Equivalence principle. Part 2

In school textbooks, Newton's laws are described as ultimate truth, though their creation required the introduction of five new entities. These five entities are the following: mass as a measure of matter, the inertial and gravitational properties of mass, gravitational forces and inertial forces. All five of these entities were introduced ad hoc (Latin. - "for this", "for the nonce", "for this purpose"). For example, the gravitational forces and gravitational properties of masses were introduced for the explanation of falling of bodies on Earth, and centrifugal force (a special case of inertial forces) explained why the cosmic bodies revolve around a common mass center and do not fall under the influence of gravity. Accordingly, the immediate task of science was the independent confirmation of the existence of each of the introduced entities. The most important step in this direction was the equivalence principle declared by Newton.

It was experimentally proved that, during the free motion of the body in the central force field, the centrifugal force is demonstrably evident, and that its value is equal in magnitude to that of the centripetal force causing the motion of a body with acceleration F_cp = F_cf = MV²/R, where M, V and R are the mass, velocity and radius of rotation circle, respectively.

Let's examine an easily realizable and well-known experiment (Figure 1).

In this experiment, a ball with a mass M rotates on a spring. A dynamometer measuring spring tension force is attached to the ball. In this system, we can measure the orbital velocity of the ball's motion, centripetal force (F_cp), centrifugal force (F_cf) and orbital radius (R).

In the course of the experiment, we can change the speed of the ball in orbit and, accordingly, the orbital radius. With increases in the rotation, the velocity, the orbital radius, the velocity of the ball's motion in orbit and the centripetal and centrifugal forces increase. Fcp = М V2/ R,        Fcf = М V2/ R. Stability condition   is   Fcp = - Fcf In Newton's time the only centripetal force believed to act at a distance (the existence of which was acknowledged by Copernicus, Galileo, Hooke and Newton) was gravitational force. Mass was a measure of the amount of substance. Accordingly, the equality of inertial mass and gravitational mass to each other and to mass as a measure of the amount of substance was and is the most weighty  proof of the real existence of all the entities mentioned above in the history of physics.

Figure 1

As stated above, the equivalence principle was proclaimed 320 years ago. At this time, the mathematical approach dominated in solving physical problems. According to this approach, the primary task of science is to formulate a mathematical equation that describes the observed phenomena. The significance of the theory was defined by the possibility of explaining and calculating phenomena which were not quantitatively described until this time.

For example, the principle of equivalence enables us to calculate the attraction force of the planets to the Sun without knowledge of the physical nature of gravitation. The first, second and third space velocities of bodies were and are calculated under the principle of equivalence.

Therefore, the questions "Why is centrifugal force is equal to centripetal force?" and "Why is inertial mass equal to gravitational mass?" had not even arisen. So, the equivalence principle was actually Newton's fifth law. However, it should be mentioned that the resolution of these issues in Newton's time was hardly possible. Meanwhile, the internal contradictions of the laws proposed by Newton were evident even in those times. Newton attributed the forces arising in a rectilinear motion and centrifugal forces to inertial forces. The reason for this assention was that these forces are calculated by analogous equations. In modern mechanics, it is accepted that a body rotating while affected by the gravitational force drops constantly towards the source of gravity. But, since inertia resists change in velocity of a falling body, the body is forced to follow an  elliptical, hyperbolic or parabolic trajectory. In the case of circular motion, the centrifugal force is equal to the force that caused the motion of the body with acceleration, which was confirmed experimentally. Accordingly, we expected that in the case of a linear accelerated motion, the inertial force must be equal to the force that caused the acceleration. However, in this case, it would be impossible, in principle, to accelerate the body moving linearly; it is unclear why the pair of forces, which are at any moment of time exactly equal and opposite to each other, cause no uniform motion.

The law of universal gravitation (universal interaction between any types of matter), discovered by Newton in 1678, assumed momentary long-range action between bodies. The main drawback of this law was that, in this case, the systems (primarily the solar system) constructed according to this law (and according to the rest of Newton's laws) will be unstable and may not always exist in principle. Because of the orbital perturbations of meteors and comets and other gravity features, either the fall of bodies to the center of gravity or their separation and recession must occur. This contradicted the ideas of eternity and immutability of the movements of celestial bodies.

As is well known, there are three equilibrium states in mechanics: indifferent, stable and unstable equilibrium (see the article "About gravitation"). 

In spatial and atomic systems described only by the laws of Newton, in case of a small deviation of a body from the equilibrium position, forces emerge which strive to increase this deviation, because both gravitational and Coulomb forces are inversely proportional to the squared distance between the planets for space objects and atoms, and the centrifugal forces are inversely proportional to the first degree of the distance. It would seem, in case of the slightest deviation of an electron rotating around a proton (an increase or decrease of the distance between the proton and electron), that the centripetal forces increase this deviation. The only account of formal conservation laws (of energy, impulse and angular impulse) enables us to solve this problem only by way of mechanics, and, even then, it enables us to solve only a limited number of simple cases. But even in these cases, the physical mechanism stabilizing the orbits remains mysterious and incomprehensible.

According to Wikipedia, Newton knew about this contradiction and therefore, in his theory of gravitation, confined himself to the following phrase: "... the maintenance of the present view of the solar system requires the intervention of some extraneous supernatural powers."

Attempts to resolve this contradiction have been made over the last 300 years. A survey of works on this subject is described in the book by V.G. Demin the founder of the Astronautics Academy of the USSR (V.G. Demin, The fate of the solar system. The popular essays on celestial mechanics [1969]). In the preface to this book, we read: "A little less than two centuries separate us from the days when eminent French scientists Joseph Louis Lagrange and Pierre Simon de Laplace, whose names excite a reverent and admiring respect among the scientists of all times, continuing the great work of Isaac Newton and the glorious galaxy of its followers, created a magnificent building of celestial mechanics. Over nearly half a century, maintaining continuous communication with each other, in the spirit of constant creative rivalry, Lagrange and Laplace devotedly toiled at the general problem of the construction of a theory of motion of major planets. They are both rightly considered to be authors of the statement of a famous problem in mechanics: the problem of stability of the solar system that generated a number of more specific problems in the evolution of the orbits and the figures of celestial bodies. Lagrange and Laplace first breach at the unassailable mathematical fortress - that is the problem of the stability of the solar system, and the scientists have received its first approximate solution.

For many decades, the most prominent mathematicians and engineer assailed the Lagrange-Laplace problem. Slowly, step by step, the scientists moved forward, forced to surmount the complicated mathematical hurdles constantly coming up on their way, pulling out of nature, one after another, secrets of the motion of celestial bodies. Specific issues of the evolution of movement of individual bodies in the solar system were solved one after another. But the strict solution of the famous problem still remained as distant as in the times of Lagrange and Laplace."

One hundred years ago, it was proved that all material bodies are composed of charged particles (electrons, protons and neutrons) and that their number in each body is directly proportional to the body's weight.

During these years, a new science - electrodynamics - emerged and was firmly established. It brought into physics new entities and new regularities. Electrostatic and electromagnetic forces became newly accepted forces.

In electrodynamics, it was proved that a charge moving with acceleration causes the emergence of an electromotive force (hereafter EMF) acting on this charge with a force equal in magnitude to the force that caused the charge's motion with acceleration and directly opposite to this force in direction (Lenz rule).

The magnitude of the electromotive force does not depend on the type of force that caused the motion of a charge with acceleration. This force can be the Newtonian gravitation force (the attraction of masses), or can be the Coulomb force (the interaction of charges) or the Lorentz force (the interaction of charges moving with an acceleration with magnetic and electric field). Acceleration can also be arbitrary (centripetal or linear).

Electrodynamics gives us the answer to the question: Why isthe centrifugalforceequa to the gravitational force?

In experiments demonstrating the equality between the centrifugal force and centripetal force, charges (of which a body consists) are moving with acceleration. This leads to the emergence of EMF acting on these charges with a force equal in magnitude and opposite in direction to the centripetal force. It is possible to give the answer to the question "Why is the centrifugal force equal to the centripetal one?" without the using mass and its properties. Since gravitation causes acceleration of charged particles that make up any body and the result of this acceleration is self-induction, which is exactly equal and opposite in direction to the force that caused acceleration, the inertia of a body caused by gravitation is in a cause-effect relationship with this gravitation. Causeand effectare alwaysseparatedalbeitnegligiblebutnon-zerotime term. Thatis whygravitation, despite the opposition of inertia, canstillchange thetrajectories ofbodies!

It is not only qualitative but quantitative evidence that mass as an entity and, accordingly, its gravitational and inertial properties, are not the original (fundamental) concepts.

Our explanation of the equivalence principle is an illustration of a well-known statement of Bacon's that the new theory is a child of time but not a child of authority.

As a result of work carried out during the last 15 years, we have come to the conclusion that such an entity as mass (gravitational or inertial mss) is as much a tribute to time as phlogiston and calorie. Laws and the equivalence principle opened by Newton appeared two hundred years before Faraday's law, atomic and molecular structure of substances and the determination of the nature of chemical bonds. We have been writing about it already for over 15 years, in both English and Russian​​. Difficulties with the perception of our explanations are related primarily to inertia of thinking. We hope that simple and clear explanations can shorten the time required to overcome the thinking inertia of the scientific community.