See Figure 2. The experiments Coulomb did, with the primitive equipment then available, were difficult. No exceptions have ever been found, even at the small distances within the atom. Figure 2. Compare the electrostatic force between an electron and proton separated by 0. This distance is their average separation in a hydrogen atom. Finally, we take a ratio to see how the forces compare in magnitude.
The charges are opposite in sign, so this is an attractive force. This is a very large force for an electron—it would cause an acceleration of 8. Here m and M represent the electron and proton masses, which can be found in the appendices. Entering values for the knowns yields. This is also an attractive force, although it is traditionally shown as positive since gravitational force is always attractive.
This is a remarkably large ratio! Note that this will be the ratio of electrostatic force to gravitational force for an electron and a proton at any distance taking the ratio before entering numerical values shows that the distance cancels.
This ratio gives some indication of just how much larger the Coulomb force is than the gravitational force between two of the most common particles in nature. As the example implies, gravitational force is completely negligible on a small scale, where the interactions of individual charged particles are important. On a large scale, such as between the Earth and a person, the reverse is true. Most objects are nearly electrically neutral, and so attractive and repulsive Coulomb forces nearly cancel.
Gravitational force on a large scale dominates interactions between large objects because it is always attractive, while Coulomb forces tend to cancel. Figure 3. Schematic representation of the outer electron cloud of a neutral water molecule.
Use Figure 3 as a reference in the following questions. Einstein's generalization of Newton's theory of gravitation in the general theory of relativity led not only to small quantitative differences between gravitational effects in the relativistic theory and the Newtonian theory, but also to essentially new phenomena and effects peculiar to the relativistic theory and absent in the Newtonian theory.
This difference is so large that the gravitational interaction in Einstein's theory even altered the attraction-only property, characteristic of Newton's theory, the law of universal gravitation, and became both attractive and repulsive.
It is notable that the nature of the repulsion in gravitational interaction already appears in the simplest case of a spherically symmetric isolated body. Einstein's equations admit for a spherical body a solution whose physical interpretation uniquely indicates the repulsive nature of a gravitational field inside the body, if the number of particles that make up the body is sufficiently large.
The structure of such a body, density distribution of the number of particles, mass, and pressure, is determined in the equilibrium state by the pressure of the substance, the gravitational attraction of peripheral layers toward the center, and the gravitational repulsion of inner layers of matter away from the center. As a result of the gravitational repulsion of matter away from the center inside the body there appears a cavity, free of the matter making up the body and its electromagnetic radiation.
If the body is cold, then the volume of the world tube of the cavity can differ from zero. In the opposite case, the world tube of the cavity reduces to the world line of the center, which is inaccessible to particles of matter and to electromagnetic radiation.
Gravitational repulsion, on the other hand, is a result of the existence of a field singularity at the center of the body, whose world line is time-like.
This is a preview of subscription content, access via your institution. Rent this article via DeepDyve. Landau and E. Google Scholar. Wheeler, B. Harrison, M. Vacano, and K. Arifov, Dokl. Nauk Uzb. Currently the Universe composed of approximately 10 22 or 2 73 stars. A final black hole will have a mass of 0. The remaining mass of the universe 98 per cent will be converted into gravitational waves. How the total gravitational mass of the Universe changes when a significant fraction of the mass of the black holes is converted into gravitational waves?
An answer to this question depends on treating the gravitational field sources i. At the earlier stage of the development of the general theory of relativity years — Einstein supposed that the energy of matter and the energy of gravitational field are equivalent as a source of gravitational field and included the gravitational energy in the right part of his equations.
Such a contradiction was considered to be insignificant at that time because the effect of emitting gravitational waves on the total mass of a system was supposed to be negligible.
However, the LIGO discovery proved non-negligible loss of the total mass of two black holes due to gravitational waves and made a question about the contribution of gravitational waves to the total mass of a system be an urgent problem. In this paper we consider a model of the universe with decreasing gravitational mass in the framework of the general theory of relativity.
It should be noted that the model of the universe with varying gravitational mass is not quite new. For example, Tolman admitted that the total gravitational mass of the universe changes due to transformation of matter to gravitational waves.
A model of the universe with growing gravitational mass was considered by Hoyle Kutschera considered a model of relativistic fireball with variable gravitational mass. We suppose that gravitational waves do not contribute to the total mass of a system.
We show that reduction of the gravitational mass of the system due to emitting gravitational waves leads to a repulsive gravitational force. The observed expansion of the Universe suggests that the repulsive forces were extremely large in the past.
Recent observations showed that the Universe currently expands with acceleration Riess et al. To explain the accelerating expansion of the Universe there were developed numerous non-Einstein's theories see e.
Bamba et al. We show that a repulsive gravitational force can be derived in the framework of the classic general theory of relativity. It should be noted that Hilbert was the first who showed that the equation of acceleration of a probe particle in the general theory of relativity contained two terms which represented both attractive and repulsive forces. The existence of the attractive and repulsive forces in the general theory of relativity has been discussed for about a century — see a review by McGruder We derive a general expression for the acceleration of a probe particle in a given diagonal metric that contains repulsive terms.
Our analysis is carried out for the case of weak gravitational field and non-relativistic velocity of a probe particle. We show that the cosmic repulsive force can be explained in the classic general theory of relativity without additional hypotheses. In this approximation, equation 14 describes the ordinary Newton gravity. For better visualization, the curve 2 shifted on 1 unit versus the curve 3; the curve 1 shifted on 2 units versus the curve 3.
The gravitational acceleration is always directed against the gradient of the gravitational potential. That is why the curves of the gravitational potential in Fig. It can be seen in Fig. Similar to Fig. A shape of the gravitational potential is often illustrated by a funnel made of rubber film where a heavy ball is located in the centre. In this model a fast decrease of the gravitational mass corresponds sharp ascent of the ball.
The film attached to the ball forms a cone-type hill in the centre of the funnel. Light balls on the central cone run away from the centre. The central cone expands fast but keeps it exterior slope; this corresponds to long-term repulsive force. Recently, there is a debate about back-reaction of small-scale inhomogeneities e. Buchert et al.
We demonstrate that the mergers of inhomogeneities like black holes, resulting in emissions of gravitational waves, can generate a repulsive gravitational force. These mergers act as an effective dark energy, if the total mass of the universe is decreased. Let us consider a model of the universe with cycling periods of compression and expansion. Gravitational waves caused by the compression may not disappear at the stage of expansion and form the relic gravitational radiation. Because the gravitational radiation does not contribute to the total gravitational mass of the Universe, a level of the energy of the relic radiation is not limited and may be very high.
According to Grishchuk a maximum of the energy of the relic gravitational radiation is at GHz frequencies. This wavelength corresponds to a frequency of hundreds GHz. We showed that a cosmic repulsive force can be explained in the classic general theory of relativity without additional hypotheses.
The repulsive force originates from a metric with the varying gravitational mass of a system. The repulsive force occurs at some distances from the quasi-spherical system which depend on time lapsed from the beginning of the change of the mass.
The repulsive force quickly decreases with radius but does not disappear. We hope that our theoretical prediction about decreasing acceleration of the Universe can be verified by observations. It is logical to suppose that the found mechanism of the repulsive force may be applied to a model of the expanding universe.
This may imply that big bang and accelerated expansion of the Universe is not related to current processes in the Universe but to a relic repulsive gravitational force or to a configuration of space—time that originates in the previous cycle of the Universe when at the last stage of a collapse the intensive generation of gravitational waves resulted in sharp decrease of the gravitational mass of the Universe and may be in avoiding a singularity.
This process generated a powerful repulsive force that transformed big crunch into big bang.
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