Rotating bodies in general relativity

Synge's approximation method is used in order to obtain the gravitational field of a massive body with an axis of symmetry around which it is rotating steadily. The method is carried out to include the second approximation. This means that terms of orderm ² are retained as significant, and there is an error of orderm ³ in the field equations,m being the mass of the body.     

Rotating dust clouds in general relativity

We present an exact solution of Einstein’s equations representing interpenetrating clouds of rotating dust. The solution is a member of the van Stockum class; it contains singularities at the centres of the clouds. These are sources of the angular momentum which is displayed by the metric at infinity. It is not clear whether the rotating dust contributes to the angular momentum. In the case of two clouds there is a conical singularity between the central ones. For three clouds the conical singularity may be absent.     

Rotating frames in special relativity. II. Generalized theory

The transformation theory for rotating frames presented in a previous paper (Strauss, 1974) is generalized by replacing the usual conditionr=R for Rr=Rg( _R ) so thatr is now defined for all values ofR,0R. This generalization does not affect the kinematic transformation {,T}{^(r),{^(r)}}, and the result group structure required by the theoretical constraints previously established, provided the old parameter r (=R) is now identified throughout with eitherr orR; for physical reasons it must be identified withR. The functiong, which cannot be fixed by theoretical constraints, determines the degree of geometrical anisotropy in the rotating planez=const. More specifically, sinceg enters the expression for the ratioC/D (circumference/diameter) its choice corresponds to the choice of a congruence definition for lengths in radial and tangential directions. While on this (purely geometrical) levelg remains undetermined, it can be uniquely determined experimentally on the kinematic level, e.g., by observing in the motion of a free particle. Thus the supremacy of kinematics over geometry is explicated by a further instance. At the same time, special relativity theory (SRT) is shown to belong to the class of theories with theoretically unsolvable problems.     

Causal functions in general relativity. II

It is shown that important space-time structure conditions of stable causality and strong causality are characterized in terms of causal functions.     

The gauge in general relativity. II

The present paper and a companion work are intended as one possible realization of a non-customary gauge theory of general relativity-as set forth in only broad outline in an earlier work. In this first paper, it is found that both radar echo delay and the perihelion shift differ slightly from their customary expressions. Unfortunately, it is also found that the usual statement of the principle of equivalence does not hold in the present formulation. Finally, in the second paper, a single cosmological model is investigated that appears to be promising.This work forms part of a Ph.D. dissertation by L.S.     

Elementary particles in bimetric general relativity. II

The possibility of an elementary particle in the framework of classical bimetric general relativity is explored further. A model is considered which is filled with a pressureless primal fluid having a fixed ratio of charge density to mass density. This ratio is assumed to be0, ± ₀ , where ₀ is a universal constant <0.5. If the particle charge is assumed to be ±1/3e, the mass is a fraction of the Planck mass, the fraction being greater than0.0285.     

Rotating,charged, and uniformly accelerating mass in general relativity

A new general class of solutions of the Einstein-Maxwell equations is presented. It depends on seven arbitrary parameters that group in a natural way into three complex parameters m + in, a + ib, e + ig, and the cosmological constant λ. They correspond to mass, NUT parameter, angular momentum per unit mass, acceleration, and electric and magnetic charge. The metric is in general stationary and axially symmetric. These solutions are of type D and contain as special cases all known solutions of type D belonging to this class. The known solutions are recovered by performing limiting transitions. An appropriate limit of our solutions describes an electromagnetic field in flat spacetime. We investigate the properties of that field. Its singular region corresponds in general to two circles moving with uniform acceleration in the positive and negative directions along the axis of symmetry. One can easily extend our solutions to the complex domain. Then it turns out that the metric can be written in a double Kerr-Schild form.     

Slow motion approximations in general relativity II. Gauge transformations

When the field equations of general relativity are expanded in powers of a small parameter, the general covariance of the exact theory implies a corresponding gauge invariance of the equations obtained in the expansion. In a slow motion expansion, the derivation of this gauge transformation is complicated by the fact that the time coordinate is singled out for special treatment. In a previous paper, a new (3 + 1)-dimensional decomposition of the field equations was obtained which is particularly suitable as a starting point for slow motion approximations. The present paper gives a systematic method, again using covariant techniques throughout, for obtaining the corresponding gauge transformations to arbitrarily high accuracy. The calculations are explicitly carried out as far as is required in the 2 1/2-post-Newtonian approximation.     

Gravitational interaction of two spinning particles in general relativity. II

Within the framework of the previous paper, we complete the set of equations of motion by including the spin propagation equation at first order. We check this equation with the known result on the precession, which is obtained by means of slow motion approximation of our result. A new scheme of expanding equations of motion is also introduced. It will be useful to undertake higher-order calculations.     

On the occurrence of naked singularities in general relativity. II

It is shown that naked shell crossing singularities can occur in the gravitational collapse of a spherically symmetric ball of perfect fluid for a large family of equations of state in which the pressure has an (arbitrarily large) upper bound, and, moreover, that this behaviour is stable with respect to spherically symmetric perturbations of the initial data, as well as with respect to perturbations of the equation of state.Work supported by the Deutsche Forschungsgemeinschaft.     

A rotating dust cloud in general relativity II

I give a stationary solution of Einstein's equations representing a rotating cloud of dust. The solution is asymptotically flat, and has no curvature singularities. However, embedded in the cloud is a rotating surface layer of negative mass which precisely cancels the mass of the dust cloud. The solution throws light on the van Stockum class of rotating dust solutions, of which it is a member.     

On isoperimetric surfaces in general relativity,II

We determine the optimal way to enclose volume in a class of domains inside certain Friedmann–Robertson–Walker metrics. The method employed is an adaptation of the Bray–Morgan isoperimetric comparison procedure to the Lorentzian setting. We also make some remarks on isoperimetric comparison in the Riemannian setting, for rotationally-symmetric space-like slices in non-vacuum space-times.     

A new approach to the theory of relativity. II. The general theory of relativity

The considerations of Part I are extended and the experimental data and hypotheses that led to the establishment of the general theory of relativity are analyzed. It is found that one of the fundamental assumptions is that light is propagated homogeneously; i.e., by using arbitrary systems of coordinates, propagation of light can be represented by a homogeneous quadratic form. This is shown to be an assumption that can be verified by experiment, at least in principle. As a result of adding a number of further assumptions to this, the usual formalism of the general theory of relativity can be established. In the above point of view, the general theory of relativity—like any other theory—cannot be built upad hoc, but is built on distinct physical hypotheses, each of which can be subjected to test by experiment.     

Rotating frames in special relativity

A uniformly rotating frame is defined as the rest frame of a particle revolving with constant velocityω in a circle about theZ-axis of an inertial frame Σ⁰. Under the conditionz=Z,r=R, theoretical constraints are established for the solution of the transformation problem Σ⁰→Σ^ω r,Σ^ω r being the cylindrical subframe of Σ^ω. The unique solution of the problem in cylindrical coordinates is isomorphic to the special Lorentz transformationL _x, withβ=v/c replaced byβ _r=ωr/c. Hence the intrinsic geometry on the surface of a rotating cylinder is Euclidean. Though there exists no complete intrinsic geometry on the surface of a rotating disk, the geodesics on it are straight lines while the circumference of a concentric circle isK _r2πr as predicted by Einstein.     

Non-trivial field configurations in five-dimensional relativity

We note the possibility of the existence of “twisted” real scalar field configurations in certain Kaluza-Klein five-dimensional spacetime models, and propose, as a consequence of the requirement of classical field stability, that they must be considered in preference to the corresponding untwisted structure when modelling charge quantization in terms of the five- dimensional Klein - Gordon equation.     

Space tensors in general relativity II: Physical applications

The general theory of space tensors is applied to the study of a space-time manifoldsV ₄ carrying a distinguished time-like congruence Γ. The problem is to determine a physically relevant spatial tensor analysis over (V ₄, Γ), in order to proceed to a correct formulation of Relative Kinematics and Dynamics. This is achieved by showing that each choice of gives rise to a corresponding notion of ‘frame of reference’ associated with the congruence Γ. In particular, the frame of reference (Γ, ∇*) determined by the standard spatial tensor analysis is shown to provide the most natural generalization of the concept of frame of reference in Classical Physics. The previous arguments are finally applied to the study of geodesic motion inV ₄. As a result, the general structure of the gravitational fields in the frame of reference (Γ, ∇*) is established. This work was assisted by funds from the C.N.R. under the aegis of the activity of the National Group for Mathematical Physics.     

From massive gravity to modified general relativity II

We continue our investigation of massive gravity in the massless limit of vanishing graviton mass. From gauge invariance we derive the most general coupling between scalar matter and gravity. We get further couplings beside the standard coupling to the energy–momentum tensor. On the classical level this leads to a further modification of general relativity.     

Metric-affine variational principles in general relativity II. Relaxation of the Riemannian constraint

We continue our investigation of a variational principle for general relativity in which the metric tensor and the (asymmetric) linear connection are varied independently. As in Part I, the matter Lagrangian is minimally coupled to the connection and the gravitational Lagrangian is taken to be the curvature scalar, but we now relax the Riemannian constraint as far as possible—that is, as far as the projective invariance of the assumed gravitational Lagrangian will allow. The outcome of this procedure is a gravitational theory formulated in a volume-preserving space-time (i.e., with torsion and tracefree nonmetricity). The vanishing of the trace of the nonmetricity is due to the remaining vector constraint. We also discuss the physical significance of the relaxation of the Riemannian constraint, the possible relaxation of the vector constraint, the notion of the hypermomentum current, and its possible relation to elementary particle physics.