presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971. Professor Petrov died in the Summer of 1972. He was one of the members of the International Committee on GRG and founder of the Journal on GRG. We are happy to publish this contribution of his and shall respectfully keep him in our memory. (The Editor.)     

Spontaneously broken symmetries in gravitational fields

In the present work we study models with material and gravitational fields unified in a self-consistent manner [2, 3]. We use a semiclassical approach, where the gravitational field is classical, but the other fields may be quantum ones. The mechanism of spontaneous symmetry breaking is due to the conformai generalization of the Higgs fields in a curved spacetime. Gravitation will be considered as a gauge field and then in the usual Einstein version.     

Gas dynamics in strong gravitational fields

The steady and axially symmetric flow of a perfect fluid is studied in the context of general relativistic gas dynamics. It is assumed that the flow occurs in the background field of a rotating black hole (or any compact object). The hydrodynamic equations are referred to a locally nonrotating frame and their characteristics are found. The equations describing oblique shock waves are also obtained.     

Gauge fields and gravitational field equations

It is shown that the gravitational field equations in free space have a similar form to the free Yang-Mills field equations, where the group SL (2, C) replaces the group SU(2). The Ricci rotation coefficients take the role of the Yang-Mills like potentials, whereas the Riemann tensor takes the role of the gauge fields.     

A dilemma in the physics of gravitational fields

It is shown that improper use of local quantities for nonlocal situations in fields leads to traditional errors. Nonlocal theoretical quantities referred to standards in a fixed field are defined in order to obtain reliable results. Nonlocal properties of gravitational fields and matter located in it are deduced with the help of physical principles and an electromagnetic model for matter. In spite of the fact that the local velocity of light should be constant, the field is a space of variable nonlocal velocity of light, which accounts for its properties. Matter and light virtually propagate themselves without exchanging energy with the external field, in disagreement with traditional assumptions. Matter becomes contracted by the field. The results are self-consistent and consistent with the observed facts. Bodies withr2GM would be different from black holes and they may account for the peak of highest energy of cosmic radiation and other astronomical facts.     

TypeN twisting vacuum gravitational fields

For vacuum, typeN, twisting gravitational fields the Einstein field equations reduce to partial differential equations for two functions, one real, the other complex, which may be regarded as initial data on a localJ ⁺. If it is assumed that in some frame these initial data take on a certain product form, one factor involving only a spatial variable, the other only a retarded time variable, then these equations become relatively tractable and reduce further to two ordinary differential equations. Rejecting all solutions which lead to Minkowski space or to zero twist leaves just two possibilities. One corresponds to the only explicitly known spacetime of the kind, namely that of Hauser. The other leads to new typeN twisting metrics. However, these metrics can be constructed explicitly only once a single nonlinear third-order ordinary differential equation has been solved.