Showing papers on "Gravitational field published in 1972"

Role of conformal three-geometry in the dynamics of gravitation.

Abstract: The unconstrained dynamical degrees of freedom of the gravitational field are identified with the conformally invariant three-geometries of spacelike hypersurfaces. New results concerning the action principle, choice of canonical variables, and initial-value equations strengthen this identification. One of the new canonical variables is shown to play the role of "time" in the formalism.

Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields

Abstract: A nearly spherical star collapses through its gravitational radius. Nonspherical perturbations exist in its density, pressure, electromagnetic field, and gravitational field, and in other (hypothetical) zero-rest-mass, integer-spin fields coupled to sources in the stars. Paper I analyzed the evolution of scalar-field and gravitational-field perturbations. This paper treats fields of arbitrary integer spin and zero rest mass, using the Newman-Penrose tetrad formalism. The analysis of each multipole ($\mathrm{order}=l$) of each field ($\mathrm{spin}=s$) is reduced to the study of a two-dimensional wave equation, with a "curvature potential" that differs little from one field to another. The analysis of this wave equation for the scalar case ($s=0$) carries over completely to fields of arbitrary spin $s$. In particular, any radiatable multipole ($l\ensuremath{\ge}s$) gets radiated away completely in the late stages of collapse; if the multipole is static prior to the onset of collapse, it will die out as ${t}^{\ensuremath{-}(2l+2)}$ at late times. Nonradiatable multipoles ($lls$) are conserved. This paper also treats gravitational perturbations in the Newman-Penrose framework, and supplies some technical details missing in the gravitational-perturbation analysis of Paper I.

Structure of the Gravitational Field at Spatial Infinity

Abstract: A formalism is introduced for analyzing the structure of the gravitational field in the asymptotic limit at spatial infinity. Consider a three‐dimensional surface S in the space‐time such that the initial data on S is asymptotically flat in an appropriate sense. Using a conformal completion of S by a single point A ``at spatial infinity,'' the asymptotic behavior of fields on S can be described in terms of local behavior at A. In particular, the asymptotic behavior of the initial data on S defines four scalars which depend on directions at A. Since there is no natural choice of a surface S in a space‐time, the dependence of these scalars on S is essential. The asymptotic symmetry group at spatial infinity, whose elements represent transformations from S to other asymptotically flat surfaces, is introduced. It is found that this group, which emerges initially as an infinite‐dimensional generalization of the Poincare group, can be reduced to the Lorentz group. A set of evolution equations is obtained: These equations describe the behavior of the four scalars under the action of the asymptotic symmetry group. The four scalars can thus be considered as fields on a three‐dimensional manifold consisting of all points at spatial infinity. The notion of a conserved quantity at spatial infinity is defined, and, as an example, the expression for the energy‐momentum at spatial infinity is obtained.

On a quadratic first integral for the charged particle orbits in the charged Kerr solution

Abstract: Associated with the charged Kerr solution of the Einstein gravitational field equation there is a Killing tensor of valence two. The Killing tensor, which is related to the angular momentum of the field source, is shown to yield a quadratic first integral of the equation of the motion for charged test particles.

Advanced Least-Squares Methods,

Abstract: : A general least-squares method (collocation) which encompasses, as special cases, least squares adjustment and least-squares prediction, is presented in detail and applied to various problems occurring in geodesy and photogrammetry, such as interpolation and coordinate transformation. In particular, this method permits an optimal simultaneous determination of geodetic positions and of the terrestrial gravity field by combining different data of any kind--terrestrial angle, distance and gravity measurements as well as data from advanced satellite techniques. To provide an adequate statistical background, an alternative statistical interpretation of the anomalous gravity field in terms of covariance analysis of individual functions is given, and its relation to the usual interpretation as a stochastic process on the sphere is discussed. (Author)

Theoretical frameworks for testing relativistic gravity. IV - A compendium of metric theories of gravity and their post-Newtonian limits.

Abstract: Metric theories of gravity are compiled and classified according to the types of gravitational fields they contain, and the modes of interaction among those fields. The gravitation theories considered are classified as (1) general relativity, (2) scalar-tensor theories, (3) conformally flat theories, and (4) stratified theories with conformally flat space slices. The post-Newtonian limit of each theory is constructed and its Parametrized Post-Newtonian (PPN) values are obtained by comparing it with Will's version of the formalism. Results obtained here, when combined with experimental data and with recent work by Nordtvedt and Will and by Ni, show that, of all theories thus far examined by our group, the only currently viable ones are general relativity, the Bergmann-Wagoner scalar-tensor theory and its special cases (Nordtvedt; Brans-Dicke-Jordan), and a recent, new vector-tensor theory by Nordtvedt, Hellings, and Will.

The Gravitational Field of Flat Galaxies

Abstract: A set of bi-orthogonal pairs of functions is derived which is especially suited for the calculation of the gravitational field of flat galaxies. The functions are Hankel tranforms of associated Laguerre functions, or ‘Hankel-Laguerre functions’. They can be calculated by means of recurrence relations, which are derived from a generating function. Numerical computations are most efficiently performed using a related set of Chebyshev type functions, which have more economic recurrence relations. The necessary algorithms for efficient calculation of the field are derived using the properties of Hankel-Laguerre functions. This field calculation, can be used for a relatively cheap and simple computer simulation of galaxies.

Hamiltonian formulation of spherically symmetric gravitational fields.

Abstract: Hamiltonian methods are used to study spherically symmetric gravitational fields with electromagnetism and a massless scalar field as sources. The constraints reduce to a first-order linear ordinary differential equation which is solved to yield the dynamical Hamiltonian, or the complete solution when no dynamics is present in the problem.

Recent results on the mass, gravitational field and moments of inertia of the moon

Abstract: Doppler tracking data from the Lunar Orbiter series of spacecraft have been used in a more complete analysis of the spherical harmonic coefficients of the lunar gravitational field through thirteenth degree and order. The value obtained for the mass of the Moon,GM = 4902.84 km3 s−2, is in good agreement with previous results and with results obtained by alternate procedures. Acceleration contour plots, derived from the gravitational coefficients, show correlations with surface features on the near side of the Moon, but are of questionable validity for the far side because of the lack of direct tracking data on the far side. Based on the most recent gravitational field data, the current estimate for the polar moment of inertia of the Moon isC/Ma 2 = 0.4019-0.002 +0.004. This value indicates that the interior of the Moon can be homogeneous, but some results presented strongly suggest that the Moon is differentiated, with an excess of mass in the direction toward the Earth.

Mapping onto Solutions of the Gravitational Initial Value Problem

Abstract: 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 weight 5 3 . These objects, in turn, give irreducible representations of the coordinate and scaling groups which are used to effect solutions of the initial value equations.

Mariner 9 celestial mechanics experiment: gravity field and pole direction of Mars.

Abstract: Analysis of the Mariner 9 radio-tracking data shows that the Martian gravity field is rougher than that of earth or the moon, and that the accepted direction of the Mars rotation axis is in error by about 0.5 deg. Contours of equivalent surface heights deduced from a sixth-degree solution for the Martian gravity field are presented. These contours represent the deviations from sphericity of a uniformly dense body with an external potential which is given by the first sixth-degree solution. In addition to Doppler observations, ranging or group-delay measurements have been made regularly since orbit insertion.

Perturbation theory for gauge invariant fields

Abstract: A method is developed for the manifestly covariant quantization of gaugeinvariant fields by means of a functional integration. It is shown that for the fields with non-Abelian gauge groups (the Yang–Mills and gravitational fields) fictitious particles appear naturally in the diagram technique, which are not present in the initial Lagrangian. An appearance of these particles restores the transversality of scattering amplitudes and the unitarity of the S-matrix.

Quasi-periodic orbits about the translunar libration point

Abstract: Analytical solutions for quasi-periodic orbits about the translunar libration point are obtained by using the method of Lindstedt-Poincare and computerized algebraic manipulations. The solutions include the effects of nonlinearities, lunar orbital eccentricity, and the sun's gravitational field. For a small-amplitude orbit, the orbital path as viewed from the earth traces out a Lissajous figure. This is due to a small difference in the fundamental frequencies of the in-plane and out-of-plane oscillations. However, when the amplitude of the in-plane oscillation is greater than 32,379 km, there is a corresponding value of the out-of-plane amplitude that will produce a path where the fundamental frequencies are equal. This synchronized trajectory describes a 'halo orbit' of the moon.

Post-newtonian gravitational radiation from point masses in a hyperbolic kepler orbit.

Abstract: The energy and the angular momentum radiated away in the form of gravitational waves from a system of two point particles with positive total energy are calculated in the lowest nonvanishing post-Newtonian approximation. By these radiations a particle arriving from infinity can be captured, and the cross section for such captures is determined as a function of the energy at infinity. The radiation from elliptic (bound) orbits can also be inferred, and is found to be in agreement with known results.

Some physical properties of neutrino-gravitational fields

Abstract: A new solution of the Einstein-neutrino field equations is given. This solution is of Plebanski class [4n]3 and describes a beam of neutrinos propagating along straight geodesies but possessing an inherent angular momentum density. Another previously known solution is also examined, and using some calculations given by Bonnor it is concluded that a uniform beam of neutrinos is gravitationally stable and that two such beams radiating in the same sense do not interact.

Gravitational radiation and neutrinos

Abstract: A class of neutrino-gravitational fields with zero energy-momentum tensor is defined. These space-times may also be interpreted as describing gravitational waves and are of Petrov typeD orN. A wave-like example of the latter is given.

Rayleigh Taylor instability of a viscous Hall plasma with magnetic field

Abstract: The influence of finite Larmor frequency on the stability of a viscous, finitely conducting liquid in a downward gravitational field under the influence of a uniform magnetic field directed along or normal to gravity, is investigated. The solution in each case is shown to be characterized by a variational principle Based on the variational principle, an approximate solution is obtained for the stability of a layer of fluid of constant kinematic viscosity and an exponentia density distribution. It has been found that finite resistivity and finite Larmor frequency do not introduce any instabifity in a potentially stable configuration. However, for a potentially unstable configuration we find that, for an ideal Hal plasma, the results depend on the orientation of the magnetic field, though the instability persists for all wave-numbers in the presence of non-ideal (finite resistivity and viscosity) effects. For the field aligned with gravity, it is found that a potentially unstable field-free configuration is stabilized if the buoyancy number B ( = gβ/12 V2) is less than unity. For B > 1, the instability arises for wave-numbers exceeding a critical value, which decreases on allowing for Hall terms in the generalized Ohm's law, suggesting a destabilizing influence of finite Larmor frequency. For an ambient horizontal magnetic field, it is found that an ideal plasma is stable, even for B > 0, for perturbations confined to a cone about the magnetic field vector. The angle of the cone of stable propagation, however, decreases on account of finite Larmor frequency.

The Uniform Transparent Gravitational Lens

Abstract: An elementary discussion is given of the gravitational deflection of light due to radially and cylindrically symmetric masses. The effect of the deflection on apparent luminosity of distant sources is also considered. All results are limited to weak, static, asymptotically flat gravitational fields. Emphasis here is on observations of sources aligned behind the disc of the deflecting mass so that the possible transparency of this mass is decisive in image formation. Detailed calculations are made for the simple case of the uniformly dense transparent sphere and comparisons are made with the opaque mass sphere. If they have the same mass and radius, the maximum light deflection produced by these lenses is nearly equal. However, their effects on the area of narrow light beams may be quite different. The uniform transparent sphere does not produce multiple images of one source and, to first order, introduces no image distortion.

Orbital and vortical motion in the Kerr metric

Abstract: The motion is analysed, in general, in the Kerr metric with the use of first integrals. Some of the high-energy particles and photons are found to move in a giant vortex around the axis of symmetry above and below the equatorial plane, dragged by the gravitational field.

Gravitational fields with groups of motions on two-dimensional transitivity hypersurfaces in a model with matter and a magnetic field

Abstract: For gravitational fields with metrics which admit of groups of motions multiply — transitive on 2-dimensional space-like invariant varieties, the exact solutions of the Einstein gravitational equations are given for the case when the sources of the gravitational field are dust-like matter and a magnetic field. A magnetic field is orientated along a direction orthogonal to transitivity hypersurface. The solutions contain arbitrary functions. In the case of transitivity hypersurface of positive curvature and in the absence of a magnetic field, the solution is reduced to the Tolman spherically symmetric solution for dust-like matter. The conditions are studied under which the solutions with a magnetic field become asymptotically isotropic and approach the flat and the open Friedmann models. The case of transitivity hypersurfaces with signature (+ −) is also considered.

The dispersion of gravitational waves

Abstract: A definition is given of a plane gravitational wave in a curved background space-time manifold. For a particular background metric, a dispersion relation for the waves is derived analogous to that satisfied by plane electromagnetic waves in a dilute plasma.

Exact gravitational field of the infinitely long rotating hollow cylinder

Abstract: The vacuum line element inside an infinitely long rotating hollow cylinder is the usual flat space line element. It is fitted in a most general way to the general cylindrical vacuum field outside at the singular hypersurfaceR0=const, representing the infinitely thin hollow cylinder. With the use of the jump conditions atR0=const the surface densities τ μ ν , of which the energy-momentum-stress tensor τ μ ν of the shell consists, are calculated. The physical properties of the cylinder, as derived from the eigenvalues and -vectors of τ μ ν , and the generated gravitational field are discussed in full detail.

Convection of a plasma in a gravitational field

Abstract: It is shown that the interchange instability of a magnetoplasma in a gravitational field can be stabilized by the combined effects of viscosity and resistivity.