Complexified Vaidya's metric

It is shown that the complexification of Vaidya's metric results in a metric that bears the same relation to Vaidya's metric as does the Kerr metric to the Schwarzschild metric. However, the energy-momentum tensor consists of two parts: (1) radiative and (2) nonradiative. The nonradiative part also corresponds to a trace-free field.     

Einstein-Maxwell Equations: Gauge Formulation and Solutions for Radiating Bodies

The subject of the SL(2, c) gauge theory of gravitation is reviewed. A detailed discussion is given on the differential geometry and the fibre bundle structure of such a theory. The coupling of Maxwell's field equations to those of gravitation is also given. The field equations obtained, which are shown to be equivalent to the coupled Einstein-Maxwell equations, are subsequently solved. The solutions sought after are radiating type ones of the kind of the Kerr metric, but with the mass of the body being variable and is a function of the retarded time. A generalization of the Kerr metric is presented and its energy-momentum tensor is analyzed in detail. The classification of the field obtained according to the Petrov scheme is also given.     

Radiating Kerr particle in Einstein universe

A generalized Kerr-NUT type metric is considered in connection with Einstein field equations corresponding to perfect fluid plus a pure radiation field. A general scheme for obtaining the exact solutions of these field equations is developed. Two physically meaningful particular cases are investigated in detail. One gives the field of a radiating Kerr particle embedded in the Einstein universe. The other solution may probably represent a deSitter-like universe pervaded by a pure radiation field.     

Spontaneous creation of massive spin half particles by a rotating block hole

The techniques of second quantization in Kerr metric for the scalar and neutrino (massless) fields are extended to the massive spin half case. The normal modes of Dirac field in Kerr metric are obtained in Chandrasekhar’s representation and the field is quantized as usual by imposing equal-time anti-commutation relations. The vacuum expectation value of energy-momentum tensor is evaluated asymptotically, leading to the result that a Kerr black hole spontaneously creates, in addition to scalar and neutrino quanta, massive Dirac particles in the classical superradiant modes.     

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.     

Some new radiating Kerr-Newman solutions

Three exact non-static solutions of Einstein-Maxwell equations corresponding to a field of flowing null radiation plus an electromagnetic field are presented. These solutions are non-static generalizations of the well known Kerr-Newman solution. The current vector is null in all the three solutions. These solutions are the electromagnetic generalizations of the three generalized radiating Kerr solutions discussed by Vaidya and Patel. The solutions discussed by us describe the exterior gravitational fields of rotating radiating charged bodies. Many known solutions are derived as particular cases.     

On quadratic first integrals of the geodesic equations for type {22} spacetimes

It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.     

The field of a rotating charged body emitting null fluid

An exact solution of Einstein's equations for an Electromagnetic field plus pure radiation is given in terms of the Kerr-Schild metric. The radiating Kerr solution and the Debney-Kerr-Schild solution are derived as particular cases.     

A family of rotating disks as sources of the Kerr metric

A method of constructing a family of regular rotating disks as sources of the Kerr metric is discussed. The algebraic type of the energy-momentum tensor is analyzed, and it is found that none of the disks satisfies the dominant energy condition and, in some cases, even the weak energy condition is violated. Therefore, if the constructed family includes the totality of sources for the Kerr metric with a disk-like topology, then any physically satisfactory source will not have a disk structure.     

Asymptotic behaviour of neutrino fields in curved space-time

It is shown that, if a solution of the Weyl equation in some Riemann space is asymptotically smooth and describes a neutrino field with causal behaviour, than the order of the radiation term in the corresponding energy-momentum tensor predominates asymptotically the orders of the other terms.Dedicated to Prof. Dr. Helmut Hönl on the occasion of his retirement.     

Two theorems on energy-momentum conservation

The mathematical meaning of the law of conservation of energy-momentum is examined. A distinction is made between the intrinsic properties of the metric tensor (i.e., those properties that are independent of the coordinate system), and the nonintrinsic properties of this tensor (i.e., those properties that depend upon the coordinate system). The covariance of the energy-momentum law is used to demonstrate that if one is given (a) any analytic contravariant energy-momentum tensor density in a given coordinate systemx and (b) an analytic specification of the intrinsic properties of the metric tensor, no matter what these properties may be, one can always choose the nonintrinsic properties of the metric tensor in such manner as to satisfy the law of conservation of energy-momentum in the coordinate systemx and thereby in every coordinate system. This result is proved only in the case where the contravariant components of the energy-momentum tensor density are given. Neither the covariant, nor the mixed energy-momentum tensor densities are considered. Other theorems similar to that described above are also derived. Many of the results obtained are nontrivial even when space-time is flat.     

A nonstationary generalization of the Kerr-Newman metric

A new metric depending on three arbitrary parameters is presented by the method of complex coordinate transformations. It gives the gravitational field of a radiating rotating charged body. The metric is algebraically special of Petrov type II according to classification of the Weyl tensor, with a twisting, shear-free, null congruence identical to that of the Kerr-Newman metric. The new metric bears the same relation to the Kerr-Newman metric as the Bonner-Vaidya metric does to the Reissner-Nordstrom metric.     

Some solutions of the Einstein-Maxwell equations in general relativity

A solution of the Einstein-Maxwell equations corresponding to source-free electromagnetic field plus pure radiation is obtained. The solution is algebraically special. A particular case of the solution is considered which encompasses many known solutions. Among them is a radiating Ruban metric.     

Localization of gravitational energy

In the general relativity theory gravitational energy-momentum density is usually described by a pseudo-tensor with strange transformation properties so that one does not have localization of gravitational energy. It is proposed to set up a gravitational energy-momentum density tensor having a unique form in a given coordinate system by making use of a bimetric formalism. Two versions are considered: (1) a bimetric theory with a flat-space background metric which retains the physics of the general relativity theory and (2) one with a background corresponding to a space of constant curvature which introduces modifications into general relativity under certain conditions. The gravitational energy density in the case of the Schwarzschild solution is obtained.     

Der Zerfall des Gd153

Several algebraic characterizations of vacuum type III metric fields are discussed. A covariant integral conservation law is obtained by introducing a divergence free vector density that is uniquely determined by the metric and Riemann tensors of a type III metric. In a region where the gravitational field is of type III almost everywhere, the vector density vanishes at a point if and only if the Riemann tensor vanishes there. The conserved quantity has the dimensions of energy but is probably not simply related to energy in the ordinary sense. The conservation law is interpreted as aHuyghen's principle for an intensity measured by measuring the relative accelerations due to the gravitational field. It is compared to a previously derived action conservation law for a classical, general relativistic electromagnertic field and with a covariant action conservation law that is valid in null (degeneate type II) metrics. Further propagation laws for null and type III waves are given under the assumption that the wave vector is hypersurface orthogonal. It is shown that in linear approximation the far and semi-far fields of a radiating quadrupole are null and type III respectively; the form of the conservation laws in these linearized metrics is discussed. In an appendix a “strongly” conserved form of the tensor ofBel andRobinson is suggested.     

Radiating fluid distribution interacting with scalar field

Taking the combined energy-momentum tensor for a perfect fluid, radially expanding the radiation and zero-mass scalar field, we investigate their interaction and obtain five new analytic solutions in a spherically symmetric Einstein universe. For the corresponding models various physical and geometrical properties are discussed. In one case an interesting equation of state is derived.     

Nonlinear massive spin-2 field generated by higher derivative gravity

We present a systematic exposition of the Lagrangian field theory for the massive spin-2 field generated in higher-derivative gravity upon reduction to a second-order theory by means of the appropriate Legendre transformation. It has been noticed by various authors that this nonlinear field overcomes the well-known inconsistency of the theory for a linear massive spin-2 field interacting with Einstein’s gravity. Starting from a Lagrangian quadratically depending on the Ricci tensor of the metric, we explore the two possible second-order pictures usually called “(Helmholtz-)Jordan frame” and “Einstein frame.” In spite of their mathematical equivalence, the two frames have different structural properties: in Einstein frame, the spin-2 field is minimally coupled to gravity, while in the other frame it is necessarily coupled to the curvature, without a separate kinetic term. We prove that the theory admits a unique and linearly stable ground state solution, and that the equations of motion are consistent, showing that these results can be obtained independently in either frame (each frame therefore provides a self-contained theory). The full equations of motion and the (variational) energy-momentum tensor for the spin-2 field in Einstein frame are given, and a simple but non-trivial exact solution to these equations is found. The comparison of the energy-momentum tensors for the spin-2 field in the two frames suggests that the Einstein frame is physically more acceptable. We point out that the energy-momentum tensor generated by the Lagrangian of the linearized theory is unrelated to the corresponding tensor of the full theory. It is then argued that the ghost-like nature of the nonlinear spin-2 field, found long ago in the linear approximation, may not be so harmful to classical stability issues, as has been expected.     

二维静态时空中Dirac场的重正化能动张量和Casimir效应

在一般渐近平直的二维静态黑洞时空中,利用重正化能动张量的一般性质, 对位于两“平行板”间满足Dirichlet条件的无质量Dirac场的重正化能动张量的真空期待值进行了分析和计算, 得到了一般表达式.利用该表达式可以给出各种具体渐近平直二维静态黑洞时空中的相应Casimir力.对于重正化能动张量及Casimir力与真空态定义(包括Boulware 真空态、Hartle-Hawking真空态和Unrum真空态三种情况)、Hawking辐射和反常迹的关系分别进行了讨论，给出了相应的表达式和计算结果. 关键词： 能动张量 Casimir 效应 黑洞 真空态     

Structure of the integrals of motion for a rotating body in the relativistic theory of gravitation

The energy-momentum tensor for the Kerr metric is calculated within the relativistic theory of gravitation. The external gravitational energy and angular momentum of a collapsing body are found. It is concluded from the analysis of their behavior near the gravitational radius that gravitational collapse is impossible.     

Local approach to dilatation invariance

Scale invariance is analyzed locally by coupling the energy-momentum tensor to a source which is the metric field of curved space-time. The resulting theory at the classical level has no mass parameters only if the general coordinate transformation group can be represented in Weyl's scheme. We further discuss the quantum extension of the theory; the Ward identities become anomalous under radiative corrections and the anomaly is shown to be connected with the instability of the classical metric field representation. The anomalies, recognized as the well-known trace anomalies for the energy-momentum tensor, are then reabsorbed by a perturbative alteration of the original metric field transformation law and we prove the modified Ward identities to be renormalizable in the flat limit. Finally we show that our approach is equivalent to the well-known parametric equations of the Callan-Symanzik type only if the dilatation invariance is not spontaneously broken. In the presence of spontaneous scale breaking we derive a functional equation which will be applied to cases of physical interest in a forthcoming paper.