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Showing papers on "Gravitational field published in 1978"


Journal ArticleDOI
K.S. Stelle1
TL;DR: In this article, the dynamical content of the linearized field is analyzed by reducing the fourth-order field equations to separated second-order equations, related by coupling to external sources in a fixed ratio.
Abstract: Inclusion of the four-derivative terms ∫RμνRμν(−g)1/2 and ∫R2(−g)1/2 into the gravitational action gives a class of effectively multimass models of gravity. In addition to the usual massless excitations of the field, there are now, for general amounts of the two new terms, massive spin-two and massive scalar excitations, with a total of eight degrees of freedom. The massive spin-two part of the field has negative energy. Specific ratios of the two new terms give models with either the massive tensor or the massive scalar missing, with correspondingly fewer degrees of freedom. The static, linearized solutions of the field equations are combinations of Newtonian and Yukawa potentials. Owing to the Yukawa form of the corrections, observational evidence sets only very weak restrictions on the new masses. The acceptable static metric solutions in the full nonlinear theory are regular at the origin. The dynamical content of the linearized field is analyzed by reducing the fourth-order field equations to separated second-order equations, related by coupling to external sources in a fixed ratio. This analysis is carried out into the various helicity components using the transverse-traceless decomposition of the metric.

1,209 citations


Journal ArticleDOI
TL;DR: In this paper, the authors adopt the point of view that a solution of Einstein's equations is an evolution of given initial Cauchy data, and propose a flat-space model of such elliptic equations (e.g. for maximal slicing) which suggests that this curvature leads to an exponential decrease in the proper time between time slices at late times.
Abstract: We adopt the point of view that a solution of Einstein's equations is an evolution of given initial Cauchy data. Implementing the evolution equations necessarily requires a determination, not directly dictated by the field equations, of the kinematics of the observers in terms of which the evolution is represented. In this paper we study the observers' kinematics (velocities and accelerations) in terms of the geometry of their congruences of world lines relative to families of time slicings of spacetime, which contrasts with the more usual approach of imposing particular "gauge" or "coordinate conditions." The types of conditions we suggest are adapted to the exact Einstein equations for general strong-field, dynamic spacetimes that have to be calculated numerically. Typically, the equations are three-dimensionally covariant, elliptic, and linear in the kinematical functions (the lapse function and shift vector) that they determine. The gravitational field enters in nonlinear form through the presence of curvature in the equations. We present a flat-space model of such elliptic equations (e.g. for maximal slicing) which suggests that this curvature leads to an exponential decrease in the proper time between time slices at late times. We show how the use of maximal slicing with minimal-distortion observers generalizes the notion of a stationary rest frame to dynamical asymptotically flat spacetimes. In cosmological spacetimes the use of minimum-distortion observers is shown to differentiate between those universes which contain only kinematic time dependence (e.g. open Kasner universe) and those in which dynamical degrees of freedom are present (e.g. mixmaster universe). We examine many examples and construct new coordinate systems in both asymptotically flat and cosmological solutions to illustrate these properties.

267 citations


Journal ArticleDOI
TL;DR: In this paper, the authors propose to replace the path integrals over the terms in the Taylor series by a discrete sum of the exponentials of all complex solutions of the Einstein equations, each solution being weighted by its one-loop term.
Abstract: The path-integral method seems to be the most suitable for the quantization of gravity. One would expect the dominant contribution to the path integral to come from metrics which are near background metrics that are solutions of classical Einstein equations. The action of these background metrics gives rise to a new phenomenon in field theory, intrinsic quantum entropy. This is shown to be related to the scaling behavior of the gravitational action and to the topology of the gravitational field. The quadratic terms in the Taylor series of the action about the background metrics give the one-loop corrections. In a supersymmetric theory the quartic and quadratic but not the so-called logarithmic divergences cancel to give a one-loop term that is finite without regularization. From the one-loop term one can obtain the effective energy-momentum tensor on the background metric. In the case of an evaporating black hole, the energy-momentum tensor will be regular on the future horizon. The usual perturbation expansion breaks down for quantum gravity because the higher (interaction) terms in the Taylor series are not bounded by the quadratic (free) ones. To overcome this I suggest that one might replace the path integrals over the terms in the Taylor series by a discrete sum of the exponentials of the actions of all complex solutions of the Einstein equations, each solution being weighted by its one-loop term. This approach seems to give a picture of the gravitational vacuum as a sea of virtual Planck-mass black holes.

177 citations


Journal ArticleDOI
TL;DR: In this paper, the effects of unphysical degrees of freedom from a quantized massless Rarita-Schwinger field interacting with a (quantized or classical) gravitational field was analyzed on the one-loop level.

155 citations



Journal ArticleDOI
TL;DR: In this paper, the authors considered linear perturbations of black hole models by a variety of fields and derived the crucial condition th at the field be singular on the inner horizon.
Abstract: Linear perturbations of black hole models by a variety of fields are considered. Perturbing fields include the zero rest mass scalar field in the case of Reissner-Nordstrom, and gravitational, electromagnetic and zero rest mass scalar perturbation in the case of the Kerr model. The analysis deals with the Ψ 0 components (in the Newman-Penrose (1962) formalism) of non-zero spin fields. The symmetry properties of the models are used to derive the crucial condition th at the field be singular on the inner horizon. This condition is independent of the field propagation equation. Initial data are then given in terms of incoming radiation from f - is shown that there exist wellbehaved initial data sets for which the resultant fields are singular on the inner horizon. It is emphasized that this instability result is dependent only on the global symmetries and causal structure of the models considered, and is independent of the precise nature of the perturbing field.

96 citations


Journal ArticleDOI
TL;DR: In this paper, the action which describes the interaction of gravitational and electron fields is expressed in canonical form, and the corresponding Hamiltonian constraints are derived and their (Dirac) bracket relations given.

95 citations



Journal ArticleDOI
TL;DR: In this paper, the conformal 2-structure is proposed as a way to represent the two gravitational degrees of freedom of the Einstein field equations, which is a solution to the Cauchy spacelike initial value problem.
Abstract: In this paper, we suggest that what we shall call the conformal 2‐structure may, in an appropriate coordinate system, serve to embody the two gravitational degrees of freedom of the Einstein (vacuum) field equations. The conformal 2‐structure essentially gives information concerning the manner in which a family of 2‐surfaces is embedded in a 3‐surface. We show that, formally at least, this prescription works for the exact plane and cylindrical gravitational wave solutions, for the double‐null and null‐timelike characteristic initial value problems, and for the usual Cauchy spacelike initial value problem. We conclude with a preliminary consideration of a two‐plus‐two breakup of the field equations aimed at unifying these and other initial value problems; and a discussion of some aspirations and remaining problems of this approach.

73 citations



Book
01 Jan 1978
TL;DR: In this article, the Hamiltonian Formulation of the Electrodynamic Equation of Motion has been used to describe the structure of space and time in the universe and its properties.
Abstract: 1 Introduction.- 1.1 Equations of Motion.- 1.2 The Mathematical Language.- 1.3 The Physical Interpretation.- 2 Analysis on Manifolds.- 2.1 Manifolds.- 2.2 Tangent Spaces.- 2.3 Flows.- 2.4 Tensors.- 2.5 Differentiation.- 2.6 Integration.- 3 Hamiltonian Systems.- 3.1 Canonical Transformations.- 3.2 Hamilton's Equations.- 3.3 Constants of Motion.- 3.4 The Limit t ? I +- ?.- 3.5 Perturbation Theory: Preliminaries.- 3.6 Perturbation Theory: The Iteration.- 4 Nonrelativistic Motion.- 4.1 Free Particles.- 4.2 The Two-Body Problem.- 4.3 The Problem of Two Centers of Force.- 4.4 The Restricted Three-Body Problems.- 4.5 The N-body Problem.- 5 Relativistic Motion.- 5.1 The Hamiltonian Formulation of the Electrodynamic Equation of Motion.- 5.2 The Constant Field.- 5.3 The Coulomb Field.- 5.4 The Betatron.- 5.5 The Traveling Plane Disturbance.- 5.6 Relativistic Motion in a Gravitational Field.- 5.7 Motion in the Schwarzschild Field.- 5.8 Motion in a Gravitational Plane Wave.- 6 The Structure of Space and Time.- 6.1 The Homogeneous Universe.- 6.2 The Isotropic Universe.- 6.3 Me according to Galileo.- 6.4 Me as Minkowski Space.- 6.5 Me as a Pseudo-Riemannian Space.

Journal ArticleDOI
TL;DR: In this article, the possibility of generating or exponentially increasing the number of waves in the metric of an ultrarelativistic rotating body has been demonstrated, depending on time amplification of the waves.

Journal ArticleDOI
TL;DR: In this article, the asymptotic structure of spacetime fields at large spacelike separation from sources is studied in terms of a three-dimensional boundary manifold representing spacelikes infinity.
Abstract: The asymptotic structure of gravitational fields at large spacelike separation from sources is studied. Limits of spacetime fields are discussed in terms of a three‐dimensional boundary manifold representing spacelike infinity. The boundary is endowed with the metric of a timelike unit hyperboloid. With sufficiently stringent conditions on the asymptotic spacetime geometry, the total energy–momentum and angular momentum emerge as integrals over any cross section of the hyperboloid at infinity. It is possible to identify physically relevant weaker conditions under which the energy–momentum, but not the angular momentum, is well defined. Under still weaker conditions, the energy–momentum also loses its meaning even though the spacetime admits a Minkowskian asymptote.

Journal ArticleDOI
TL;DR: In this article, the second-order differential equations of motion of spinning test particles (tops) were derived from a variational principle in a given gravitational background defined by a Riemannian metric and a torsion tensor.
Abstract: The (second-order differential) equations of motion of spinning test particles (tops) are derived from a variational principle in a given gravitational background defined by a Riemannian metric and a torsion tensor. The mass and (magnitude of) the spin of the top are conserved. There exists a Regge trajectory linking the mass and the spin of the top. Constants of the motion associated with Killing vectors of the metric along which the Lie derivative of the torsion tensor vanish are found.

Journal ArticleDOI
TL;DR: In this article, the Euler dynamical equations for the motion of the axis of figure of the rigid Earth are integrated analytically by the method of variation of parameters, and the mutual gravitational potential of n solid bodies is expanded without approximation in terms of harmonic coefficients of each body.
Abstract: The mutual gravitational potential ofN solid bodies is expanded without approximation in terms of harmonic coefficients of each body. As an application the Euler dynamical equations for the motion of the axis of figure of the rigid Earth are integrated analytically by the method of variation of parameters.

Journal ArticleDOI
Vincent Moncrief1
TL;DR: In this article, the problem of quantizing the gravitational fluctuations about a symmetric vacuum background spacetime with compact Cauchy surfaces was studied and it was shown that the allowed physical states must all be invariant under the symmetry transformations of the background space.
Abstract: We study the problem of quantizing the gravitational fluctuations about a symmetric vacuum background spacetime with compact Cauchy surfaces. In the context of lowest-order perturbation theory we show that the allowed physical states must all be invariant under the symmetry transformations of the background spacetime. This constraint does not unduly restrict the range of allowed states and is consistent with temporal evolution (in the presence of a timelike symmetry) or spacial localization (for a spacelike symmetry) provided the evolution or localization is interpreted intrinsically rather than with reference to the background spacetime.

Journal ArticleDOI
TL;DR: In this article, a line-of-sight gravity measurements derived from spacecraft-tracking data can be used for quantitative subsurface density modeling by suitable orbit simulation procedures such an approach avoids complex dynamic reductions and is analogous to the modeling of conventional surface gravity data.
Abstract: It is proposed that line-of-sight gravity measurements derived from spacecraft-tracking data can be used for quantitative subsurface density modeling by suitable orbit simulation procedures Such an approach avoids complex dynamic reductions and is analogous to the modeling of conventional surface gravity data This procedure utilizes the vector calculations of a given gravity model in a simplified trajectory integration program that simulates the line-of-sight gravity Solutions from an orbit simulation inversion and a dynamic inversion on Doppler observables compare well (within 1% in mass and size), and the error sources in the simulation approximation are shown to be quite small An application of this technique is made to lunar crater gravity anomalies by simulating the complete Bouguer correction to several large young lunar craters It is shown that the craters all have negative Bouguer anomalies


Journal ArticleDOI
01 Aug 1978-Nature
TL;DR: In this article, the authors used the zero frequency limit (ZFL) quantum technique of Weinberg to obtain a classical source which gives correct results in this limit and applied the ZFL method to the gravitational bremsstrahlung process.
Abstract: SUPERNOVAE, the violent collapses of massive stars into neutron stars, are considered to be the strongest known sources of gravitational radiation. The quadrupole (and higher) moments of the hydrodynamic motions of the infalling matter generate the radiation. A burst of neutrinos of total energy ∼0.1 M⊙c2 and of duration between ∼10−3 and 1 s is thought to accompany the collapse1. If this burst has nonspherical angular distribution, it will generate additional gravitational radiation— possibly comparable in magnitude to that caused by the collapse. That the neutrino burst might produce gravitational waves had been overlooked until recently, when Epstein suggested the idea2. This problem is of interest in its own right because a massless field is acting as the source of a radiation field. In electromagnetism there are no charged, massless particles; in fact, the existence of such a particle would lead to infrared divergences3. The gravitational effects of neutrinos have been studied previously2,4–6, but the approach we report here is new and has a nice analogue in electromagnetism. The emission of a neutrino is a quantum process and so we use the zero frequency limit (ZFL) quantum technique of Weinberg3 to obtain a classical source which gives correct results in this limit. Because the time scale of the burst (∼10−3–1 s) is much longer than that of the neutrino emission process (∼10−22 s) only the ZFL is needed in the calculation. The ZFL technique has been used in an analogous way in electromagnetism to calculate the electromagnetic radiation accompanying β-decay or electron capture7. Smarr has applied the ZFL method to the gravitational bremsstrahlung process8.

Journal ArticleDOI
TL;DR: In this paper, the observations that would be made by a uniformly accelerated observer are considered in detail, including the observer's event horizon, the variation of clock rates with position (which implies that the gravitational red shift is not totally dependent on curvature), and the effects to be expected when the motion of a freely falling object is followed.
Abstract: The observations that would be made by a uniformly accelerated observer are considered in detail. These include the observer’s event horizon, the variation of clock rates with position (which implies that the gravitational red shift is not totally dependent on curvature), and the effects to be expected when the motion of a freely falling object is followed. Consideration of the similarities between this reference frame and a real gravitational field is also given. A geometrical method for the construction of the reference frame of the accelerated observer is described in an Appendix.


Journal ArticleDOI
TL;DR: In this paper, the authors describe a procedure for generalizing static axially symmetric solutions of the vacuum Einstein field equations, and employ this procedure to append an external gravitational field to the C-metric black hole solution.
Abstract: We describe a procedure for generalizing static axially symmetric solutions of the vacuum Einstein field equations, and employ this procedure to append an external gravitational field to the C‐metric black hole solution. We conjecture that if the strength of the appended gravitational field is chosen appropriately, our generalized C‐metric will serve as a better model for an accelerating black hole than the original C‐metric.

Journal ArticleDOI
TL;DR: In this paper, the hydrodynamics of two droplets submerged in an unbounded arbitrary velocity field are studied by solving Stokes' equations for the flow fields in and around the droplets by means of the reflection method.

Journal ArticleDOI
TL;DR: In this paper, a full derivation of the one parameter family of twisting gravitational fields was given, which was previously reported by the author, and the coordinate ranges were specified for all possible cases.
Abstract: A full derivation is given of that one parameter family of type N twisting gravitational fields which was previously reported by the author. Explicit forms of this solution are obtained, and the coordinate ranges are specified for all possible cases. The general problem of the search for other type N twisting gravitational fields is also discussed; the differential equations required for that search are derived, and the extension of these equations to type (3,1) is given without derivation.

Journal ArticleDOI
TL;DR: In this paper, the potential of a single electron in a quasi-uniform field was derived from the partial differential equation satisfied by the electrostatic potential in terms of generalised harmonic functions.
Abstract: In a recent paper, Prof. Whittaker has discussed the effect, according to the general theory of relativity, of gravitation on electromagnetic phenomena. In particular, he has discussed electrostatics in gravitational fields of two kinds, namely (i) the field due to a single gravitating mass, in which case space-time has the metric discovered by Schwarzschild, and (ii) a limiting case of this, called a quasi-uniform field, in which the gravitational force is, in the neighbourhood of the origin, uniform. Whittaker’s general method, so far as electrostatical problems were concerned, was to solve the partial differential equation satisfied by the electrostatic potential in terms of generalised harmonic functions, and then, from these, to build up other solutions. In this way, he succeeded in finding an algebraic expression which represents the potential of a single electron in the quasi-uniform field; he did not, however, obtain a corresponding algebraic expression for the potential of an electron in the Schwarzschild field.

Journal ArticleDOI
TL;DR: In this paper, 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 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.
Abstract: 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).

Journal ArticleDOI
TL;DR: In this article, Teukolsky's equations are expressed directly in terms of the basic derivative operators of the theory and a preferred gauge in which two of the components of the Weyl tensor are governed by the same equations as a Maxwell field is suggested.
Abstract: As a preliminary towards a complete integration of the Newman-Penrose equations governing the gravitational perturbations of the Kerr black hole, the perturbations in the spin coefficients and in the components of the Weyl tensor, which vanish in the stationary state, are considered. The manner of treatment of the basic equations yields Teukolsky's equations expressed directly in terms of the basic derivative operators of the theory and, further, suggests a preferred gauge in which two of the components of the Weyl tensor are governed by the same equations as a Maxwell field. Various identities and relations that are needed in subsequent work are assembled. In two appendixes, the solution of Maxwell's equations in Kerr geometry and the perturbations of the charged Kerr-Newman black hole are considered.

Journal ArticleDOI
TL;DR: In this article, a modification of the Palatini Lagrangian for the free gravitational field that yields the vanishing of the torsion as a result of the field equations and requires only the assumption of the symmetry of the metric.
Abstract: We give a modification of the Palatini Lagrangian for the free gravitational field that yields the vanishing of the torsion as a result of the field equations and requires only the assumption of the symmetry of the metric. We transcribe this Lagrangian into the tetrad formalism and show how the tetrad form of the Einstein field equations follows from it. Some remarks on possible generalization to a theory with nonvanishing torsion in the presence of matter conclude the paper.

Journal ArticleDOI
TL;DR: In this paper, a simple discussion of the energy changes involved in faulting within a self-gravitating body is presented, where the authors show that the energy released by the body is a relatively small difference between large changes in the potential energy of the elastic and gravitational fields, as recognized by Dahlen.
Abstract: A simple discussion of the energy changes involved in faulting within a self-gravitating body shows that the energy released in faulting is simply ∝ ( p′ ij + p ij /2) u i dS j , where p′ ij is the ambient stress before faulting, p ij and u i are the stress change and displacement associated with faulting and the integral is over the fault surface. This result was obtained earlier by Dahlen. A new expression for the work done against gravity (− ∝ ρ g i u i dv where ρ is density, and g i is the acceleration of gravity, and the integral is over the entire volume of the body) is given in the form of an integral over the fault surface. For dip-slip faulting in the earth, the gravitational energy change may be several orders of magnitude greater than the energy released; that large change in gravitational energy is compensated for by a similar change in stored elastic initial energy. In such cases the energy released is the relatively small difference between large changes in the potential energy of the elastic and gravitational fields, as recognized by Dahlen.

Journal ArticleDOI
TL;DR: In this paper, a new formulation of the stationary axisymmetric vacuum gravitational field equations which is substantially different from the well known formulations of Lewis and Ernst is presented, and three methods are given for the construction of the full metric from e2γ.
Abstract: A new formulation of the stationary axisymmetric vacuum gravitational field equations which is substantially different from the well known formulations of Lewis and Ernst is presented. The basic variable is e2γ = -g11g44 and satisfies a field equation of the fourth differential order which may be interpreted as the condition that a certain 2-space has constant curvature, K = -1. The principal motivation is that for many known solutions and all known asymptotically flat (non-static) solutions, e2γ takes a much simpler functional form than either the metric coefficients, g44, g34 and g33, or the Ernst potentials, E and ξ . Three methods are given for the construction of the full metric from e2γ. A duality principle is invoked to provide a very similar field equation for the metric coefficient, e2γ-2u = -g11.