Vacuum Gravitational Field Research Articles

The Kerr metric and stationary axisymmetric gravitational fields

A treatment of Einstein’s equations governing vacuum gravitational fields which are stationary and axisymmetric is shown to divide itself into three parts: a part essentially concerned with a choice of gauge (which can be chosen to ensure the occurrence of an event horizon exactly as in the Kerr metric); a part concerned with two of the basic metric functions which in two combinations satisfy a complex equation (Ernst’s equation) and in one combination satisfies a symmetric pair of real equations; and a third part which completes the solution in terms of a single ordinary differential equation of the first order. The treatment along these lines reveals many of the inner relations which characterize the general solutions, provides a derivation of the Kerr metric which is direct and verifiable at all stages, and opens an avenue towards the generation of explicit classes of exact solutions (an example of which is given).

On the variational method of derivation of equations of state for a material medium and a gravitational field: PMM vol 42, no. 5, 1978, pp. 771–780

The formulation of equations of state for a continuous medium and gravitational field is considered in the framework of the general theory of relativity, using the fundamental variational equation. The arbitrariness of determination of equations of state is examined in detail, and problems of determining the energy-momentum tensor for the medium and gravitational field are clarified in the case of invariant dynamic and generally physical Euler's equations. Obtained formulas and deductions are specifically defined for the gravitational field in vacuum. The developed theories make possible a proper evaluation of the essence of the various assumptions prevailing in the theory of the energ-ymomentum pseudotensor and in the general problem of the possible form of equations of state for physical models.

Is gravitation a result of Mach's inertial interaction?

According to Sciama's interpretation of Mach's principle, when a body of massm moves with acceleration α relative to a second massM, the latter experiences a force of the formF=GmMα/c2r, whereG is the gravitational constant,c is the speed of light, andr is the distance between the two masses. In the present paper we propose than, if Sciama's interpretation of Mach's principle is correct, the Newtonian force of gravity can be understood as resulting from the acceleration of particles within matter. It is shown that the structure of Newtonian vacuum gravitational fields is consistent with this proposal, that the random motion of a particle confined to a small volume generates a Newtonian-type gravitational force, and that the acceleration of a Dirac electron in a stationary state leads to a gravitational force of the correct magnitude.

Classification of unquantized Yang-Mills fields

A classification of unquantized Yang-Mills fields is given according to their algebraic and differential properties. A method is used which is similar to the one employed by Petrov and Schell in their classification of the vacuum gravitational field. It is proved that there is no inherently nonlinear solution to a sourceless Yang-Mills field in a vacuum.

On the initial singularity in the scalar-tensor anisotropic cosmology

In connection with the problem of the initial singularity in the scalar-tensor anisotropic cosmology of Jordan-Brans-Dicke, the dynamics of homogeneous models of Bianchi type I is examined on the basis of the general analytic solutions in vacuo and in the presence of gravitating matter with state equations P=nɛ (0 ⩽ n ⩽ 1). It is shown that the scalar homogeneous ϑ-field, as an effective source of the V4geometry, has to influence essentially the dynamics of the early anisotropic stage of the Universe's expansion, and significantly modifies the character of the initial singularity. At negative ω (ω < −6), the sourceless scalar ϑ-field may remove the singularity and provide regular ‘bouncing’ in the models with matter (P ⩽ ɛ/3) if it prevails over the tensor anisotropic mode of the vacuum gravitational field.

Unphysical Solutions of Yang's Gravitational-Field Equations

A static, spherically symmetric solution of Yang's vacuum gravitational field equations is given which predicts incorrect values for experimental observations. It is argued that Yang's field equations must be supplemented by further restrictions on the class of allowable space-times.

?Electric? and ?Magnetic? gravitational fields in vacuum

Gravitational tensor fields Eij and Hij in vacuum are introduced and the case of radiation is formulated, in analogy with electromagnetism.

Change of Petrov type under generation of solutions of Einstein’s equations

A method given by Geroch [J. Math. Phys. 12, 918 (1971)] for generating new solutions of the vacuum gravitational field equations from known solutions is investigated in order to find out how the Petrov type of the new metric will be related to that of the old metric. Conditions are found under which algebraically special metrics will generate algebraically special metrics. It is shown how the Petrov type of these metrics will change under the generation method of Geroch.

Einstein-Maxwell null fields with non-zero twist

It is shown that iffa is the propagation vector of a null Maxwell field in a space-time with metric0gab, then it is also the propagation vector in the space-timegab=0gab+2fiaib. This result, together with the Robinson metric for vacuum gravitational fields and Hughston's generalization to radiating fields, is used to set up equations for combined gravitational and electromagnetic null-fields with special reference to fields with non-zero twist.

All Stationary Vacuum Metrics with Shearing Geodesic Eigenrays

The general solution of the field equations of stationary vacuum gravitational fields possessing geodesic eigenrays with nonvanishing shear is obtained. Nontrivial solutions exist only if the eigenrays do not rotate. The resulting metrics fall into two classes: either there is a functional dependence among the field quantities (this class belongs to the Papapetrou solutions), or the quantity γ0, which in the shear-free case has been interpreted as the central mass, is uniquely determined. This latter class consists of two space-times. The curvature invariants vanish in the r→∞ limit for both solutions; however, the metrics exhibit singular behavior in this limit.

Mapping onto Solutions of the Gravitational Initial Value Problem

Two approaches to the Einstein initial value problem for vacuum gravitational fields are considered. In the first, the metric of a spacelike slice is prescribed arbitrarily and it is shown that momenta satisfying the constraints can be constructed by exploiting the well-known relation of three of the four constraints to the three-dimensional coordinate transformation group. Specifically, it is shown that there exists a coordinate mapping of a certain specified set of functions onto momenta satisfying the constraints for a specified 3-metric. A further interpretation of this procedure is discussed. In the second approach the 3-geometry of a spacelike slice is specified up to a conformal factor. It is shown that, using a coordinate transformation method similar to the above, transverse traceless momenta can be constructed and that this construction depends essentially only on the conformal geometry of the spacelike hypersurface. The remaining constraint is satisfied by a choice of the conformal factor. As a result, it follows that the initial value equations can be satisfied by mapping certain specified sets of functions onto solutions by using coordinate transformations and a group of scale transformations which include conformal transformation of the metric. This is significant because the unconstrained initial data (gravitational degrees of freedom) are represented by a pair of scale-invariant transverse, traceless tensors of weight53. These objects, in turn, give irreducible representations of the coordinate and scaling groups which are used to effect solutions of the initial value equations.

The general stationary gravitational vacuum field of cylindrical symmetry

The general stationary vacuum gravitational field of cylindrical symmetry as recently found by Davies and Caplan is even static. The possible Petrov types of the Riemann tensor areI,D orO. In spacelike infinity the spacetime becomes necessarily flat.

Analysis of the Newman-Unti Integration Procedure for Asymptotically Flat Space-Times

An analysis is given of the procedure of Newman and Unti for solving the vacuum gravitational field equations for all space-times in which Ψ0 = O(r−5). It was found empirically by Newman and Unti that when the nonradial equations and three of the u-derivative equations have been satisfied to their lowest nontrivial order in r−1, they are then found to be identically satisfied to all orders. A general proof of this result is given which avoids the need for direct verification.

3-Dimensional “relativity” for axisymmetric stationary space-times

The equivalence of the axisymmetric stationary vacuum gravitational field problem to a 3-dimensional “relativity theory” in the presence of a certain scalar matter field is shown. An invariant classification can be achieved with respect to the algebraic structure of the 3-dimensional trace-free Ricci-tensor. The extension of these results to electrovac spaces is also discussed.

Coupled weyl gravitational and zero-rest-mass scalar fields

The Einstein field equations are solved for the metric representing static coupled gravitational and zero-rest-mass scalar fields that depend on at most two variables. Several metrics are then analysed to determine what changes occur in their singular structure when a zero-rest-mass scalar field is coupled to a vacuum gravitational field. In certain cases it is found that the addition of a zero-rest-mass scalar field can have the effect of changing some of the topological properties of the singularities of the corresponding vacuum field.

Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Effective Stress Tensor

The high-frequency expansion of a vacuum gravitational field in powers of its small wavelength is continued. We go beyond the previously discussed linearization of the field equations to consider the lowest-order nonlinearities. These are shown to provide a natural, gauge-invariant, averaged stress tensor for the effective energy localized in the high-frequency gravitational waves. Under the assumption of the WKB form for the field, this stress tensor is found to have the same algebraic structure as that for an electromagnetic null field. A Poynting vector is used to investigate the flow of energy and momentum by gravitational waves, and it is seen that high-frequency waves propagate along null hypersurfaces and are not backscattered by the lowest-order nonlinearities. Expressions for the total energy and momentum carried by the field to flat null infinity are given in terms of coordinate-independent hypersurface integrals valid within regions of high field strength. The formalism is applied to the case of spherical gravitational waves where a news function is obtained and where the source is found to lose exactly the energy and momentum contained in the radiation field. Second-order terms in the metric are found to be finite and free of divergences of the $\frac{(\mathrm{ln}r)}{r}$ variety.

Erhaltungss�tze f�r die Wirkung in elektromagnetischen und gravischen Strahlungsfeldern

Two generally covariant integral conservation laws concerning electromagnetic null fields and vacuum gravitational fields of type II with vanishing Riemanneigenvalue respectively are proved. The conserved quantities both have the dimensions of action. The electromagnetic law shows that with respect to the propagation of energy a null field may be represented by a stream of photons. The gravitational conservation theorem is in many respects similar to the electromagnetic one; it admits to calculate the change of intensity along a gravitational ray.