Showing papers on "Gravitation published in 1970"

Effective potential for even parity Regge-Wheeler gravitational perturbation equations

Abstract: The Schr\"odinger-type equation for odd-parity perturbations on a background geometry has been extended to the even-parity perturbations. This should greatly simplify the analysis for calculations of gravitational radiation from stars and from objects falling into black holes.

Post-Newtonian metric for a general class of scalar-tensor gravitational theories and observational consequences

Abstract: Space-time Riemann metric for scalar-tensor gravitational theories with arbitrary omega parameter

Spherical Symmetry and Mass‐Energy in General Relativity. I. General Theory

Abstract: The mass‐energy of spherically symmetric distributions of material is studied according to general relativity. An arbitrary orthogonal coordinate system is used whenever invariant properties are discussed. The Bianchi identity is shown to imply that the Misner‐Sharp‐Hernandez mass function is an integral of two combinations of Einstein's equations for any energy‐momentum tensor and that mass‐energy flow is conservative. The two mass equations thus found and the mass function provide a technique for casting Einstein's field equations into alternative forms. This mass‐function technique is applied to the general problem of the motion of a perfect fluid and especially to the examination of negative‐mass shells and their relation to singular behavior. The technique is then specialized to the study of a known class of solutions of Einstein's equations for a perfect fluid and to a brief treatment of uniform model universes and the charged point‐mass solution.

The 2½-post-Newtonian equations of hydrodynamics and radiation reaction in general relativity

Abstract: In this paper the equations of hydrodynamics in the 2½-post-Newtonian approximation to general relativity are derived. In this approximation all terms of O(c-5) are retained consistently with Einstein's field equations; it is also the approximation in which terms representing the reaction of the fluid to the emission of gravitational radiation by the system first make their appearance. The paper is in four parts. In Part I (by S. C.) the lowest-order terms in the metric coefficients are derived which are consequences of the imposition of the Sommerfeld radiation-condition at infinity. It is shown (following an early investigation of Trautman) that these terms are of O(c-5) in g00, of O(c-6) in g0α, and of O(c-5) in gαβ. Unique expressions are obtained for these terms. They are found to be purely of Newtonian origin. In Part II (by S. C. and F. P. E.) the equations of motion governing the fluid in the 2½-post-Newtonian approximation are derived. In addition to the coefficients already determined, these equations depend on a knowledge of the term of O(c-7) in goo. This term is determined by an explicit appeal to the field equation. It is further shown that this approximation brings no change to the density (c2ρμ0√-g) and the linear momentum (πα) that are conserved in the second post-Newtonian approximation. In Part III (by S. C.) it is shown that the terms of O(c-5) in the equations of motion contribute principally to the dissipation of the energy and the angular momentum conserved in the second post-Newtonian approximation. The rates of dissipation of energy and of angular momentum that are predicted are in exact agreement with the expectations of the linearized theory of gravitational radiation. Finally, in Part IV (by S. C. and F. P. E.) the energy, θ00-c2ρμ0√-g, to be associated with the 2½-post-Newtonian approximation is derived by evaluating the (0, 0)-component of the Landau-Lifshitz complex and the conserved density in the 3½-post-Newtonian approximation.

Derivation of the Equations of Motion of a Gyroscope from the Quantum Theory of Gravitation

Abstract: Previous work on the gravitational two-body problem is surveyed Next, we present a new approach, which we consider to be simpler and more transparent than the usual methods because it is based on a gravitational potential energy This enables us to carry out our calculations using only the familiar tools of Newtonian mechanics and the Euler-Lagrange equations Starting from a gravitational potential energy derived from Gupta's quantum theory of gravitation, the classical motion of a spherical gyroscope in the gravitational field of a much larger mass with a quadrupole moment is found The results of the precession of the spin are compared with those of Schiff, and a detailed derivation of the results of O'Connell for the effect of a quadrupole moment (and higher moments) on the precession of the spin is presented In addition, we present some new results First, we show that the quadrupole moment manifests its presence in another way, which also contributes to the precession of the gyroscope a term that is about ten times larger than what could be detected Second, with regard to the precession of the orbit, in addition to the usual contributions, our results include the effects of the spin of both particles (which enables us to calculate the effect of the rotation of Mercury on the precession of its perihelion)

Colliding gravitational waves.

Abstract: NOWHERE should the nonlinear features of general relativity show up more clearly than in the collisional interaction of two gravitational waves. One of the direct consequences of the linearity of Maxwell's equations is that electromagnetic waves pass straight through each other, and this is probably one of the best attested facts of physics. It may readily be demonstrated1 that such a principle of superposition can never apply to gravitational waves travelling in nonparallel directions, but the precise way in which the waves will diffuse through each other has not hitherto been understood. The problem may have a distinct bearing on observational phenomena, because the gravitational fluxes observed by Weber2 appear to be large enough to contribute significantly to the curvature of the universe3,4, hence large enough to make the linearized approximation invalid (a high frequency approximation has, however, been successfully applied to the cosmological problem by Isaacson5). We shall describe here a coordinate system in which the problem of colliding plane waves may be discussed in general relativity; a particular solution expressible in elementary functions and representing at least a portion of such a collisional situation will be given.

Theory of Particles with Variable Mass. I. Formalism

Abstract: The equivalence principle (through the mechanism of the gravitational red shift) allows one to set up a classical formalism with the proper time as an extra degree of freedom, independent of the coordinate time, and with an immediate physical interpretation. Then proper time and mass occur as conjugate variables in a canonical formalism, leading to a gravitational theory of particles with variable mass. The nonrelativistic theory and a relativistic vector theory of gravity are described as models. The theory is capable of providing a dynamical framework for cosmological models with the creation of matter. Some simple examples are discussed, including the steady‐state universe with continuous creation, where the correct relation between the density of matter and the Hubble constant appears automatically, with no free parameters.

Theory of Particles with Variable Mass. II. Some Physical Consequences

Abstract: The formalism of the previous paper, in which mass and proper time are treated as independent dynamical variables in a canonical formalism, is shown to imply certain physical consequences. There will exist a mass vs proper time uncertainty relation; trajectories and proper time will be exactly determinable in an external gravitational field, while mass will be determinable in an external electromagnetic field; and conventional quantum mechanics will imply that equivalence is invalid for low‐lying quantum states. This leads to a second possible way to quantize a system in a gravitational field, which introduces a fundamental length. It is shown that it is possible to test for quantum interference effects of gravitational systems with present technology and conventional techniques, using the earth's gravitational field.

Measuring the gravitational interaction of elementary particles

Abstract: Elementary particles gravitational interactions, identifying tensor field with local gravitation

Elasticity Theory in General Relativity

Abstract: Elasticity theory in general relativity formulated from classical nonlinear three dimensional theory, discussing thermodynamics and weak field limit

Mach’s Principle and Einstein’s Theory of Gravitation

Abstract: The increased interest of physicists in Einstein’s theory of gravitation over the past years has been accompanied by a number of discussions of Mach’s suggestion concerning a relation between inertial forces and distant stars. There has been, however, no accord on the mathematical formulation of Mach’s ideas and on their meaning in General Relativity. My purpose is to review, in the framework of the present-day understanding of Einstein’s theory, the changes in interpretation which Mach’s ideas have experienced and to sort out different statements collectively labeled as Mach’s principle.

On the singularities of cosmological solutions of the gravitational equations.

Abstract: Publisher Summary This chapter examines the general properties of the cosmological solutions of the gravitational equations near a time singularity. The customarily used (Friedmann) cosmological solution of Einstein's gravitational equations is based on the assumption that matter is distributed in space homogeneously and isotropically. This assumption is very far-fetched mathematically, apart from the fact that its fulfillment in a real world can at best be only approximate. The solution obtained for the centrally symmetrical problem is actually a particular case of a more general class of solutions. In the absence of matter, there is naturally no centrally-symmetrical solution as the free gravitational field cannot have such symmetry.

Newman‐Penrose Constants and Their Invariant Transformations

Abstract: The origin and significance of the Newman‐Penrose (N‐P) constants of the motion are examined from the point of view that constants of the motion generate invariant transformations. Here the calculation makes use of a generalization of Green's theorem to a situation applicable to the coupled Einstein‐Maxwell fields in general relativity. One finds strictly electromagnetic constants generated by an incoming electromagnetic shock wave, with dipole symmetry, at future null infinity. The gravitational constants contain an admixture from the electromagnetic field. They are generated by an incident quadrupole gravitational shock wave at future null infinity (J+). Both the electromagnetic dipole field and the gravitational quadrupole field behave like linearized fields at J+. All higher‐multipole fields do not uncouple from the nonlinear corrections induced by the self‐coupling of the gravitational field and its coupling with the electromagnetic field. It is shown that the gravitational constants are related to t...

Some effects of gravitational tides on a model Earth's core

Abstract: A simple two-density earth model, with no dissipation and a point moon, is examined for the net effects of lunar tides and precession on the dynamics of the fluid core. The problem is discussed in terms of the fundamental parameters of the system, which determine the tidal configuration and the precession rate. The resultant fluid flow is generally nontrivial, and has been shown by other experimental and theoretical studies to drive geophysically significant interior deviations from constant vorticity. Criteria are deduced that show when the tides can cancel precessional effects, a case not existing in the earth. Because of the elementary nature of the model, the problem remains completely analytic, although it is thus limited in its relation to the earth.

Asymptotic analysis of oscillatory mode of approach to a singularity in homogeneous cosmological models

Abstract: Publisher Summary This chapter presents the evolution of the metric in the oscillatory mode of approach to a singularity in homogeneous cosmological models. It is possible to carry out analytic and statistical investigation of the evolutions of a model with appreciable fullness in the asymptotic limit of times that are arbitrarily close to the singularity. It is appropriate to recall that although the physical applicability of Einstein's equations in their present form can be clarified under the indicated singular conditions only by a future synthesis of physical theories, the existing gravitational theory itself does not lose its logical cohesion at any density of matter. The process of evolution of the metric on approaching the singular point consists, consequently, of successive periods, during each of which the distance scales oscillate along two spatial axes and decrease monotonically along the third axis. On going from one era to another, the direction along which the monotonic decrease of the distances takes place is transferred from one axis to another.