Attractive Gravitational Force Research Articles

Chaotic Dynamics in Charged‐Particle Beams: Possible Analogs of Galactic Evolution

During the last couple of years of his life, Henry Kandrup became intensely interested in using charged-particle beams as a tool for exploring the dynamics of evolving galaxies. He and I recognized that both galaxies and charged-particle beams can exhibit collisionless relaxation on surprisingly short time scales, and that this circumstance can be attributed to phase mixing of chaotic orbits. The chaos is often triggered by resonances caused by time dependence in the bulk potential, which acts almost identically for attractive gravitational forces as for repulsive electrostatic forces superposed on external focusing forces. Together we published several papers concerning evolving beams and galaxies, papers that relate to diverse topics such as the physics of chaotic mixing, the applicability of the Vlasov-Poisson formalism, and the production of diffuse halos. We also teamed with people from the University of Maryland to begin designing controlled experiments to be done at the University of Maryland electron ring. This paper highlights our collaborative findings as well as plans for future investigations that the findings have motivated.

Chaotic Collisionless Evolution in Galaxies and Charged‐Particle Beams

Both galaxies and charged particle beams can exhibit collisionless evolution on surprisingly short time scales. This can be attributed to the dynamics of chaotic orbits. The chaos is often triggered by resonance caused by time dependence in the bulk potential, which acts almost identically for attractive gravitational forces and repulsive electrostatic forces. The similarity suggests that many physical processes at work in galaxies, although inaccessible to direct controlled experiments, can be tested indirectly via controlled experiments with charged particle beams, such as those envisioned for the University of Maryland electron ring currently nearing completion.

HYDRODYNAMIC LIMIT FOR THE VLASOV–POISSON–FOKKER–PLANCK SYSTEM: ANALYSIS OF THE TWO-DIMENSIONAL CASE

We consider the hydrodynamic limit for the VPFP system in dimension two, dealing with general initial data having finite mass, energy and entropy. The limit equation consists in a drift-diffusion equation, where the drift velocity is defined by means of the Poisson relation. Our result is twofold. In the case of repulsive (electrostatic) forces, we prove the convergence globally in time in a weak L1setting. Considering attractive (gravitational) forces, the same result applies provided a certain scaling parameter is large enough. This is precisely the assumption which prevents the formation of Dirac masses in finite time in the limit equations, as recently shown by Dolbeault–Perthame.

Obtaining equations of motion for charged particles in the (v/c)3-approximation by the Einstein-Infeld-Hoffmann method

We consider some principal methodological problems that appear when the Einstein-Infeld-Hoffmann method is used to find approximate solutions of the general relativity equations and to obtain information about the motion of particles whose interaction force is much greater than the gravitational attraction force. Among these problems are normalizing approximate expressions by expanding exact solutions written in the same coordinate conditions used in the Einstein-Infeld-Hoffmann method, assigning the smallness orders depending on relations between the smallness parameters in play, and verifying cancellations of divergent terms arising in surface integrals. Solving these questions in accordance with the internal logic of the Einstein-Infeld-Hoffmann method results in new tools and techniques for applying the method. We demonstrate these tools and techniques in the example of the problem of the motion of two electrically charged pointlike particles in the (v/c)3-approximation.

On the intrinsic gravitational repulsion

Newton’s hesitation to publish his Principia was presumably caused by his inability to explain why volumes were irrelevant to radial action of gravity, which appeared somewhat incomplete without possible nonradial effects. His doubts about completeness of his gravitational theory are justified by results from several 20th century experiments, which suggested presence of a nonradial impact of gravity in addition to the radial attractive force that he has introduced. Existence of nonradial effects is implied by mathematical and physical preconditions, if total energy is to be conserved, and it has already been confirmed by experiments. Nonradial potentials do not affect directly the radial Newtonian potentials, which generate the radial force of gravitational attraction between bodies, but they decrease energy of the rays that run across gravity fields. The nonardial potentials also contain repulsive effect of gravity that becomes dominant at very large distances where the usual attractive radial force practically vanishes. They provide thus global mechanism for an accelerated expansion of the universe and so they explain the lack of superclusters in it. They also imply gravitational contraction, yet another new long-range effect that may explain formation of galaxies.

The construction of periodic orbits close to triangular libration points for the three-body problem in a resistive medium

A circular, restricted three-body problem is considered when a passively gravitating point also experiences the action of small resistive forces, acting in the opposite direction to the absolute velocity vector. The nature of the loss of stability of the triangular libration points is studied using the Poincaré method in whcih the ratio of the magnitude of the resistive force to the force of gravitational attraction serves as the small parameter. Asymptotically stable periodic orbits are constructed. Two Lyapunov families of periodic orbits, which exists in the neighbourhood of the libration points of the classical theory, are the generating families. Calculations were carried out for mass ratios corresponding to the Earth-Jupiter and Earth-Moon systems, with different values of the parameters characterizing the law of resistance.

On the possibility of gravitational collapse in the relativistic theory of gravity

In the example of dust matter, it is shown that a gravitational attractive force does not lead to the formation of “black holes” in the relativistic theory of gravity. It is proved that in the absence of matter, the gravitational field is also absent. Therefore, a vacuum is not a source of a gravitational field. The mechanism of energy production in the process of the accumulation of matter into massive objects is discussed.

Instability of Synchronized Binary Neutron Stars in the First Post-Newtonian Approximation of General Relativity

We investigate the stability property of binary neutron stars (BNSs) just before the merging in the first post-Newtonian(PN) approximation. Stability analysis is performed making use of equilib­ rium configurations for synchronized BNSs which are obtained by the numerical scheme developed in a previous paper. NSs are modeled by means of the polytropic equation of state with the polytropic exponent r=2 and 3. From numerical calculations, we find that as in the Newtonian case, in the PN approximation, the secular instability will occur for synchronized BNSs at a critical angular velocity .!krlt before the surfaces of the two stars come into contact. The PN correction changes not only the gravitational attraction force between two NSs, but also the configuration of each NS of a binary system. As a result, Jhlt in the PN approximation is ~10-15 % larger than that in the Newtonian case for a NS of mass MADM~ 1.4 M. and radius rA ~ 10-15 km. The implication of this property to the orbital evolution of BNSs just before merging is discussed.

Equilibrium Condition in the Axisymmetric N-Reissner-Nordstrom Solution

We solve the source-free Einstein-Maxwell equations in a static and axisymmetric spacetime by using the inverse scattering method that is one of the soliton techniques to solve the gravitational field equation. We obtain an exact solution that describes n Reissner-Nordstrom black holes located along the symmetry axis. We define the mass and charge of each black hole and study the condition for static equilibrium of black holes. We show that the equilibrium is realized only when all the black holes are extremal and the spacetime becomes Majumdar-Papapetrou type. Ever since the soliton techniques were first applied to solving the gravitational field equation we have been able to study the structure of many-body system in general relativity using exact solutions. Although the obtained solutions are static or stationary it may be important to study them as a first step toward considering a dynamical process such as black hole collision. The simplest example of many-body system is the axisymmetric n-Schwarzschild solution, which is obtained as a 2n­ soliton solution to the vacuum Einstein equation.ll This solution includes conical singularities between Schwarzschild black holes along the symmetry axis. These conical singularities, which we call struts, are necessary to uphold the gravitational attractive force between the black holes. Static balance without struts could be realized by introducing negative masses but naked singularities appear. Another way of static balance is to introduce some fields with repulsive force. Scalar fields act attractively and also break event horizons of black holes, which brings about naked singularities again. 2 > Repulsive Coulomb force in electromagnetic field can balance attractive gravitational one and leads the system to static equilib­ rium. The solution given by Majumdar 3 > and Papapetrou 4 > is known as such a solution. This solution is an exact solution of the source-free Einstein-Maxwell equations in a conformastatic spacetime and describes a system of extremal black holes.

RESTRICTIONS ON THE HYPOTHETICAL LONG RANGE INTERACTIONS FROM THE CASIMIR-TYPE NULL EXPERIMENT WITH THREE TEST BODIES

A realistic null experiment is suggested in which the Casimir force between a plane plate and a spherical lens is compensated by the force of gravitational attraction. This configuration is shown to be very sensitive to the existence of additional hypothetical forces of Yukawa-type or power laws. From the suggested null experiment the restrictions on the Yukawa constant α can be strengthened by a factor up to 1000 in a wide range 10−8 m < λ < 10−4 m and by a factor of 10 for λ from several centimeters to several meters. For power law interactions the strengthening of restrictions by a factor of 20 is possible for the force decreasing as r−5.

Multi-Weyl Solutions to the Einstein Equation Coupled with Scalar Field

We study the static Einstein equation coupled with a scalar field with axial symmetry. The zero curvature equation both for the scalar field and a part of the metric in the scalar-Einstein equation enables one to obtain soliton solutions for both the scalar field and the metric. Of the infinite series of solutions for the metric and the scalar field we focus on the solution composed of 4-soliton solution of metric and 4-soliton solution of scalar field. We give a physical interpretation of the solution that two Weyls with scalar charges are exactly sustained by the tension of a lying between the two Weyls. One of the most interesting problems of black holes is the study of bifurcation of a black hole and coalescence of multi black holes_ Before the dynamical behaviors of black holes is understood through the exact solutions of the Einstein equation, we should first understand the static and stationary solutions of multi black holes. In the preceding papers!) we have studied the vacuum soliton solutions to the static Einstein equation which includes the ring-type solutions and the multi-Weyl solution. In the 4-soliton solution or two-Weyl solution, what is peculiar in the solution is the exis­ tence of a conical singularity which may be interpreted as a cosmic string or strut between two Weyls. We showed that the tension cancels the gravitational attractive force. There are two possible solutions which satisfy the asymptotic flatness: (i) The masses of two Weyls could be both positive while the tension or the line density of the strut is negative; (ii) One of the masses of two Weyls should be negative if the tension has the positive sign_ Either case comprises physical difficulty: negative mass or negative energy density. Classically there can be several ways for particles to balance despite of their attractive gravitational forces. The first is the introduction of strut to sustain their gravitational forces. The second is the introduction of negative mass to the system of particles. The third is the introduction of another force which can cancel the gravitational attractive force. As for the third one of the realization in the general relativity is the work by Papapetrou and Majumdar (the PM solution)_2) They introduced the coulomb force to cancel the gravitational attractive force. However the electric charge and the mass of each black hole are not independent charges but should satisfy the relation which realizes the extremal case of the Reissner-Nordstrom solution. What if the constraint between the mass and the electric charge are removed? This has not been answered so far. In the general relativity what could be the correspondence to the strut and the negative mass for balancing? This can be answered by showing the exact solutions to the Einstein equation. As far as we consider the vacuum Einstein equation we could not find physically

Existence of black hole solutions for the Einstein-Yang/Mills equations

We consider the Einstein/Yang-Mills equations in $3+1$ space time dimensions with $\SU(2)$ gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang/Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

Smooth static solutions of the Einstein-Yang/Mills equations

We consider the Einstein/Yang-Mills equations in 3 + 1 space time dimensions with SU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang/Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

Smooth static solutions of the Einstein/Yang-Mills equations

We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions withSU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

Possible effects of anisotropy ofG on celestial orbits

Will (1971) has discussed a possible anisotropy in the gravitational constantG. Suppose that the attractive gravitational force between two particles of massesm1 andm2 is given by the usual expressionF=−Gm1m2r/r3, wherer is the separation vector. Ifc is the velocity of light in vacuo and if 1r≡r/r, he expresses the anisotropy byG=G∞[1+e(v·1r/c)2], whereG∞ is a constant,v is identified practically as the velocity of the Sun around the galaxy, and e≈1. Will's suggestion is to look for such an effect in the laboratory.

Optimization problems in gravitational attraction

Three types of problems in gravitational attraction are considered. In the first type, two isoperimetric ones are solved elementarily by means of a known inequality concerning integral means. In the second one, it is shown that if a homogeneous ellipsoid is divided into two measurable sets, then the gravitational attractive force between them is greatest when the two sets are hemiellipsoids formed by a plane containing the two largest principal axes. In the third one, it is shown that the motion of a free particle in a straight tunnel through a homogeneous ellipsoidal planet is simple harmonic. Furthermore, the periods achieve their maximum and minimum values when the tunnels are parallel to the largest and smallest axes, respectively.

Reformulation of general relativity in accordance with Mach's principle

It is argued that Einstein's Theory of General Relativity as it stands incorporates Mach's Principle. The boundary conditions for Machian solutions are stated in a coordinate system in which the cosmological background is described by a conformally flat metric. The metric tensor g μν is then written as a product of the scalar density ϕ 2 and a tensor density γ μν with unit determinant. In the coordinate system that has been so chosen ϕ describes the cosmological structure, while γ μν refers to gravitational phenomena. This becomes clear when Einstein's fundamental equations are rewritten in terms of ϕ and γ μν . Then κϕ −1 is seen to play the role of the gravitational constant instead of κ in the weak field approximation. The quantity κϕ −1 can be expressed in terms of the radius and the total mass of the universe and the sign of the forces between inhomogeneities of the metric is determined by the requirements of Mach's principle. The forces which derive from ϕ are found to be repulsive for the cosmological background, leading to the expansion of the universe, while attractive gravitational forces arise from the deviations of γ μν from the Minkowski metric. Various statements associated with Mach's Principle are discussed in the light of this reformulation of Einstein's Theory.

Acceleration of Interplanetary Grains.

As is well known, the repulsive force of solar radiation pressure and the gravitational attractive force are significantly competitive for small interplanetary particles. For a homogeneous sphere of radius s and density p the ratio of the acceleratioi~s, radiation pressure to gravitation, is 5.7 X l()-5/ps. An additional repulsive force that can achieve importance is that of the solar wind. Let the outbound solar plasma at any moment be characterized by a number density n of protons and electroi~s each, a kinetic temperature T, and a radial velocity v. For v of order 300 km/sec and T in the rai~ge 1050 to 1060K, the momentum of the electrons is largely thermal and that of the protons comes chiefly from the wind velocity. Because of their higher mobility the electrons impact more frequently and the grain rapidly acquires an equilibrium negative charge. Thereafter the capture cross section of the grain is much reduced for solar electrons and increased by a small factor for solar protons. The solar wind inter- acts with the grain in two ways. First, the direct bombardment of the grain by solar protons has a strength compared with solar radiatioi~ pressure that is approximately twice the ratio of the momentum flux of solar protons to the radiation momentum flux. Normally the effect of direct proton bombardment is very small. Of greater importance by a factor of perhaps five is the transfer of radial momentum from nonimpacting protons that curve around the grain and go away on non radial orbits. This force is only about one percent as strong as solar radiation pressure under quiet-sun conditions when the wind has n =10cm-3 and v = 300 km/sec. Under hurricane conditions, however, with n = 10~ cm-3 and v = 1000 km/sec, the effect can temporarily be 100 times as strong as radiation pressure and may play a significant role in governing the motion of dust tails in comets. The relative importance of electrostatic Poynting-Robertson drag to radiative Poynting-Robertson drag is greater than the above ratios by a factor of about c/v. Because c/v lies in the range 300-1000, electrostatic drag may be the dominant perturbing force governing the dynamics of the zodiacal cloud.

The exosphere and upperFregion

The view is presented that the outer portion of the earth's ionosphere, through which radio whistlers propagate, has its origin in the earth's atmosphere rather than in interplanetary space. The outer portion of the earth's neutral-particle exosphere is dominated by neutral atomic hydrogen which is escaping steadily from the earth's atmosphere. Some of this neutral atomic hydrogen reacts with atomic oxygen ions near the base of the exosphere, giving rise to hydrogen ions. Diffusive equilibrium is the dominant factor controlling the relative concentration of ions and neutral particles near the base of the exosphere. At higher levels, the geomagnetic field exerts an influence in that it causes the ions and electrons to rotate with the earth, and the centrifugal force modifies the force of gravitational attraction appreciably. The ion electron plasma through which the radio whistlers propagate is hydrostatically supported within the magnetic tubes of force.

Observations on the fundus oculi during black-out.

ONE OF the consequences of flying at high velocities is that changes in direction or speed may produce profound physiological effects in the pilot. These accelerative forces are known as g forces (expressed as multiples of the gravitational attractive force). Depending upon how the force is vectored through the body of the subject, it is referred to as a positive, negative, or transverse g force. Military maneuvers, such as gunnery and dive bomb runs and steep inside turns, produce a characteristic group of symptoms culminating in black-out and unconsciousness which are known collectively as the physiological effects of positive g.* Many authors use the words "black-out" and "unconsciousness" interchangeably. In an effort to counteract this semantic error, the term amaurosis fugax has been substituted as a more accurate and all-inclusive description.† However, this is not fully warranted, because clinically amaurosis fugax refers to a temporary loss of vision, such as