Showing papers in "General Relativity and Gravitation in 1991"

Cosmological models with variable gravitational and cosmological “constants”

Abstract: We consider the Einstein field equations with perfect fluid source and variable Λ andG for the Robertson-Walker metric. When conservation of energy momentum is postulated and the deceleration parameter is assumed constant, we find perfect gas equation of state models in the Euclidean and non-Euclidean cases. The resulting models offer an alternative to the inflationary scenario; they also explain the huge value of the cosmological term in the early universe.

Null geodesics in black hole metrics with non-zero cosmological constant

Abstract: We study the radial motion along null geodesics in the Reissner-Nordstrom-de Sitter and Kerr-de Sitter space-times. We analyze the properties of the effective potential and we discuss circular orbits. We find that: 1) the radius of circular photon orbits in the Reissner-Nordstrom-de Sitter space-times does not depend on the cosmological constant. We show also how this is related to properties of the optical reference geometry. 2) For a specific range of the cosmological constant, photons with high impact parameter may travel radially between the cosmological horizon and the black hole horizon in the equatorial plane of the Kerr-de Sitter space-times.

Particle production from signature change

Abstract: We consider the (massless) scalar field on 2-dimensional manifolds whose metric changes signature and which admit a spacelike isometry. Choosing the wave equation so that there will be a conserved Klein-Gordon product implicitly determines the junction conditions one needs to impose in order to obtain global solutions. The resulting mix of positive and negative frequencies produced by the presence of Euclidean regions depends only on the total width of the regions, and not on the detailed form of the metric.

Cosmology withG and Λ coupling scalars

Abstract: Cosmology with the gravitational and cosmological constants generalized as coupling scalars in Einstein's theory is considered. A general method of solving the field equations is given. Fifteen exact solutions for zero pressure models satisfyingG=G0(R/R0) n are given in the Appendix; they are briefly discussed.

Comments on the computation of Liapunov exponents for the Mixmaster universe

Abstract: To explain the discrepancy between recently computed vanishing Liapunov exponents for the evolution of Mixmaster universes and the positive Liapunov exponent for the associated 1-dimensional map first discussed by Belinskii, Khalatnikov, and Lifshitz, the Liapunov exponents computed from a numerical universe evolution are compared using several time variables. Previous numerical results of vanishing Liapunov exponents were obtained with time variables which increased roughly exponentially in each epoch. Here it is found that minisuperspace proper time, which increases by a fixed amount during each epoch, yields nonvanishing Liapunov exponents within the limited number of epochs numerically accessible. The map parameteru as measured along the trajectory attains the values predicted by the map to very high accuracy (except near the maximum of expansion) even though the metric coefficients deviate in some cases from idealized Mixmaster behavior. The number of consecutive single epoch eras is shown to be related to the presence of u in an interval bounded by ratios of Fibonacci numbers.

Wormholes and the construction of compactified phases

Abstract: In the context of Classical General Relativity, assuming the positive energy condition, we study the possibility of connecting a region where some dimensions are compact to another region where compactification does not exist. In some simple models, where compactification is achieved through the condensation of gauge fields, we find that the matching of these two regions implies the existence of a wormhole. We also discuss the possibility of spontaneous quantum nucleation of a compactified phase in the midst of an uncompactified space.

Five-dimensional Kaluza-Klein black holes coupled to massive scalar particles

Abstract: Spherically symmetric solutions coupled to massive scalar particles in five-dimensional Kaluza-Klein theory are obtained. The solutions contain two event horizons. The inner horizon corresponds to the Schwarzschild black hole and the outer one is a new horizon which is produced by the massive scalar particles. It is found that the massive modes contribute an effective cosmological constant to the four-dimensional Einstein theory.

Algebraic invariant curves in cosmological dynamical systems and exact solutions

Abstract: The Einstein field equations for a number of classes of cosmological models have previously been written as polynomial systems of ordinary differential equations. We show that, for restricted parameter values, these equations admit algebraic invariant curves, which, in turn, lead to exact solutions of the field equations. This property explains the recent discovery of a number of exact solutions and is used to produce additional ones.

The age problem in inhomogeneous universes

Abstract: Recent determinations of globular cluster ages imposes severe problems for the standard cosmological models. Before introducing a cosmological constant, other possibilities of solving the problem should be investigated carefully. In this paper, the backreactions of inhomogeneities in realistic, clumpy universes and their influence on age determinations are studied. Applying a recently developed approximation scheme, it is shown that the backreactions lead to an underestimation of the age of the universe as inferred from a measurement of today's Hubble parameter. Observations of the cosmic microwave background radiation impose constraints on the parameters. For a simple model within the framework of pancake theory for structure formation on a flat expanding background, it is shown that the age problem may be solved by taking into account the backreactions of inhomogeneities in an averaged sense. No cosmological constant is needed.

Shear-free spherically symmetric solutions with conformal symmetry

Abstract: Dyer, McVittie and Oattes (1987) presented the field equations for shearfree perfect fluid spacetimes which are spherically symmetric and admit a conformal symmetry. Two special solutions of these equations are found. Furthermore, in the general case, one field equation is solved in terms of a Painleve transcendent, while the remaining equation is reduced to an Emden-Fowler equation.

Fifth force from Kaluza-Klein unification

Abstract: We discuss the possible modification of Einstein's theory of gravitation due to the fifth force generated by the Kaluza-Klein dilaton. In particular we discuss the gravitational redshift, the deflection of light, the precession of perihelia, and the time-delay of radar echo around a spherically symmetric black hole in a 7-dimensional Kaluza-Klein unification which hasE2 as the internal isometry. It is shown that the long-range effect of the higher-dimensional fifth force is characterized by the dilatonic charge carried by the black hole even when it is neutral.

Null charts and naked singularities in spherically symmetric, homothetic spacetimes

Abstract: We give a unified derivation of a null chart for all spherically symmetric, homothetic spacetimes. These spacetimes contain an interesting class of naked singularities which we are also able to elucidate. Much use is made of graphical representation; in particular a chart of the spacetimes based on their homothetic group motions is introduced. Dust spacetimes, and two homogeneous examples with non-zero pressure (flat Robertson-Walker and a Kantowski-Sachs example) are studied in detail. We show the horizon structure in the null atlas, in comoving coordinates, in terms of the areal radius and comoving time, and in the homothetic diagrams. The critical delay between comoving observers for the onset of “nakedness” is interpreted in terms of a decreasing mass concentration in the spirit of Thorne's “hoop” conjecture. We also give a simple criterion for the existence of apparent horizons isolating the various singularities, and study in detail how this criterion is circumvented in the naked examples. We conclude that this type of naked singularity is a consequence of the imposed homothetic symmetry, by showing it to be generally present and timelike in the homothetic group chart even when it is not visible at comoving infinity (before the onset of criticality). It is the delayed final collapse of initially distant observers in inhomogeneous spacetimes that causes the initial singularity to become visible at comoving infinity. We conclude that these examples do not present an obstacle to the “Event Horizon Conjecture” as summarized by W. Israel (1984). That is, one can formulate criteria for the formation of apparent horizons that do not imply that all singularities are necessarily so enclosed. It is still possible that all singularities stronger than homothetic are isolated by an apparent horizon, in the spirit of Tipler's conjecture.

Complex coordinates and Kähler potential for the Atiyah-Hitchin metric

Abstract: In the case of self dual metrics, we detail and put together the Geroch three-dimensional formulation of the Einstein equations and the Boyer-Finley results on Killing vectors to obtain a set of complex coordinates and a Kahler potential for the Atiyah-Hitchin metric.

Remarks about late stage homogeneous cosmological dynamics

Abstract: Some simple observations are made to clear up misconceptions regarding the behavior of spatially homegeneous monotonically expanding general relativistic cosmological models at very late times, as well as to emphasize some simple features of the dynamics in the direction away from an initial singularity even in the case of recollapse. A few examples illustrating these ideas are discussed, including the derivation of some approximate solutions of the late stage field equations.

Conservation of vector-valued forms and the question of the existence of gravitational energy-momentum in general relativity

Abstract: Recently, Dalton has shown that conserved vector-valued currents have nullcovariant (rather than partial) derivatives and has discussed the corollary that there is no gravitational energy-momentum in general relativity (GR). An equivalent statement on conservation of vector-valued currents was given by E. Cartan in 1922. Given the potential implications of the Dalton corollary, Cartan's contribution to this problem is reproduced here in more modern but very powerful and relatively simple language. For completeness, his accompanying work on the current of GR is also included. The question then arises of whether or not the absence of a gravitational source in GR is a problem. For the purpose of comparison, we consider the gravitational sector of teleparallelism-withtorsion theories. These exhibit field equations of the formGμν =Tμν, whereGμνis the Einstein tensor for the Levi-Civita (part of the total) connection, which is conserved. The source now has two distinct parts, one depending on both the metric and the torsion and the other metric independent. They can be viewed as being respectively gravitational and non-gravitational. In addition, the conservation law of these theories is global. It thus follows that several unesthetic (and certainly controversial) features of GR should be viewed as real problems, and not as necessary concomitants of the geometrization of the physics.

A BRST-like operator for space with zero curvature but non-zero torsion tensor

Abstract: A new type of BRST-like operatorQ has been constructed for spaces with zero Riemann curvature tensor but with non-zero torsion. It is invariant under local coordinate transformation, and its proof depends upon the validity of the first Bianchi identity. Also, there exists a supersymmetry associated withQ. Finally, the cohomology induced byQ has been investigated.

Gauge-invariant variations in the redshift and cosmic background radiation anisotropies

Abstract: The gauge-invariance of the calculations determining anisotropics in the cosmic microwave background radiation (the Sachs-Wolfe effect) is re-examined. It is shown that the results obtained are gauge-invariant only if a physically-based definition of the surface of last scattering is implemented, in a context where perturbations of the surface of last scattering as well as of the space-time are taken into account. Any physical interpretation of the results based on their splitting into ‘scalar’, ‘vector’, and ‘tensor’ parts, is unique only if non-local (unverifiable) conditions are imposed; locally, any such interpretation is non-unique. The physical meaning of the Sachs-Wolfe potentialB and associated redshift formula depends on implementing a very specific gauge, without a clear physical or geometric meaning; its implications do not extend to other, more usual, gauges.

Interaction between gravity and moving superconductors.

Abstract: We propose a unified phenomenological theory to investigate the interaction between arbitrarily moving superconductors and gravitational fields including the Newtonian gravity, gravitational waves, vector transverse gravitoelectric fields, and vector gravitomagnetic fields. In the limit of weak field and low velocity, the expressions for the induced electromagnetic and gravitational fields in the interior of a moving superconductor are obtained. The Meissner effect, London moment, DeWitt effect, effects of gravitational wave on a superconductor, and induced electric fields in the interior of a freely vibrating superconductor are recovered from these two expressions. We demonstrate that the weak equivalence principle is valid in superconductivity, that Newtonian gravity and gravitational waves will penetrate either a moving superconductor or a superconductor at rest, and that a superconductor at rest cannot shield either vector gravitomagnetic fields or vector transverse gravitoelectric fields.

Gauge invariant electromagnetic coupling with torsion potential

Abstract: Electromagnetism is coupled to torsion in a gauge invariant manner by relaxing minimal coupling and introducing into the Lagrangian a term bilinear in the electromagnetic field tensor and the torsion potential. The resulting coupling between electromagnetism and torsion is examined and a solution corresponding to traveling coupled waves is given.

Constructing physically valid fluid spheres in general relativity

Abstract: We extend a method by Goldman to include conditions which are both necessary and sufficient for construction of physically acceptable relativistic fluid spheres. We thus give the conditions for having finite and non-negative pressure and finite and positive density inside the fluid sphere. We also give conditions for the pressure gradient and the density gradient to be negative and for the speed of sound to be less than the speed of light in vacuum. We further give the condition for the trace of the energy-momentum tensor to be positive and for the relativistic adiabatic index to be larger than 4/3. A model given by Goldman is examined, and we find that all these conditions are fulfilled. We further show that this model is stable. However, we also give a class ofnew exact solutions, and show that these models are physically valied. These models are also stable with respect to small radial disturbances. We calculated the total mass, the density and the physical radius for these fluid spheres. Some of these spheres turn out to be extremely compact. They can have a radius of only several centimeters, but typically contain the mass of a planet. We put these models forward as tentative exact solutions for spheres which could contain the hidden dark matter in the universe.

Quartic equations and classification of Riemann tensors in general relativity

Abstract: For space-times in general relativity, the Petrov classification of the Weyl conformai curvature and the Plebanski or Segre classification of the Ricci tensor each depend on the properties of the roots of quartic equations. The coefficients in these quartic equations are in general complicated functions of the space-time coordinates. We review the general theory of quartic equations, and discuss algorithms for determining the existence and values of multiple roots. We consider practical implementation of an algorithm and the consequent Petrov classification. Tests of programs embodying this algorithm, using the computer algebra system CLASSI based on SHEEP, are reported.

The interplay between relativistic gravitational, centrifugal and electric forces: a simple example

Abstract: We show that some striking properties of the circular motion of charged, ultrarelativistic particles close to the circular photon orbit around a charged black hole (found by Balek, Bicak and Stuchlik) can be easily understood in terms of the centrifugal force vanishing at the circular photon orbit. We also comment briefly on a possible non-uniqueness of the recently-proposed operational definition of the centrifugal force in static spacetimes.

Reduction of Einstein's field equations for twisting typeN-vacuum fields with anH 2

Abstract: The problem of vacuum typeN solutions of Einstein's field equations with Killing vector and homothetic group can be reduced to a single ordinary differential equation of the third order for a single real function. Hauser's solution [4] is an exceptional case. If the homothetic parameterN takes the valueN=2 the third order differential equation becomes surprisingly simple. This caseN=2 is therefore the most promising one for the search of exact solutions.

Finsler spaces with Riemannian geodesics

Abstract: In Finsler spaces the intervalds=F(x i ,dx i ) is an arbitrary function of the coordinatesx i and coordinate incrementsdx i withF homogeneous of degree one in thedx i . It is shown that for Riemannian spacesds R 2=g ij dx i dx i which admit a non trivial covariantly constant tensorH i .(x k ) there is an associated Finsler space with the same geodesic structure. The subset of such Finsler spaces withH i .(x k ) a vector or second rank decomposable tensor is determined.

Gravitational collapse with a cosmological constant

Abstract: We give a simple example of a spacetime in which one region undergoes gravitational collapse while another undergoes inflation. The spacetime is a spherical ball of dust in a region with a cosmological constant. The implications of this example for the “cosmic no hair” conjecture are discussed.

An investigation of the dynamical evolution of a class of Bianchi VI−1/9 cosmological models

Abstract: We consider the evolution of a subclass of the orthogonal spatially homogeneous cosmologies of Bianchi type VI−1/9. Expansion normalized variables are introduced to write the Einstein field equations for these models as a three-dimensional autonomous system of ordinary differential equations. This system is analyzed qualitatively using the techniques of dynamical systems, and a cosmological interpretation of the phase portraits is given.

The frequency of light propagating in gravitational wave spacetimes

Abstract: We study the propagation of light signals in two exact solutions of vacuum Einstein's equations, in the geometric optics approximation. The gravitational wave affects the frequency of light propagating in it (a known effect) even if the metric componentsgtt andgti (i=1, 2, 3) are zero. Moreover we find that red-/blue-shift depends on the relative spatial position of the light source and the observer, even if the wavefront is perfectly uniform, leading to a characteristic redshift pattern.

String dust solutions of the Einstein field equations with spherical or static cylindrical symmetry

Abstract: The Einstein field equations are solved assuming spherical or static cylindrical symmetry and a string dust source. All solutions which may be matched onr =a to an external vacuum solution having the same symmetry are found.