Static Gravitational Field Research Articles

Equations of motion of test bodies in a spherically symmetric, static gravitational field

We consider a class of theories of gravity in which the motion of test particles is governed by the path equation (with respect to a general affine connection Г). When we restrict attention to a spherically symmetric, static gravitational field, the path equation is characterized by three arbitrary functions of the gravitational field (as opposed to two in the case of metric theories where Г={ }). We find that there are essentially only two constraints on the three functions by appealing to solar system experiments. Therefore, we must supplement the equation of motion with other physical laws to obtain a value for the third arbitrary function (this, of course, is not necessary in the case of metric theories). If we consider theories in which both Г andg play a physical role we find in certain circumstances that this “third” condition is sufficient to prove that the theories under investigation reduce to their “metric” form.

A simple example of a classical gauge transformation

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Clocks and gravity

We consider the assumption that clocks measure proper time-that is, in a gravitational field ideal clocks are governed by the equationds2=g ij dxi dxj-and give some theoretical and experimental constraints on clock measurements. In particular, we find that if we assume that clocks are governed by an equation of the formds4=c ijkl dxi dxj dxk dxl, then this equation must reduce to the quadratic equation in a weak, spherically symmetric, static gravitational field (at least to first order in the Newtonian gravitational potentialU), otherwise “additional” contributions to the time-delay effect of radar propagation (that are not observed) are predicted.

Schiff's Conjecture on Gravitation

Considered here is a class of theories of gravity characterized by a set of equations which represent the gravitational and electromagnetic structure of the theories in a spherically symmetric and static gravitational field. If one demands that the weak equivalence principle (WEP) and the principle of universality of gravitational red shift (UGR) be satisfied, it is found that the theories under investigation must be metric. This result lends support to the current version of Schiff's conjecture that WEP+UGR\ensuremath{\rightarrow}EEP, where EEP refers to the Einstein equivalence principle.

Globally stationary but locally static space-times: A gravitational analog of the Aharonov-Bohm effect

It is well known that gravitational fields may be locally the same but globally distinct due to differences in the topology of their underlying manifolds. Globally stationary but locally static gravitational fields provide an example of gravitational fields which are locally the same but globally distinct in spite of the homeomorphism of their underlying manifolds. Any static metric on a space-time manifold with nonvanishing first Betti number ${R}_{1}$ is shown to generate an ${R}_{1}$-parameter family of such solutions. These fields are seen to provide a gravitational analog of the electromagnetic Aharonov-Bohm effect. The exterior field of a rotating infinite cylinder of matter is discussed as an exactly soluble example.

Static gravitational fields in a general class of scalar–tensor theories

The static field equations are investigated within the framework of a general class of scalar-tensor theories of gravitation proposed by Nordtvedt. In the Brans-Dicke and Barker theories, it is shown that in vacuum g00 is functionally related to the scaler field. Two families of exact solutions are also obtained for a static spherically symmetric gravitational field in Barker theory.

Self-interaction of a point charge in the Kerr space-time

Determines the electric and magnetic self-field of a point charge at rest on the symmetry axis of the Kerr space-time in the coordinate system in which the metric describes locally a constant, static and homogeneous gravitational field. The result differs from that of a uniformly accelerated point charge in Minkowski space-time because of the influence of global boundary conditions. It follows that one can determine the induced self-force on the point charge.

Mass-energy in static gravitational fields

The mass-energy equation in static gravitational fields is shown to beE =g 44 mc 2, which agrees with the expressionE =m 0 c 2(dx/ds)ξμ for the energy in a gravitational field possessing a timelike Killing vector ξ. For the Schwarzschild field this leads toE s ⋞m 0 c 2 + 1/2m 0 v 2 −km 0 M/r. For the Reissner-Nordstrom field an additional term describing the interaction between the mass and the charge is found to be 2πkm 0 Q 2/c 2 r 2. In the Kerr-Newman case more terms are found due to the central rotating gravitating mass.

Axially Symmetric and Stationary Solution of New General Relativity

In New General Relativity and axially symmetric and stationary gravitational field equation agrees with that of General Relativity with ω replaced by ω = (1 + x/ λ)1/2ω, provided the integrated orbital angular momentum L and the integrated spin angular momentum S are connected by L = (λ/x −1)S at spatial infinity. Here λ is a new parameter, which might agree with the Einstein constant x.

On the analyticity of stationary gravitational fields at spatial infinity

It is proved that all stationary, vacuum solutions of Einstein’s equations which satisfy certain weak differentiability conditions characterizing asymptotic flatness, possess an analytic structure near spatial infinity. This analyticity theorem implies the existence of a multipole expansion whose coefficients can be expressed in terms of the Geroch–Hansen multipole moments defined at the point at infinity on the conformal manifold. This proves a longstanding conjecture that these moments uniquely determine the local structure of a stationary, asymptotically flat, vacuum metric.

Axially Symmetric, Stationary Gravitational Field Equations and Pseudospherical Surfaces

Axially Symmetric, Stationary Gravitational Field Equations and Pseudospherical Surfaces

Gravitational lens

A plastic lens is described which will bend light rays parallel to the lens axis in the same way as a static (Schwarzschild) gravitational field. This allows one to observe the optical distortions caused by a (nonrotating) black hole directly. The double-imaging property is discussed with particular application to the recently announced double quasar which is suspected to be a gravitational double image of a single quasar. A detailed cosmological model calculation is performed in order to place limits on the mass of the galaxy acting as the lens. It is seen that if the double quasar is a double image and if the lens is a massive galaxy then we will have established that the quasar redshift is cosmological rather than gravitational. The gravitational lens thus occupies a key role in resolving this question.

Magnetic field creation by electron-weak interaction in a gravitational field

We give rough arguments showing that a stationary gravitational field should produce a magnetic field by vacuum polarization effects on neutrinos and electrons.

Escape of photons from an infinite cylinder in Brans-Dicke theory of gravitation

Solutions for a cylindrically symmetric static gravitational field in Brans-Dicke theory are obtained. It is found that the condition for escape of photons emitted from the surface of the source depends on two parameters, instead of only the mass parameter as in the case of Mardar's solution in Einstein's theory.

Static gravitational and Maxwell fields in the general scalar tensor theory

The expression for g00 as a function of the scalar field Ψ is obtained in the general scalar tensor theory of gravitation proposed by Nordtvedt and later discussed by Barker, assuming that there exists a functional relationship between them. Exact solutions for a plane symmetric static gravitational field are also obtained in this theory. Further the calculations are extended for the static electrovac with the assumption that here both g00 and the scalar field Ψ are functions of the electrostatic potential φ, and the results are different from those previously obtained in the corresponding situation of Brans–Dicke theory.

The stationary gravitational field near spatial infinity

Any stationary, asymptotically flat solution to Einstein's equation is shown to asymptotically approach the Kerr solution in a precise sense. As an application of this result we prove a technical lemma on the existence of harmonic coordinates near infinity.

Complex Ernst-Harrison potential and nonrelativistic limits of stationary vacuum gravitational fields

Expressions are derived for determining the potentials of the nonrelativistic gravitational fields corresponding to stationary vacuum gravitational fields described by given complex Ernst-Harrison potentials E or ξ on a definite region of the three-space ¯V3 with given metric tensork (¯V3, is determined by the set of trajectories of a timelike Killing field). The dependence of the form of the nonrelativistic potentials on the parameters of the symmetry group SL(2, R) of the equations of the stationary vacuum gravitational fields with respect to transformations of the form of E is found.

The static gravitational field near spatial infinity I

A static solution to Einstein's equations, which is asymptotically flat near spatial infinity, is shown to be asymptotically Schwarzschildian. As an application of this result we prove a technical lemma on the existence of asymptotically flat harmonic coordinates near infinity.

Ring laser and ring interferometer in accelerated systems

The purely geometric optical and systematic method of treating the general Sagnac effect is investigated. The four-velocity of the ray propagating in a moving isotropic medium is expressed in terms of the wave four-vector and the medium four-velocity. An explicit form of Fermat's principle applicable for a moving isotropic medium in a stationary gravitational field is derived. The expressions of a fringe shift and a frequency shift are developed to deal with various types of ring interferometers and ring lasers with an isotropic medium in the beam path.

The Newtonian theory of gravitation and its generalization

The classical, Newtonian theory of gravitation is generalized in two stages. First, a theory is constructed which is valid for a class of strong, static gravitational fields. This theory, which we call the newtonian theory, is compatible with all the classical tests of relativistic gravitation. The newtonian theory is then rewritten in a covariant, geometrical form and is generalized to give a complete theory of gravitation whose post-Newtonian approximation is in agreement with all observations.