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Lorentz-Invariant Gravitation Theory

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Lorentz-Invariant Gravitation Theory Alexander G. Kyriakos Lorentz-invariant gravitation theory. 1 2 1 2 A 4 2 4 2 A с 2 t 2 с 2 t 2 c 2 2 rs 2 2 dr 2 2 2 2 4 N M s a 2 ds 1 c dt r sin d d 2 sin ddt r rs c r 1 r 2016 2 Annotation The modern theory of gravity, which is called General Theory of Relativity (GTR or GR), was verified with sufficient accuracy and adopted as the basis for studying gravitational phenomena in modern physics. GR is the geometric theory of gravitation, in which the metric of Riemannian space-time plays the role of relativistic gravitational potential. Therefore it has certain features that make it impossible to connect it with others physics theories in which geometry plays only a supporting role. Another formal feature of general relativity is that the study and the use of its mathematical apparatus require much more time than the study of any of the branches of modern physics. This book is an attempt to build a non-geometrical version of the theory of gravitation, which is in the framework of the modern Lorentz-invariant field theory and would not cause difficulties when teaching students. A characteristic feature of the proposed theory is that it is built on the basis of the quantum field theory. Table of contents From author……………………………………………………………………………….. 2 Notations……………………………………………………………………………………3 Chapter 1. Statement of problem………………………………………………………….4 Chapter 2. Origin of the gravitation field source ………………………………………... 6 Chapter 3. The axiomatics of LIGT and its consequences………………………………. 9 Chapter 4. Connection of electromagnetic theory and gravitation……………………...11 Chapter 5. Electromagnetic base of relativistic mechanics………………………..……. 17 Chapter 6 . The equation of motion in LIGT………….………………………………. 26 Chapter 7 . Geometry and physic of LIGT and GTR………………………………….. 31 Chapter 8 . The equivalence principle and metric tensor of LIGT…………………….. 36 Chapter 9 . Solution of the Kepler problem in the framework LIGT…………………. 44 Chapter 10 . The solution of non-cosmological problems in framework of LIGT…… 48 Chapter 11. Cosmological solutions in framework of LIGT………………………….. 57 Chapter 12 . Quantization of gravitation theory……….………………………………. 58 References ……………………………………………………………………………… . 60 From author The Lorentz-invariant theory of gravitation (LIGT) is the conditional name of the proposed theory of gravity, since Lorentz-invariance is a very important, although not the only feature of this theory. Note that our approach was used in the past in relation to the gravitational theories that have some similarities with our theory. Therefore the results obtained by well-known scientists are widely cited in the book. However, for posing the problem and for some of the basic elements of the theory which are obtained by the author of the book, the only person responsible is the author. 2 3 NOTATIONS. (In almost all instances, meanings will be clear from the context. The following is a list of the usual meanings of some frequently used symbols and conventions). Mathematical signs - Greek indices range over 0,1,2,3 ,,,... g - metric tensor of curvilinear space-time i, j, k… - Latin indices range over 1,2,3 g GR - metric tensor of GR space-time ˆ ˆ , - Dirac matrices R - Riemann tensor A - 3-dimensional vector R - Ricci tensor R A - 4-dimensional vector A - Tensor components R - Ricci scalar R - Covariant derivative operator G - Einstein tensor 2 - Laplacian - Minkowski metric � - d'Alembertian operator 2 2 2 h - Metric perturbations t - Lorentz transformation matrix Physical values j - Current density u - velocity J - Angular momentum tensor a - 4-acceleration du d N - Newton's constant of gravitation - 4-momentum p L - Lorentz factor (L-factor) T - Stress-energy tensor m - mass of particle F - Electromagnetic field tensor M S - mass of the star (Sun) M , L - angular momentum Abbreviations: NQFT - nonlinear quantum field theory; LIGT - Lorentz-invariant gravitation theory; QED - quantum electrodynamics. EM - electromagnetic; EHE - Einstein-Hilbert equation EMTM - electromagnetic theory of matter; HJE – Hamilton-Jacobi equation EMTG - electromagnetic theory of gravitation; GTR or GR - General Theory of Relativity SM - Standard Model; L-transformation - Lorentz transformation QFT – quantum field theory L-invariant - Lorentz-invariant Indexes g - gravitational e - electrical ge - gravito-electric, m - magnetic, gm - gravito-magnetic em - electromagnetic, N - Newtonian 3 4 Chapter 1. Statement of problem 1.0. The place of gravitation theory in a number of other physical theories The first classical mechanics theory was created by Newton. Two types of laws of mechanics exist: laws of motion of material bodies under the action of forces, and laws that define these forces (which are often called equations of sources). In frames of Newton's theory, his second law is the primary law of motion, while Newton’s gravitational law defines the force of gravity. It should also be mentioned that during the further development of mechanics, numerous mathematical formulations of the original laws of Newtonian mechanics were found, which are physically almost equivalent, including the ones that use energy characteristics of the motion of bodies, rather than force. 2.0. Relativistic theories As was revealed later, Newtonian mechanics is valid for speeds, well below the speed of light c 300000 km/sec. Mechanics, which is valid for speeds from zero to the speed of light was conditionally named relativistic mechanics (detailed overview of the theory see (Pauli, 1981)). Under the condition c , Newton's laws have very high accuracy. The definition "relativistic" is equivalent to the requirement "to be invariant under Lorentz transformations". Therefore we will use the definition of "relativistic" equally with the definition of "Lorentz-invariant (briefly "L-invariant"). In relativistic mechanics, there are also several forms of equations of motion and equations of sources. As the relativistic law of motion (including the theory of gravity) the relativistic Hamilton-Jacobi equation is often used. 2.1. How non-relativistic mechanics is related to the relativistic mechanics Non-relativistic theories give correct predictions at speeds much less than the speed of light. The relativistic theories give exact values in the entire range of speed from 0 until the speed of light c 300000 km/sec. The inaccuracy of non-relativistic theories compared to the 2 2 relativistic can be attributed to the Lorentz factor L 1 1 c , a factor of the Lorentz transformation (see in reference book the diagram of Lorentz-factor L as a function of speed). As seen from the graph, factor is not very different from the unit, up until the velocity of the particle reaches the 1/10 of the velocity of light (i.e., about 30000 km/c). The maximum speeds of the planets and the massive bodies on the Earth and in the solar system are: projectile - 1.5 km/s, the rocket - 10-12 km/s, meteorites - 18-25 km/s, the Earth around the Sun - 30 km/s, the Sun in the direction to the galactic center - 200 km/s, our galaxy - up to 400 km/s. Higher speed is achieved only by elementary particles in cosmic space or in accelerators, but they do not play any role in the theory of gravity. Thus, the value 2 c 2 in real problems of mechanics is very small and the Lorentz factor is not very different from unit. This means that Newtonian mechanics is valid in practical applications with great accuracy. This was already understood by one of the founders of the Lorentz-invariant physics – A. Poincare, who had warned (Poincaré, 1908): ―I tried in a few words to give the fullest possible understanding of new ideas and explain how they were born ... In conclusion, if I may, I express a wish. Suppose that in a few years, this new theory will be tested and come out victorious from this test. Then, our school education is in serious danger: some teachers will undoubtedly want to find a place to new theories .... And then [the students] will not grasp the usual mechanics. Is it right to warn students that it gives only approximate results? Yes! But later! When they will be permeated by it, so to speak, to the bone, when they will be accustomed to think only with 4 5 its help, when there will not be a risk that they forget how to do this, then we can show them its borders. They will have to live with the ordinary mechanics, the only mechanic that they will apply. Whatever the success of automobilism would be, our machines will never reach those speeds where ordinary mechanics is not valid. Other mechanics is a luxury, but one can think about a luxury only when it is unable to cause harm to the necessary.‖ 3.0. The general theory of relativity The modern theory of gravitation, called the general theory of relativity (GTR or GR), refers to classical mechanics. As the equation of source is considered to be the Einstein-Hilbert equation (EHE) of general theory of relativity (GTR) or (GR), which was found by these researchers almost independently and almost simultaneously (Pauli, 1981; Vizgin, 1981). As the basis for theory building, Hilbert used a variational principle. The approach of Einstein was heuristic, emanating from the experimental fact of equality of gravitational and inertial masses (note that this equivalence is also valid in nonrelativistic theories). A very difficult question, is whether the GTR and its equation are relativistic in terms of the Lorentz invariance. Strictly speaking, it is not (Katanaev, 2013, p. 742). Einstein assumed that the general covariance of the equations of general relativity includes special relativity. As is known, EHE is very different from other equations of mechanics, since it is based on the Riemann geometry in general system of coordinates. Besides, GR has several disadvantages, which have not been overcome to date (Fock, 1964; Rashevskyi, 1967); Logunov, 2002).
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