Static Gravitational Field Research Articles

Classification of stationary axisymmetric gravitational fields

A gravitational stress tensor is defined for an arbitrary axisymmetric stationary metric, and the stationary solutions of Weyl, Lewis, Papapetrou, Kerr, Marek and Newman, Unti and Tamburino are classified by expressing them in a single canonical co-ordinate system.

New conservation laws for zero rest-mass fields in asymptotically flat space-time

Some recently discovered exact conservation laws for asymptotically flat gravitational fields are discussed in detail. The analogous conservation laws for zero rest-mass fields of arbitrary spin s = 0,½,1,...) in flat or asymptotically flat space-time are also considered and their connexion with a generalization of Kirchoff’s integral is pointed out. In flat space-time, an infinite hierarchy of such conservation laws exists for each spin value, but these have a somewhat trivial interpretation, describing the asymptotic incoming field (in fact giving the coefficients of a power series expansion of the incoming field). The Maxwell and linearized Einstein theories are analysed here particularly. In asymptotically flat space-time, only the first set of quantities of the hierarchy remain absolutely conserved. These are 4 s + 2 real quantities, for spin s , giving a D ( s , 0) representation of the Bondi-Metzner-Sachs group. But even for these quantities the simple interpretation in terms of incoming waves no longer holds good: it emerges from a study of the stationary gravitational fields that a contribution to the quantities involving the gravitational multipole structure of the field must also be present. Only the vacuum Einstein theory is analysed in this connexion here, the corresponding discussions of the Einstein-Maxwell theory (by Exton and the authors) and the Einstein-Maxwell-neutrino theory (by Exton) being given elsewhere. (A discussion of fields of higher spin in curved space-time along these lines would encounter the familiar difficulties first pointed out by Buchdahl.) One consequence of the discussion given here is that a stationary asymptotically flat gravitational field cannot become radiative and then stationary again after a finite time, except possibly if a certain (origin independent) quadratic combination of multipole moments returns to its original value. This indicates the existence of ‘tails’ to the outgoing waves (or back-scattered field),which destroys the stationary nature of the final field.

The gravitational field of planes in general relativity

From the requirement of the plane symmetry the form of the exterior and static gravitational field of the plane shell is found. Its geometry is demonstrated by embedding the selected sections into the plane space. The exterior solution is linked up with the plane and with the couple of plane-parallel planes at rest. It is shown that the existence of strong pressures in the planes (which are necessary for the manifestation of the rest of both planes) makes it difficult to compare it directly with the corresponding classic case. If the pressure is relaxed, the planes will move towards one another by hyperbolic motion. It is shown that the hyperbolic acceleration of the plane is the same as in the classical case. The gravitational field outside the plane remains static so that no gravitational radiation arises.

Simple wave in a conductive medium in a stationary gravitational field: PMM vol. 32, n≗5, 1968, pp. 960–962

Simple wave in a conductive medium in a stationary gravitational field: PMM vol. 32, n≗5, 1968, pp. 960–962

Mechanics in euclidian terms giving all the three einstein effects

The first part of this paper is concerned with test particles in a radially symmetric and stationary gravitational field. It is shown that if the special theory of relativity is valid in the infinitesimal neighborhood of any point of such a field, then a single new assumption will suffice to yield equations that predict all the three Einstein effects. Notably, the equations suggest also an energy process associated with the motion of a test particle in a gravitational field that may give an explanation to such phenomena as novae and radio stars. The second part of this paper shows that without further assumptions the equations for the radially symmetric and stationary field may be extended to a general scalar field theory. A simple solution of the latter theory reveals effects that may prove to be of interest to astronomy. The paper contains also a brief discussion of a necessary modification of the corpuscular theory of light in order to bring it in line with the present theory.

On the energy of a static gravitational field

On the energy of a static gravitational field

Semigenerally covariant equations of motion—II the significance of the “tail” and the relation to other equations of motion

The semigenerally covariant equation of motion of a particle of finite mass moving in an “external” gravitational field, obtained in the previous paper, is further discussed. At first we decompose it into four scalar equations along four orthonormal vectors moving with the particle. It is shown that if the particle moves along a geodesic line in the external field, then the “tail” vanishes. The vanishing of the “tail,” on the other hand, implies a motion that tends to be geodetic, without the appearance of any runaway solution. The “tail” term is then determined for the case of slow motion (to an accuracy up to the seventh order in c −1) of two finite mass particles. It is shown that its structure is very similar to that obtained for the (test) charged particle moving in a static gravitational field, i.e., it depends on the velocity of the particle and the Riemann tensor of the “external” field. We then show that both the Einstein-Infeld-Hoffmann and the Havas-Goldberg equations of motion can be obtained as limiting cases of the semigenerally covariant equation when one expands the latter in c −1 and G, respectively.

Stationary gravitational fields due to symmetrical systems of bodies

By a method of successive approximations given in earlier papers, it is possible to find the gravitational field of a single body, at rest or with stationary internal motion. That method involved the proof of certain critical conditions by an argument dependent on a topological fact—the surface of a body can be continuously transformed into the infinite sphere without passing through matter. This proof fails when the system consists of more than one body. The present paper supplies a proof of quite a different kind, based on the symmetry of the system, and so extends the method of successive approximations to cover systems con­sisting of several bodies, subject to certain conditions of symmetry.

Stationary gravitational fields due to single bodies

In previous papers it has been shown how stationary gravitational fields due to single bodies may be calculated by a method of successive approximations. The method is now improved and made more compact by using the contravariant components of the energy tensor instead of the covariant components.

The energy-momentum tensor of a static gravitational field

The method of a tetrad field proposed by Rayski was used to determine the energy-momentum tensor for an approximate solution of the Einstein equations in the static case, from which it follows that the static gravitational field has non-zero mass but zero momentum, as was to be expected.

Falling charges

The radiative damping force on an electrically charged particle falling freely in a static weak gravitational field is computed in the nonrelativistic limit of small velocities. It is shown that, in this limit, the force separates naturally into two components, a conservative part which arises from the fact that the mass of the particle is not concentrated at a point but is partly distributed as electric field energy in the space surrounding the particle, and a nonconservative part which depends linearly on both the velocity and the Riemann tensor. The conservative force is shown to correspond to a repulsive inverse square potential and to make a retrograde contribution to the perihelion precession. The nonconservative part is shown to produce an average energy loss identical with that of the traditional formula which is used for accelerations caused by nongravitational forces. Because the nonconservative force depends on the velocity rather than its second derivative, however, the phenomenon of preacceleration does not occur with gravitational forces. The questions answered by this investigation are of conceptual interest only, since the forces involved are far too small to be detected experimentally.

The principle of equivalence

Various statements identified with the principle of equivalence are not acceptable because they are either not generally valid or are simply the definition of an inertial coordinate system. The accepted statement refers, roughly speaking, to the equivalence of inertial mass with passive and active gravitational mass. The clarification of this problem is greatly eased by a detailed study of the static homogeneous gravitational field which precedes the discussion of the equivalence principle. The role of electromagnetic phenomena, and, in particular, the presence of a charge in such a field is analyzed in detail.

Static gravitational fields in general relativity

Static gravitational fields are defined in terms of the existence of a set of observers mutually « at rest » with respect to each observer’s Fermi coordinate frame. The Fermi co-ordinate frame is chosen because it is the only « non-rotating » frame for an observer. It is shown that this definition is equivalent to the usual one, and the time-reversibility of static fields is derived from the new definition. Application of Fermi cylindrical co-ordinates to an axially symmetric static field is discussed.

Note on the harmonic oscillator in general relativity

The theory of relativity shows that the times measured by two observers will in general be different if they are in relative motion, so that their respective times between any two coincidences will differ. Bergmann [1] has investigated the problem of a particle moving in a small simple harmonic motion in a static gravitational field, and has found that the time difference for this particle and an observer at rest becomes zero whenever the particle passes through the centre and limits of its swing. This problem will now be dealt with in a different manner, using Schwarzschild's interior solution of the gravitational equations. The exterior solution for a point mass is not suitable in the present case, due to the singularity of the field at a point in the path of the particle.

Phenomenological linear theory of gravitation: Part I. Classical mechanics

Reasons are given why a linear theory of gravitation, if not the final word on gravity, yet is of considerable interest. Starting from a theory of considerable generality, we narrow down this generality by choosing the various constants in the theory on considerations of convenience as well as adjustment to experimental data; two of the constants are made zero. The number of facts predicted by the theory exceeds the number of adjustable constants appearing in it. Our theory differs from both Birkhoff's or Moshinsky's theory and from the linearization of Einstein's theory. In the present paper we explain the choice of the gravitational Lagrangian and discuss the choice of interaction of gravity with particles and other fields. We calculate the static gravitational field around the sun assumed to be at rest. Gravitational effects due to stresses inside the sun are taken into account as they are of a slightly larger magnitude than the relativistic effects leading to the advance of the perihelion. We discuss the equations of motion of a point mass in this gravitational field and consider both slow particles ( e.g. Kepler orbits of planets) and fast particles ( e.g. photons passing the sun). When the slow-particle case is expanded to second-order terms in v 2 c 2 , the advance of the perihelion is found. The results are compared with available experimental evidence, and the choice of the constants of the theory is given as a function of the more accurate results which future experiments may procure. Some possible choices of the constants are indicated.

Temperature Equilibrium in a Stationary Gravitational Field

Temperature Equilibrium in a Stationary Gravitational Field

On the Thermodynamical Equilibrium of Fluids in Gravitational Fields

It is shown that the thermodynamical equilibrium condition for a fluid at rest in a static gravitational field is, apart from the Tolman condition $T{({g}_{44})}^{\frac{1}{2}}=\mathrm{const}.$, governed by the condition $\ensuremath{\alpha}{({g}_{44})}^{\frac{1}{2}}=\mathrm{const}.$, where $\ensuremath{\alpha}$ is the chemical potential of the fluid (including mass energy) regarded as a function of the space coordinates. This relation constitutes the relativistic generalization of the well-known Gibbs condition for the equilibrium in a gravitational field.

A Particle at Rest in a Static Gravitational Field

It is shown that, for a particle to be at rest in a static, axially symmetric gravitational field, the force on the particle must vanish. This result is then generalized to the case of an arbitrary static gravitational field.

Thermal Equilibrium in a General Gravitational Field

In the absence of any appreciable gravitational field the conditions satisfied by the temperature of a medium having no thermal flow from one portion to another, are well known to be those of uniformity throughout the system. In the presence of a gravitational field, however, the conditions imposed at thermal equilibrium on the proper temperature -- as measured at different points of the medium by local observers -- are considerably more complicated, owing to the association of inertia and weight with all forms of energy and the consequent tendency for heat to flow from regions of higher to those of lower gravitational potential. Nevertheless, for the special case of static gravitational fields, these more complicated conditions have already been obtained,(1,2) and it is the purpose of the present article to investigate the conditions for thermal equilibrium in the general case of any kind of gravitational field.

XXIV.The axially symmetric stationary gravitational field

XXIV.<i>The axially symmetric stationary gravitational field</i>