JPH04506242A - Apparatus and method for generating anti-gravity forces - Google Patents

Abstract

(57) [Summary] This bulletin contains application data before electronic filing, so abstract data is not recorded.

Description

[Detailed description of the invention] Apparatus and method for generating anti-gravity forces field of invention The present invention provides a method and apparatus for providing repulsion, particularly for providing propulsion and levitation. A method and apparatus for providing a compatible anti-gravity repulsion.

Background of the invention The gravitational force of attraction has been the subject of research for several centuries. Development and current state of understanding is illustrated by Appendix I. Traditionally, gravitational attraction has been associated with atomic and subatomic The large-scale aspects of spacetime and cosmology are distinguished from currently supported theories of the structure of Illustrated by abstracts submitted in the field of astrophysics applying astrophysics. but However, inconsistencies in scale-based analysis of materials compared to actual experimental measurements This results in inconsistencies and unpredictability in the resulting model.

In Newtonian gravity, two grains of masses m1 and m2 separated by a distance r The mutual attraction between children is , where G is the gravitational constant and its value is 6.67XIQ−”Nm”kg− It is 2. Newton's theory gives a correct description of the force of gravity, but most of the Which aspects of natural forces are even more important (undefined? So which aspects of gravity are the most important? What we consider to be the most fundamental thing is whether it is true or not. newton's number 2 laws, F=ma (1, 2) by comparing with his law of gravity, and by using the following equation The motion of a falling object can be described: and the gravitational mass (proportional to the force of gravity) ), MΦ is the gravitational mass of the Earth, and r is taken from the center of the Earth. It is a vector of the position of the object.

The above equation is It can be rewritten as Extensive experimental coverage from Galileo's Pisa experiments to the present day is the acceleration of an object produced by the force of gravity is the same, regardless of the object selected From equations (1, 4), we can see that the values of m and /m+ are the same for all objects. It means being one. In other words, we mt/m+=universal constant (1, 5) has.

In physics, the discovery of universal constants often leads to the development of completely new theories. light A special theory of relativity was derived from the universal constant C of the velocity of; and the blank constant Quantum theory was derived from numbers. Therefore, the universal constant mg/m+ is the problem of gravity. It may be important for the topic. The theoretical difficulty of using Newton's gravity is that the relation ( 1,5) is a natural addition to (1,1) and (1,2) in Newton's theory. It just explains why it exists intrinsically as another law. In addition, astronomical observation A contradiction exists between predictions based on Newton's celestial mechanics, and Newton Einstadian gravity can be transformed into Newtonian gravity on a scale supported by the theory of They could not be reconciled until Yin's development of general relativity.

On cosmological measures, general relativity was experimentally correct: however, it Based on a flawed dynamical formulation of Galileo's law. Einstein is his As a basis for deriving the gravitational field equation, he used a law he called the "equivalence principle." The kinematics results of several species are discussed. The equivalence principle does not provide a theory of quantum gravity.

Some further discussion of Einstein's Equivalence Principle can be found in excerpts from Appendix II. provided.

Furthermore, general relativity treats matter not on an atomic scale (but on a cosmological scale). It is a partial theory. All gravitating objects are consists of matter and has fundamental particles such as electrons, which are lebutons, and protons. It consists of forks that make up the ton and neutron. Is gravity a fundamental particle? It originates from The dominant theory of atomic properties and subatomic physics is quantum mechanics. , and the combination of quantum theory and general relativity is a combination of well-known problems. contemporary In theory, there is no satisfactory theory of quantum gravity; The simplest applications of quantum mechanics face significant conceptual difficulties. quantum gravity theory is the universe It is inherent to its success that constants must be identified. universe The constant is the energy density of the vacuum, i.e. the amount of energy in a unit volume of empty space. can be most easily defined as a quantity. The gravitational equation containing the cosmological constant is (Equation 52.08) in Appendix Vl.

According to quantum mechanics, the value of the cosmological constant can be non-zero and, in principle, true The empty energy density can assume any positive or negative volume, and the grain Child physics and current ideas about gravity suggest that it is very large . The cosmological constant is the non-zero energy density of the vacuum, which is Einstein's general relative The characteristic effect that appears on the geometric structure of space-time generated by gender theory. You can roughly measure it by looking for it. Such an effect has not been discovered The limit of current observation is the discrepancy in the cosmological constant of about IQ-128. Smaller than the theoretical expected value due to staggering factor It means something. theory without denying quantum mechanics, general relativity, or both. The only way to explain this large discrepancy between expected values and experimental reality is to A very precise, Assuming that natural parameters are involved in totally incomprehensible connections It is. Research on the cosmological constant is based on quantum mechanical waves that describe the spatial geometry of the universe. To calculate the dynamic function, Stephen Hawking ing) and his collaborators. According to this theory and the quantum mechanical processes in our universe that affect our universe and each other. connect other universes through wormholes (wo rmh o Ie s). Create and destroy. Sitney consisting of Hawking's theory and the theory of variable α - Sidney Coleman's extension, where the wormhole Further discussion of influencing the values of physical parameters via α is in the Appendix Included in III.

However, Hawking's approach requires an indefinite mathematical expression and a large number of trials. It relies on untested assumptions. Although it finds the desired result, a cosmological constant of zero, , in principle, Koulman's system is a way to predict the value of α and all of nature Since all parameters are functions of these, it is a theory of parameters. Ru. This theory holds that for any other fundamental parameter, even if it is zero, Failed to give correct value.

The method and apparatus of the invention, described in further detail below, can be carried out as described. However, the following discussion of the novel theoretical basis of the invention is provided for additional understanding. available.

The action of gravity eliminates the presence of a frame because of inertia in a large area, and it Only local inertial frames are possible, where the relationship between is determined by gravity. simple Simply stated, the action of gravity is in determining the local inertial frame. Fray depends on gravity, and describes the spatiotemporal background of the motion of frame matter. therefore, gravity, unlike other kinds of forces, determines the nature of spacetime. affects the motion of matter, which is itself described by the metric of space and time. Book The inventive gravitational methods and devices cover everything from atoms to cosmologies where the cosmological constant is zero. provides metrics across all scales. Gravity is the basis of all matter in the universe It has been demonstrated that this results from a two-dimensional manifold of particle masses.

The novel atomic model of the present invention is described in co-pending U.S. applications no. 07/341.733 and and No. 07/345,628), the names of both inventions, and the energy/matter conversion Method and Structure, April 21, 1989 and April 28, 1989, respectively. The theory of general relativity is described in detail in the submission and incorporated herein by reference. Provide the correct basis for solving Einstein's field equations of theory: fundamental grains The child's gravitational mass and inertial mass are equal. Gravitational field equations are in Appendices Vl and VII derived from this principle, and the spatial two-dimensional (three-dimensional space-time) manifold When considering , atomic and cosmic gravity are integrated.

The new model of the atom is consistent with first principles and can be summarized as follows.

An electron is described by the product of two angular functions, and said angular functions are the harmonics of the sphere, the time scale with the sum function and the delta function of the radius forming a particular radially quantized orbit. be. In order to distinguish the atomic structure of the new atomic model from the conventional atomic model, these The radially quantized orbit is called a "Mills" orbit. atomic structure When applied to , the Mills orbit is the orbit of an electron. The angular momentum and energy are Maxwell's equations are preserved without being ignored. The energy of an electron is its storage is the sum of the electrical and magnetic energy generated. From the delta function of the radius, the electron Wave-like properties occur naturally. The angular harmonics are the spin and orbital angular momentum, the spin The selection rules for pin-orbit coupling and absorption of electromagnetic radiation are occurs naturally. The Mils orbit is a sphere, and the radius is the absorption of electromagnetic energy and Both will increase. When an electron is ionized, the radius of the Mils orbit becomes infinite: The child is a plane wave with de Broglie wavelength.

The electric field of an electron in a Mills orbit is zero inside the orbit, and at the origin outside the orbit is a field at a point charge at and are naturally eliminated. The radius of an orbit in an atom is Coulomb and It is calculated by setting the centripetal force equal to the sum of the magnetic forces. thus, The isolated Mils orbit is stable, where the Coulomb attraction causes the electron to decay into a nucleus. The result that there is no arises naturally. For all atoms and ions, the net charge is The center acting on each orbital is proportional to its charge (which is the charge that is not erased by other electrons) There is a Coulomb force. A positive central force exists between two unpaired electrons, with opposite Pairing occurs in the same shell with spin. Thus, Pauli's prohibition principle occurs naturally. There is a diamagnetic property between the paired electrons in the inner shell and the unpaired electrons in the outer shell. There is a repulsive central force. The four-object problem does not occur.

This is because the change in the centripetal force of the electrons in the inner shell, which is affected by the outer electrons, is This is because it is precisely balanced by the Lorentz force given by the electron's magnetic field. be. Of course, the Mils orbit has a modulation of the angular harmonic charge density and is spatially disposed. , it forms a legally symmetric charge density function with a minimum energy, so the electric charge Hund's rule naturally arises as a result of the child. Furthermore, as mentioned above, the electronic The electric field is spherically symmetric, and for radial distances greater than the radius the density function of the radius It is that of a point charge and is zero inside the shell. This applies to the patent application mentioned above. As described, it is the basis of chemical bonding and cold fusion. Properties of chemical bonds is the overlap of the Mills orbitals of the participating atoms, which is the overlap of the electric and magnetic fields. Minimize the energy stored in Energy holes absorb electrons resonantly. As the electron shell of each atom decreases due to quantized fractionation, As a result of the Coulomb force being eliminated between two nuclei for shorter internuclear distances, the colder nuclei Fusion occurs. For Shutylium, the resonance contraction energy is n/2 27 ．． 21 eV, where n is an integer.

Moreover, the new atomic model shows that electrons are made up of negatively charged fundamental particles, leptons. handle. However, the same principle applies to other elementary particles, including forks. It will be done. Wave-like properties, spin and orbital angular momentum phenomena, and selection rules are They are identical to those for leptons, and the boundary conditions that exclude radiation are also similar: The fork has a space-time charge density function equal to the Lebuton. Also weak and The strong nuclear force, as well as Lebuton spin pairing, diamagnetic, and bonding forces are properties is electromagnetism.

Mass exists as a Mils orbit, and each Mils orbit is a harmonic angular function of two spheres, time It consists of the product of an interharmonic function and a radius delta function. Determine that matter is two-dimensional The gravity of the space-time manifold (three-dimensional space-time) arises from the radius delta function. tertiary The original manifold has positive curvature. This is the angle of the geodesic triangle on the surface of the sphere. It can be determined by the fact that the sum exceeds 180°. Euclid The plane geometry of de is that in the plane, the angles of triangles sum up to 180°. Emphasize. This is the definition of a flat surface. Curvature of a normal material with positive curvature is given by K=1/re, where ro is the radius of the radius delta function. Ru. Essentially all ordinary substances exist in air when bent, and All macroscopic forms of ordinary matter exist in the form of Mils orbitals as forks When the shape is bent, it exists in the air. Furthermore, the total curvature of spacetime is The combination of contributions from, called "Mills" orbits as defined in the application filed in It is a total.

All matter exists as Mils orbits consisting of mass limited in three-dimensional space-time. Ru. The surface is a sphere: thus spacetime is curved with a constant curvature. non-local time The effect of this "local" curvature on the sky makes it the opposite reamer for Euclidean. The aim is to make sure that the This effect occurs over distances greater than the radius of the Mils orbit. is a function of the radial distance from the center of the Mils orbit for The magnitude of this effect on spacetime is proportional to the total mass of the Mils orbits. In this way, time The effect on the sky is independent of the radius of the Mils orbit for spacetime outside the orbit.

(According to the new atomic theory, this feature is neutral for radial distances larger than the radius of the shell. It is symmetrical to the electric field of an electron in a Mills orbit, which is that of a point charge in the heart. )Also , the new atomic theory is based on the general theory of relativity, which consists of the Riemannian geometry of spacetime. The derived inertial mass also includes the assumption that the total mass of the orbit is equal.

In general relativity, Einstein's field equations state that matter is the origin of gravity. We give a relational expression that determines the curvature of spacetime. The definitive form of this equation is That's right: R=gaβRR=guvR(1,7) μg μαβ, μV Here, T is the stress-momentum tensor of the material. According to Appendices VI and VII The derivation of the above equation, which is discussed in detail, is based on the novel atomic model and optical The inertial mass and the inertial mass given by the principle that all particles including and based on the principle of gravitational mass equivalence.

The Schwarzschild metric is the Einstein gravitational field applied to the Mils orbit. is the solution to the boundary value problem of the equation where the mass discontinuity is the spacetime curvature discontinuity Think of it as equivalent to continuity. Thus, the novel atomic model of the present invention and the theory of general relativity are unified in the quantum theory of gravity, which is valid at any scale, and This is henceforth referred to as the Mils theory of quantum gravity.

The present invention combines three forces, electromagnetic, gravitational, and mechanical, and mutual conversion is possible. As an example, the Meissner effect shows that the superconductor of the present invention Converting the force of force into electromagnetic force, and the strong nuclear force was discussed in the previous patent application added above. This is a phenomenon in which heat is released during the annihilation (low temperature) fusion of Coulombs. The heat is Electricity can be generated directly via thermionic electrons in a photovoltaic generator, or heat can be generated via water vapor. generates a mechanical force that rotates the generator and generates electricity. This electricity is generated by an electromagnetic The device of the present invention imparts an electrostatic, electrostatic or magnetostatic force to a material with negative curvature. , thus an anti-gravity force is generated on the object being attracted by gravity. The anti-gravity force is , provides mechanical force that is useful for propulsion or levitation.

Antigravity methods and means A preferred embodiment of the invention is a propulsion and flotation device, which device comprises a material source, this means to form a material into a spatial two-dimensional manifold of negative curvature, and a negatively curved It consists of a means of generating a force on a material, where this force is applied to a negatively bent material and an attractive Balancing the repulsive gravitational force between the attracted masses. Responds to force balance Therefore, a substance with negative curvature moves at a constant speed and responds to the gravitational field of an object that is attracted by gravity. produces useful work, where constant velocity encompassing zero velocity is of negative curvature Given the space-time manifold, which is the solution of the three-dimensional wave equation, its space-time Fourier variable The exchange does not contain a synchronous component with waves traveling at the speed of light. Therefore, this manifold is non- It is radioactive.

The space-time manifold is based on Newtonian mechanics, Maxwell's equations, atomic theory, and general relativity. theory, and based on the novel atomic model of the present invention that combines weak and strong nuclear forces (.

Thus, the present invention of propulsion and flotation devices converts electromagnetic forces from gravitational forces into mechanical forces. This mechanical force provides useful propulsion and levitation. one In embodiments, the electromagnetic force is driven by a mechanical force applied in response to water vapor. The binding energy of the strong nuclear force is obtained by a indirectly via qi or into mechanical energy, e.g. via a thermionic generator. converted. Thus, the integration and interconversion of fundamental forces is achieved with the novel atomic model of the present invention. Provided by mold and propulsion and flotation equipment.

In one embodiment, the anti-gravity propulsion and flotation means are particles, e.g. consists of a means of injecting it as a plane wave that acts as a substance, and furthermore, a plane wave Includes guides. The injected and guided negative curvature exerts a force on the material It is caused by The applied force may be one of an electric field, a magnetic field, or an electromagnetic field. Given by 2 or more. The second bushing on the negatively curved material is in the direction of the force of gravity. Added. The second force is applied by one or more of an electric field, a magnetic field, or an electromagnetic field. It will be done. In a preferred embodiment, the gravitational force is equal to the repulsive anti-gravitational force. , this anti-gravity force is due to the negative curvature of the guided object, which is attracted by the gravitational force occurs between and matter. Next, the repulsive force guide (second force source) of the object that is attracted by the gravitational force ), this guide should further accelerate or float the attached structure Transfer to the body.

In a preferred embodiment of the propulsion device, the vehicle to be accelerated Ie) is an anti-gravity flotation device and a flywheel rotating around its axis. Consists of ru. The anti-gravity force is a pure force when the force of gravity on the vehicle is overcome by an equal amount. Provides cool radial acceleration. The unbalance of the central force applied to the vehicle is it will make it lean. The tangential addition is due to the angular momentum of the rotating flywheel. velocity is generated, which conserves angular momentum. High acceleration and velocity then result in gravitational force obtained by accelerating the structure along a hyperbolic path around the object being pulled. , thus the structure is accelerated at a high velocity.

Brief description of the drawing These and further features of the invention will be apparent from the following detailed description of the invention together with the drawings. It will be better understood by reading it.

Figure 1 shows the guide and field generating means at one point along the channel of Figure 4. , a two-dimensional graph showing the cross section of the magnetic potential and the corresponding magnetic field lines (arrows). It is f.

Figure 2 is a three-dimensional graph, which is an electric potential function, xyz The magnetic of the electric force in two directions for, and the magnetic potential function, 2 for xy The direction indicates the magnitude of the magnetic force, and the electron beam propagates in two directions.

FIG. 4 is a diagram of a system of anti-gravity propulsion and flotation means according to one embodiment of the invention; It is a surface.

FIG. 5 shows gravitational and anti-gravitational forces acting on a vehicle according to one embodiment of the invention. Schematic representation of force and angular momentum.

FIG. 6 shows an embodiment of the present invention that generates negative curvature electrons that simultaneously generate anti-gravitational forces. 1 is a drawing of an experimental setup according to one embodiment; FIG.

Figure 7 shows the relativistic state of the device in Figure 6 after passing through the quadruple tangent magnetic triplet. It is a drawing showing the distribution of negative curvature and anti-gravitational force in an electron beam.

FIG. 8 shows the power generated by Coulomb annihilation (cold) fusion according to one embodiment of the present invention. This is a block diagram of the anti-gravity repulsion device.

Detailed description of the invention As mentioned above, we provide the basis for the new integrated theory of gravity that we present here. The models of fundamental particles provided, e.g. lebton and quater, are hereby cited by reference. U.S. patents before the title of the invention “Methods and structures for energy/material conversion” to be added This is a new atomic model described in the application.

The new atomic model includes an electron consisting of a negatively charged fundamental particle, the lebuton.

However, the same principle applies to other fundamental particles, including quarters. into the waves Similar properties, spin and orbital angular momentum phenomena, and selection rules exclude radiation. are symmetrical to those of Lebton; thus, the quota has a space-time charge density function equal to the Lebuton. Moreover, weak and strong nuclear forces (Levton's spin pairing, diamagnetism, and binding force) are electromagnetic in nature .

The nucleus consists of nucleons, °protons and neutrons. Each nucleon is a new atomic model As a Mills orbit similar to the case of the electron in a three-electron atom, Ru. Posons W+, W- and zO are spin-1 large photons and weak This is because of nuclear power. W4 and W- have the energy of the weak nuclear force of spin pair formation. and zO has diamagnetic weak nuclear force energy.

The strong nuclear force combines photons and neutrons with quotas and the nucleus of the atom Combines with photons and neutrons. The strong force is caused by a spirit called gluon. This is because of the field of n-1 particles. The gluon is described by a novel atomic theory. Lebuton's Mils orbit when absorbing a photon or energy hole such that analogous to a steady, moving wave of photons that exists inside the . Gluon is in Mills orbit maintain quotas within, and cause a balance of power. A new atomic model The strong nucleon binding energy released by Coulomb annihilation fusion is the Mils orbital Similar to the energy released in chemical bonds as an overlap of minimize the energy stored in the electric and magnetic fields.

The rule for the binding of quotas by gluons is confinement, and this confinement is conserve angular momentum and energy, minimize energy, and travel at the speed of light There is no space-time Fourier component of the charge density function that is synchronous with the wave that follows: thus , there is no radiation. It always binds particles together, making combinations that have no color. make a lie The "colors" of gluons and quotas are red, green, and blue. red quo The data are joined by a gluon string J to form green and blue quarts. data (red, ten green + blue = white). Such a triblet can be a photon or a new Does not constitute Tron.

The space is 120π given by the square root of the quotient of the free space permeability and permittivity. = 277 ohms. It transmits gravitational waves and electromagnetic waves Give a speed limit of C for the propagation of the waves involved.

It also provides a field that meets the boundary conditions. Matter/energy is created in space. and space acts on matter/energy. Thus, the spatial quadratic of matter The original manifold produces a gravitational field in space: the three-dimensional space-time manifold of the current produces a magnetic field in space. Create a place. A spatial two-dimensional manifold of charges produces an electric field in space. thus, General relativity and Maxwell's equations are valid at arbitrary scales. moreover, The existence of a substance with a mass determined as a charged three-dimensional space-time manifold is a spatial body. Maximize the ratio of product/surface of material. This is due to the energy of the resulting gravitational, electric and magnetic fields. Minimize rugie.

Matter/Energy is interconnected and, in essence, Matter/Energy is The same entity with different boundary values given by space-time that has a reactive effect on space-time It is. All matter/energy follows a three-dimensional wave equation, and magnetic fields, The Coulomb field, photon field, and gravitational field are This can be derived as a boundary value problem of the space-time wave equation that gives the field of The complexity of the action/reaction is obvious. The fact that space-time is four-dimensional It is clear. Because the basics of gravity and Coulomb attraction, which are time-dependent force is the area of legal symmetry that transmits force at the speed of light given by space-time. This is because it has a value divided by 1 by a distance square relational expression that is equal to the distance dependence.

The third fundamental force, the action/reaction relation of mechanical force, is gravitational field, magnetic field, and photon field. Newton gives the motion of matter, including charged matter, which can give rise to It is given by the law of Action exerted by a force in one inertial frame /reaction is valid for Euclidean spacetime and is a consequence of the limiting speed of light , given in different inertial frames by the Lorentz transformation of special relativity.

For example, the magnetic field in one inertial frame is given as a Coulomb field, and in the other given as a result of their relative motion. The existence of matter is due to the gravitational field and The geometry of space-time is biased away from the Euclidean geometry that appears as . The gravity equation is From the novel atomic model of the present invention in which spacetime is Riemannian for all scales, be guided.

The provision of the equivalence relation between inertial mass and gravitational mass by the novel atomic model of the present invention is as follows: It allows for the correct derivation of the general theory described in Book VI and VII. And things The former provision of a two-dimensional nature of quality allows for the integration of atomic and cosmological gravity. . An integrated theory of gravity is derived by first establishing the metric. general The mathematics of metric expansion as tensor analysis is given in Appendices ■ and IV; The principle of covariation of equations is described in Appendix v. About the principle of general relativity A related summary is included in Appendix V.

The curvature tensor has the following formula: %formula%) A space with constant curvature is called a space of constant curvature. is a four-dimensional generalization of hevsky) space. The constant is called the curvature constant. space-time It has been shown that curvature results from discontinuities in matter that are restricted in two spatial dimensions. Ta. This is the property of all matter as Mills orbits. isolated mils orbit and the radial distance, r from its center. For r less than ro , where ro is the radius of the Mils orbit and there is no mass; thus, spacetime is a plane or Euclidean. The curvature tensor is all about the inertial frame thus, for r less than ro, K=O0r=r.

, there is a discontinuity in the mass of the Mills orbit. This is r. greater than or or a curvature discontinuity for an equal radial distance. Below, This discontinuity is important to Einstein, who equates the properties of matter with the curvature of space and time. This gives rise to the problem of boundary values of force fields.

The equivalence relation between inertial mass and gravitational mass is directly derived from the existence of matter as a Mills orbit. arise. The present invention's induction of the general theory of relativity is based on this principle and all Based on the principle that particles follow geodesics (.Then, Mills' theory of quantum gravity is It is given by the solution of the gravitational field equations of general relativity given in Appendix VII.

According to the novel atomic model of the present invention, the energy of the vacuum is zero; thus, The cosmological constant of this model is zero, as demonstrated in Appendix Vl. This is correct This is a precise experimentally determined value. The novel atomic model of the present invention is based on the cosmological scale to integrate atomic theory and gravity. In Appendix VII, the boundary value of the Mills orbit is The problem is that the solution to Einstein's field equations is based on atomic theory and uniformity at the atomic level. A fully integrated theory of gravity for all scales of spacetime by integrating general relativity give. Applications of the relevant equations are given in Appendices Vl and Vll.

Solutions of the gravitational field equations described in Appendix Vll can yield opposite signs.

The positive result for α in equation (57, 37) in Appendix Vl arises from the positive curvature; And negative results arise from the negative curvature of the material. Thus, antigravity is the matter Mils orbit. It can be created by forcing a path to negative curvature. negative curvature A fundamental particle is an object that is attracted by gravity and is composed of matter with a positive curvature, and has a center Yes, but experience the force of repulsion. The anti-gravity force gives a mechanical force: thus the negative curve The deviation of a substance with a rate is the lebton), where the fundamental particle is the essence of the substance. is made into a plane wave. As described in a previous patent application, the Mills orbit is an energetic The material absorbs energy and has an infinite radius, giving rise to the properties of a plane wave.

an electric, magnetic or electromagnetic field, e.g. parallel to or transverse to a plane wave of matter A laser applied or completely moved in a fiber optic cable, e.g. One or more of the vanishing fields generated by internally reflected electromagnetic fields Plane waves are accelerated and shaped (or distorted) into negative curvature.

The resulting anti-gravitational force is transferred to the source means of the field and, in turn, the latter's hand on the structure. For rigid installation of the stage it is moved to the structure to be accelerated or floated.

According to the invention, the force generated by the anti-gravity flotation/propulsion means is As a boundary value problem for a three-dimensional space-time manifold with negative curvature, Ein It can be roughly calculated by solving Stein's field equations. However, the force at the limit can be obtained as follows. equation( Consider a negative solution for the variable α of 57, 37). A negative solution is a boundary with negative curvature Occurs naturally as a match to a condition. Furthermore, a material with negative curvature has a positive Compared to curvature matter, it occupies a reduced amount of four-dimensional spacetime. Surface/volume of sphere The ratio of is the minimum. After all, the integral of equation (57, 37) has a rectangular shape, where where S is the space defined by the material boundary of negative curvature.

The existence of a three-dimensional space-time manifold (spatial two-dimensional manifold) in a four-dimensional space-time is important. It occurs in the curved non-local space-time where the force originates, and in the four-dimensional space-time. The two-dimensional space-time manifold (spatial one-dimensional manifold) has infinite gravity and a black hole. occurs at a singular point. Sign of curvature.

For the case of negative curvature, an anti-gravitational force using an object attracted by gravity has a negative curvature can be increased by increasing the strength of.

Furthermore, according to the present invention, the negatively curved material "ionizes" the fundamental particles into plane waves. It is created by making it happen. As described in the previous application, the Mills orbit Roads can absorb electromagnetic energy.

When a photon is absorbed, the radius of the Mils orbit expands from the ground state with ro to nre. where n=2.3.4181. ■. When n becomes infinite, the radius r becomes infinite , and the Mills orbit becomes a plane wave. Sufficient electromagnetic energy is absorbed Ionization occurs when a plane wave is generated in a Mills orbit. ionization energy – by applying a large potential to the cathode or by heating or irradiating the cathode. It can be obtained by In the latter case, continuous waves or pulses A photocathode illuminated by a laser with a This can cause problems. Photocathodes, thermionic cathodes, and low-temperature cathodes are Described in the literature: Orttinger, P. et al., Physics Nuclear Instruments and Methods in Research and Methods in Physics Re5earch), A2 72.264-267 (, 1988) and Sheffield 1eld), R:, et al., 1bid, 222-226, which are incorporated herein by reference. I add. By crossing a selected field such as that created by the field source means. This causes the resulting plane wave to propagate through space and acquire a negative curvature. Ba Gente The stages provide one or more of an electric, magnetic, or electromagnetic field. the resulting space-time manifold is two-dimensional (spatial two-dimensional manifold and time-dependent), and according to the formula This is the solution to the three-dimensional wave equation: In addition, manifolds are objects that propagate through space and are attracted by gravitational forces, such as anti-gravitational forces. and accelerated by the propagating force provided by the source means. material A space-time manifold with negative curvature due to the effect on the space-time Fourier transform is It is like having no waves that are synchronous with waves traveling at the speed of light. Manifolds are antigravity This is the Mills orbit that gives the force.

A material of negative curvature (such as a Mils orbit) moving at constant speed can interact with a wave traveling at the speed of light. It has a space-time Fourier transform that has no Fourier component that is synchronous. 3-dimensional space-time Appearance δ[z-f(x)g(y)-K(t)] (1, 11) where K(t)=V, , V are constants.

The space-time Fourier transform is written as: where F (k,) and G(k, ) are the Fourier transforms of f(x) and g(y), respectively.

The only non-zero Fourier components exist for: VCO8θ where θ is the angle between V and K. In this way, the space-time Fourier transform is a wave at the speed of light. has no component that is synchronous with: therefore the manifold is non-radiative.

For example, the Fourier transform of the three-dimensional space-time manifold δ(z-sy-vt) is as follows: Described: π/2 It has no component that is synchronous with waves traveling at the speed of light: therefore it is non- It is radioactive.

In a preferred embodiment, the manifold is described by the following function: δ[z−x (z) y (z) - v, t] Here, v8 is one of the two directions in the balance of force. It is a constant speed. The manifold is the electric field of a quadrupole at infinity or the quadrupole at infinity. produced by a magnetic field and a constant force of equal magnitude and opposite anti-gravitational force. Thus, a material of negative curvature moves with a constant velocity Vt.

Implementation mode In one embodiment according to the invention, the device for applying an anti-gravitational force is a plane wave of electrons. and a guide means for guiding the propagation of the plane wave. Acceleration and The formation of negative curvature and negative curvature is caused by the application of one or two electric, magnetic, or electromagnetic fields by field source means. Implemented in the propagating guided electrons by adding the above. mutual The repulsive force is the relationship between the propagating electron of negative curvature and the gravitational field of the object attracted by the gravitational force. An object that is attracted by gravity is made of a material with positive curvature, and the field source is the repulsive force. give an equal and opposite force to Thus, the forces of interaction are sourced and guided. The guide further transfers the force to the attached structure to be accelerated.

In this embodiment, the anti-gravity means shown schematically in FIG. and an electron acceleration module 101, e.g., an electron gun, an electron storage ring, a radio frequency consisting of a wavenumber linear accelerator, an introductory linear accelerator, an electrostatic accelerator, or a microtron . The beam is focused by a means 112, such as a magnetic or electrostatic lens, a solenoid, Focused by a quadrupole magnet or laser beam. The electron beam 113 is a beam means 102 and 103, for example dipole magnets, for directing the electronic guide 10. Directed into channel 9. Channel 109 is directed against the direction of the anti-gravitational force. field generating means for generating a constant electric or magnetic force in opposite directions. example For example, the anti-gravitational force is negative 2 directed as shown in FIG. 9 through the linear potential provided by grid electrodes 108 and 128. For a constant electric field in the negative 2 direction, give a constant electric field in the Z direction: antigravity Assuming that the force is positive y directed as shown in FIG. , given by the top electrode 120 and the bottom electrode 121 of the field generating means 109 When the anti-gravitational force passing through the linear potential is directed towards positive y, the field generating means 109 is constant due to the constant dipole magnetic field in the X direction for the electron beam moving in two directions. gives a magnetic field in the negative y direction.

In one embodiment, the field generating means 109 distorts the electrons of the electron beam 113. an electric or magnetic field that creates an anti-gravitational force on objects that are attracted by a gravitational force Give a place. In a preferred embodiment, the electric potential of the distorting electric field is written as: xyz+cp where C is a constant and p is X1y or 2 and in the opposite direction to the anti-gravitational force: hence the corresponding The force is opposite to the anti-gravity force as mentioned above. The electric field is the negative of the potential Described by gradient. The electrical straining forces in the Z direction are shown in FIG.

In a preferred embodiment, the magnetic potential of the strain field is written as Ru: xy+cp where C is a constant and p is xly or 2, thus the pair The corresponding constant dipole magnetic field is constant in the opposite direction to the antigravitational force, as mentioned above. generates a magnetic field. The potential functions and field lines are shown in FIG. magnetic The field is given by the negative gradient of the potential. Electricity propagating in two positive directions The two-directional distortion field on a plane wave is shown in FIG.

Electric and magnetic distorting fields transform the plane wave of an electron into a plane wave of negative curvature, written as Make it a manifold δ[z-x (z) y (z)-vt] This manifold is shown schematically in Figure 3. There is.

The velocity of the manifold, Due to the equivalence with the anti-gravity force that arises as an interaction between It is. This constant force causes the manifold to become a channel of the guiding means and field generating means 109. As the object propagates along the gravitational force, it has a constant levitation against the gravitational field. or give the work of propagation. The work produced depends on its attachment to the field generating means 109. and then transferred to the means to be propelled or floated.

A constant electric or magnetic field is variable until a force balance with the anti-gravitational force is achieved. It is. In the absence of force balance, the electrons are accelerated and the beam radiation is will increase. Also, the accelerated electron will radiate: thus radiating A decrease in and/or the absence of radiation is a signal that a balance of forces has been achieved. release The radiation and/or radiation is detected by sensor means 130, e.g. a photomultiplier tube. and the signal is sent to the feedback module of analyzer-controller 140. The analyzer-controller 140 is used in Control the potential or dipole magnet in stage 109 to maximize antigravity work. Change a constant electric or magnetic field by increasing the

In other embodiments, the negative curvature of the electrons in electron beam 113 may be caused by photon source 105, e.g. e.g. by absorption of photons provided by a high intensity photon source, e.g. a laser. will be accomplished. Laser radiation is confined to the resonator cavity by mirrors 106 and 107 be able to.

In a preferred embodiment, the laser emitting or resonator cavity emits a plane wave of electrons. oriented with respect to the propagation direction, thus resulting in a quadruple tangent transition where the change in angular momentum is zero. such that the selection rule relation for is maximal for a given multipolar radiation . For example, if the propagation directions of beam 113 are two directions in FIG. 4, and the radiation is multiplexed, If it has polarity M1 (magnetic dipole radiation), then the laser radiation or the resonator radiation The orientation of the cavity is along the X or y axis (i.e. 90° to the electron beam): the beam 113 propagation directions are two directions, and the radiation is multipolar E2 (electron dipole emission). The laser emission or resonator cavity orientation is along the z-axis ( That is, O' for the electron beam).

Anti-gravity work is extracted from beam 113 propagated through field generating means 109 Afterwards, the beam 113 is directed by a beam directing device 104, e.g. a dipole magnet. Directed into line dump 110.

In a preferred embodiment, beam dump 110 dumps the remaining energy of beam 113. means to recover the energy, e.g. by recirculating the beam or by electrostatic moderation or or by deceleration in the structure of a radiofrequency-excited linear accelerator. replaced by means of These measures are described in the following documents: Feldman, D.W., et al., in the study of physics. Nuclear Instruments and Met hods in PhysfcsRe5earch), A259.26-30 (1987), which is incorporated herein by reference.

The present invention relates to high current and high beam and free electron laser systems. It consists of Such systems are described in the following references, which I add by citation: It is.

Nuclear Instruments and Methods in Physics Research ts and Methods in Physics Research), A272, (1,2L 1-616 (1988).

Nuclear Instruments and Methods in Physics Research ts and Methods in Physics Research), A259, (1,2L 1-316 (1987).

In atoms or free space, Mills orbits satisfy the balance of forces.

Thus, the electron is accelerated by the force of the electric field as a plane wave, and a non-radiative of negative curvature The radioactive Mils orbit moves at a constant speed, and the force of the absorbed photon forms/disappears. Attracted by an attractive force consisting of a force, a propagating acceleration force, and a curvature opposite (positive) to the Mils orbit It exists when the repulsive anti-gravity forces between it and the object are precisely balanced. mill The orbit carries out constant antigravity work as it propagates along the guide, and here The curvature of the object and the Mils orbit that are attracted by gravity is determined by the time of interaction of the gravitational forces is essentially constant.

For a propagating electric field strength of 10'V/m and a gravitational interaction of 1 meter, the electric The child's antigravity work is IGev.

transmitting a total of 1000 amperes, where the product of the Kerr distances for repulsive gravitational interactions is IGev. The effective propulsion force for a moving guide or one series of guides (109 in Figure 4) is as follows. is written as: A structure, e.g. a vehicle with a mass of 500,000 kg, at 100 m/s The time to accelerate to a speed of ec is written as:

Thus, by applying the anti-gravity force generated by the anti-gravity device according to the invention, a large can accelerate large vehicles or levitate arbitrary large objects .

In a further embodiment, the force exerted by the anti-gravity device according to the invention is It is central with respect to objects that are attracted by a force. The surface of an object that is pulled by gravity Acceleration in a direction tangential to a surface can be achieved through conservation of angular momentum. can. Accelerated structures, e.g. atmosphere and space that should be tangentially accelerated A vehicle can move with cylindrical or spherical symmetry and has a moment of inertia. have a capable mass, e.g. a flywheel device. Flywheel is a drive device Driven with more angular movement, this drive is powered by an electric motor and an electric energy source. Fusion reactions using stages, e.g. thermionic or steam generators or batteries Potassium is produced using power. The drive device imparts angular momentum to the flywheel. Ru. The vehicle uses anti-gravity means to levitate and overcome the gravitational pull of an object. The angular momentum vector of the flywheel is pulled by the gravitational force, which overcomes the force. This is a levitation parallel to the center vector of the gravitational force of the object being held.

The vector of angular momentum of the flywheel is the levitation (reaction) given by the antigravity device. By modulating the symmetry of the force (of gravity), the center vector of the force of gravity and a finite angle Due to the interaction of vector angular momentum of angular momentum , a torque is generated on the flywheel.

The resulting acceleration that conserves angular momentum is given by the center vector and the angular momentum vector. perpendicular to the plane formed by the Thus, the resulting acceleration is attracted by the gravitational force It is tangent to the surface of the object.

Moment of inertia 11 Spin moment of inertia ■1, total mass m1 and S The equation describing the motion of a vehicle with its flywheel spin frequency is , is given as: where θ is between the center vector and the vector of angular momentum is the angle of inclination of - the height at which the vehicle floats, and φ is the angular process resulting from said torque. It is a frequency relay. A schematic diagram is given in FIG.

The vector of angular momentum of the vehicle is achieved when tilted with respect to the center vector. The calculation of the approximate velocity is written as follows: where g=10m/5ecS l =-5000m, r=10m, , S=2.5-'The linear velocity is written as follows. Angular frequency x radius: 2π20 cycles/see (5000m)s in (45°)-4,4X105m/see According to this calculation, a large tangential velocity will finish a trajectory that is perpendicular, and then The motion is achieved by the tangent (velocity) performed by tilting the flywheel. is shown. Energy stored in the flywheel during tangential acceleration - is converted into kinetic energy of the vehicle. kinetic energy of rotation The equations for ER and the transition kinetic energy ET are written as: E11 = 121ω2 where 1 is the moment of inertia and ω is the angular rotation It is the frequency.

E t = 1 / 2 m V 2 where m is the total mass, and V is the transition It is the speed of movement.

The equation for the moment of inertia I of the flywheel is written as : ■=Σmr2 Here, m is an infinitesimal mass at a distance r from the center of mass.

As these equations demonstrate, the maximum rotational energy is Quantity can be stored by maximizing the distance of the mass from the center. child Thus, the ideal design parameters are the mass rotating around the vehicle and Cylindrical symmetry.

Further, in accordance with the method and apparatus of the present invention for providing anti-gravity forces, rapid long-distance transport The means by which transportation can be realized and propelled here, such as outer space, are attracted by gravity. This is accelerated to enormous speeds by executing a hyperbolic orbit around an object Here, the force of the object drawn by gravity and the vehicle exerted by the anti-gravity means of the present invention are Kru's anti-gravity force accelerates the vehicle to high speeds.

衷 - 憇 A high current, high energy electron beam is injected into a four-electrode magnetic field, and the beam The profile of the geometric cross-section was determined by Carlstein [Karsten (Carlstein), B.E., et al., Nuclear Instruments and Methods in the Study of Physics. Nuclear Instruments and Methods in Physics Research), A272.247-256 (198 8)]. One embodiment of the anti-gravity propulsion and levitation of the present invention is , consisting of the apparatus of FIG. 6 without the wiggler and spectrophotometer. However, Furthermore, the device of the invention comprises a channel and field generating means 1 for the electron beam in FIG. It consists of an electronic guide means consisting of 09, 4-electrode triplet, Ql, Q shown in 2 and Q3 to generate electrons of negative curvature as they propagate. Useful for power The unused antigravity is due to the frame shape of the beam as a function of the current as shown in Figure 7. (This is Figure 11 of the reference). child As shown in the data, the negative curvature Poltzmann distribution clearly This is achieved by the shape of the frame of the lofill (see Figure 7). This shape is A point on a manifold of negative curvature that gives rise to a Portzmann distribution of the gravitational force and the corresponding displacement. This is due to the Earth's constant gravitational field interacting with the Lutzmann distribution. Due to the force of anti-gravity The maximum vertical deflection of relativistic electrons is approximately 5 cm to 150 cm. is the displacement in the direction of the line. Thus, the electrostatic and electromagnetic forces of this device Comparable anti-gravity forces were achieved. The current dependence of the efficiency of negative curvature generation is Higher beam powers impede efficient coupling of electrons with polar triplets. resulting from electron-electron interaction with the current. However, significant antigravity is It was generated with a current of amperes. Thus, as our experiments show, IGev/electronic Anti-gravity work of the order of magnitude is accomplished by the method and apparatus of the present invention.

The present invention integrates three forces, electromagnetic, gravitational and mechanical, and their Enabling interaction. As a further example of the structures and methods of the invention, The ner effect is the following phenomenon. In other words, the superconductor of the present invention converts gravitational force into electromagnetic force. and convert the energy of the strong nuclear force into the Coulomb annihilation (low temperature) fusion of the present invention. Coulomb annihilation (low temperature) fusion generates thermionic or photovoltaic power. Electricity is generated directly through a generator, or the heat is rotated through a generator to generate electricity. These are illustrated in FIG. 8, which generate mechanical force via water vapor. electricity is electricity A magnetic force is generated, and this electromagnetic force causes the material to be distorted to negative curvature by the device of the present invention. , thus generating an anti-gravity force. Anti-gravity forces are useful mechanical forces for propulsion or or provide flotation.

In one embodiment shown in FIG. 8, fusion reactor 210 is a Coulomb annihilation fusion This heat is converted into water vapor in the heat exchanger 214. water vapor The air is transferred by connection 216 to a turbine 218 which converts it into water vapor. is driven to generate electricity to supply the electrical load of the anti-gravity device 224. be Alternatively, heat is transferred by connection 212 to a thermionic power converter 226, which 6 directly converts heat into electricity to supply the electrical load of the anti-gravity device 224, and is used here. Any heat that is not used is returned via connection 213. Electrical energy is sent to anti-gravity device 228 Converted into anti-gravity energy, the anti-gravity device W228 is capable of propulsion and levitation. The vehicle is provided with an anti-gravity device 228 constructed by structural connections 206. Architecturally installed. Fusion reactor 210, heat exchanger 214, turbine 218, power generator 220, and thermionic power converter 226 also propelled by the vehicle by the connections 201-206 of their respective structures to the Or floated.

Appendix■ Basic Concept in Relativistic Astrophysics Re1ativistic Astrophysics), Su Ji Fang (L i Zhi Fang) and Remo Ruffini, World 5 scientific Pubfishing Co, Pte, L td, 1983.1 and 2 chapters, apl-70, incorporated herein by reference. .

Appendix II (The following text is about the Theory of Space, Time and Gravity.

f 5pace, Time and Gravitation), 2nd revised edition. As Chapter V, Pergamon Press, pp228-233 (1965) Published and added here with 4 quotes.

In accordance with the present invention, new developments have been made from this text. Additions are underlined. , and deletions are indicated in [brackets] and by initials. ) In gravitational theory, the equivalence principle states that in a certain sense the field of acceleration is equivalent to the gravitational field. This is understood to mean the following. Equality is equal to: An appropriate coordinate system ( This is usually interpreted as an accelerated frame of reference). In this new system, the gravitational force has the appearance of the equation of motion of a free mass point. It is possible to transform the equation of motion of a mass point in a field. Thus, the gravitational field is , so to speak, can replace or even mimic the field of acceleration. Inertial property Due to mass and gravitational mass equivalence, such a transformation reduces the arbitrary mass of the photocathode to The values are the same. But it only exists in an infinitesimal area of space for that purpose. i.e. it will be strictly local.

In the general case, the transformation we describe is a number that goes to a locally geodesic system of coordinates. Scientifically equivalent. As shown by Fermi, not only at one point (, It is also possible to introduce a coordinate system that is locally geodesic along the temporal world line. Wear.

Thus, the equivalence principle relates to the law of equivalence of inertial and gravitational masses; It is not the same as that. The latter has general nonlocal properties, but the acceleration and gravitational field equivalence Particularity exists only locally, i.e. it calls only a single point in space ( More precisely, it refers to the spatial neighborhood of a point on the world line, which has the properties of a time axis.)

The equivalence principle played an important role before Einstein created his theory of gravity. Ta. Next, I will describe and analyze the argument that Einstein gave at that time.

Einstein used the example of a laboratory inside a falling elevator to explain his The equivalence hypothesis was exemplified. All objects in such an elevator are They look like heavy berets and have the same acceleration as all elevators. Even when they are not fixed to the walls of the elevator, they The relative acceleration of these vanishes. We follow Einstein and use two reference frames. one is an inertial or nearly inertial frame, fixed to the earth; and the other frame is accelerated and fixed to the elevator.

In the first inertial frame, a gravitational field exists and in the second accelerated frame And it doesn't exist. Thus, following Einstein, acceleration is caused by gravity or can be replaced by a small (and uniform) gravitational field. Einstein Expanding further. He compares both accelerated and non-accelerated frames. We propose to consider the systems as completely physically equivalent, and Point out that the inertial frame and absolute acceleration become meaningless.

Let's analyze this consideration of Einstein in more detail. The first of all problems is Cheating: What is the accelerated reference frame and how is it physically? Will it be realized? In the elevator example, the "frame of reference", so to speak, is Identified using seed boxes and elevator boxes. However, we []gravity The abstract concept of an absolutely rigid body proves untenable even when we do not consider and principal 4μ: When accelerated, every object has a different strain for different objects. will experience [. ] Here the effect is given by special relativity. However, Therefore, when dealing with boxes or rigid scaffolding [inertial frames, we refer to Section 11] [of the kind discussed in] are not useful as models for accelerated reference frames. do not have. Thus, in Einstein's reasoning, reference in accelerated motion The basic concept of a reference frame remains undefined. This difficulty is due to the magnitude of the acceleration and can be avoided by placing limits on the size of the spatial region to be considered. You will be able to do that. For example, one could ask for: forgiveness. The accelerations resulting from them should be very small, so that in the region of space considered, the be able to ignore the force and use the concept of rigid bodies. .

In that case, the approximate nature of Einstein's argument becomes clear.

Furthermore, Einstein himself believed that all of the gravitational field could be replaced by acceleration. Emphasize that this is not possible: for this to be possible, the gravitational field must be homogeneous. This also imposes a restriction on the spatial dimension of the region, and here The gravitational field and the accelerated field can be approximately equivalent. For example, terrestrial objects It is not possible to "remove" the gravitational field around the For example, one must introduce an ``accelerated degeneracy'' frame of reference. It would be nice.

Einstein also used the equivalence principle in a nonlocal way, but in 1911 In a paper published in His attempt to do so, as calculated in Appendix I, was based on his theory of gravity [see section 59]. gave a deflection of the ray of light by only half the amount resulting from. This is because the equivalence principle is about ds. The correct form (51, 11) cannot be derived in most cases, but it is best for slow movements. Combined with the fact that only formula (51°10) is valid, [. ds” in the inertial frame induced in Appendix VI. This is a minimal displacement. Thus, in the nonlocal interpretation, the approximation of the gravitational and acceleration fields Equivalence is also limited. As already mentioned, this equivalence applies to weak homogeneous fields and slow It exists only for hard movements.

Einstein used him to make sense of the indistinguishability of the fields of gravity and acceleration. We adopt the equivalence principle of His equivalent by emphasizing the impossibility of stating an absolute acceleration in He gave an expanded interpretation to the principle. Einstein responded to this with his ``general theory of relativity.'' [We discussed this in section 49*] (Appendix V): He uses the latter to correct the requirement that his equations should be generally communist. It was justified. (The concept of communism was discussed in Appendix V.) However, [we ] Such an extended interpretation seems inconsistent. The equivalence principle is appropriate for reference. This allows the introduction of locally geodesic (“free-falling”) frames. In fact, we can see the essence of the equivalence principle, and by using it, A uniform Galilean space can be defined at infinitesimal. However, this This refers to the equivalence or indistinguishability of acceleration and gravitational fields in a finite region of space. does not justify the conclusion at all.

Mathematics with such a conclusion, to explain the nature of the error made in the drawing. This example happens to have the essence of this problem directly. 2nd lead All functions limited to behave as linear functions at infinity. But long , which makes all such functions indistinguishable in a finite domain I cannot conclude that. However, similar conclusions, i.e. acceleration and gravity The conclusion that the fields of are completely indistinguishable is based solely on their local equivalence. Einstein guided it.

Such a conclusion even refutes Einstein's theory of gravity itself. fact, If perfect homodominance between the acceleration and gravitational fields did not exist, an equivalent idea would be The developed theory is purely kinematic, which is similar to Einstein's theory of gravity. Not at all in that case. Regarding the "general theory of relativity", [we [already pointed out in Section 49*] As demonstrated in Appendix V Physical principles are impossible, and also serve as the basis for the requirements of general co-operation. is unnecessary, and this co-requisite requirement is purely for theories where the coordinate system is not fixed. This is a requirement for logical agreement.

Thus, the equivalence principle can be applied in a narrow sense (approximately and locally), but , cannot be applied in a broader sense. The effects of acceleration and gravity are “small” "in small objects", i.e., although they may be locally indistinguishable, consider the boundary conditions that should be applied to the gravitational field ``in the large case''. can be distinguished without any doubt. Uniformly accelerated reference frame introduced The gravitational potential obtained when The condition is not satisfied at infinity when it tends to zero. rotating reference In the reference frame, the centrifugal potential increases with the square of the distance from the axis of rotation. , furthermore, there is Coriolika. These characteristics make such reference frames It can be immediately detected that the ``gravitational field'' in the system is virtual.

Next, let's consider the uniformly accelerated reference frame example in somewhat more detail, taking into account relativity. discuss. When doing this, how can the accelerated frame be achieved? Ignoring the question of whether it is possible to Interpret it more formally. In this sense, a frame moving with acceleration Transitioning means multiplying coordinates by a transformation involving time nonlinearly. Dew.

There is no true gravitational field, and thus the square of an infinitesimal interval is the formula%(formula%) where X', y' and Z' are Cartesian coordinates and t' is some inertial reference Suppose we have , which is the time in the frame. y to perform coordinate transformation =y　;　z　=z　(61,02) where g is a constant of the acceleration dimension. this The previous equation under the condition is can be written.

Substituting (61,02) into (61,01) gives (61,05) is obtained.

A problem arises: this equation Can it be interpreted as the square of ? The answer to this problem is also that the equivalence principle It is the answer to the question of whether and to what extent it is true.

In order to find the answer, let us define (61,05) as an approximate expression given by gravity theory. compare.

(61,06) Here U is the Newtonian potential of the true gravitational field.

under this condition The coefficient of l　gx　l　<<c"　(61,07)dt" is U = -gx (61,08) If we take the gravitational potential given by, it is approximately equal.

ds”, it is the quantity inequality %formula%) is not significantly different from 1 for intervals satisfying .

The value (61,08) for the gravitational potential is, in fact, in Newtonian mechanics leading to uniformly accelerated motion. To make the initial velocity zero, x', y' and and 2', and approximately.

[+] [This transformation was given by Mohler [20]. ] This is the coordinate (x, t ) describes uniformly accelerated motion.

We compare the two equations for the square of the interval and find that the acceleration in the absence of gravity is The frame of reference for the given motion has similarities to the inertial frame in the presence of gravity. It was shown that However, in identical comparisons, the similarity is close (but not perfect). Since there is no problem of perfect homogeneity or indistinguishability of inertial and gravitational forces, It was shown that it cannot be done. This is ``in the large'', that is, the entire space This becomes especially clear when considering equation (61,05) through the field. First, dt Since the coefficient of 2 tends to become infinite along with X, the boundary condition is satisfied. First, secondly, the coefficient and with it the speed of light is 2′ on the surface x=−c”1g. This cannot be tolerated.

An even more obvious neglect of the boundary conditions for the metric tensor is the transformation (35, 47), it happens. In Newtonian mechanics, it is a rotating coordinate It has the meaning of introducing a system. This transformation is expressed by equation (35, 48) for ds2. Lead to [. ) (35,47) and (35,48) are listed in Appendix IV. Ru. ) Here, the metric tensor not only cannot satisfy the boundary conditions (, and also at large distances from rotation, as established in [Section 35] Appendix IV Ignore inequalities. <35.48) as a gravitational field. What cannot be interpreted (i.e., in the spirit of the "equivalence hypothesis") is the Coriolis It is obvious even from a local point of view due to the existence of the force.

The example just discussed shows that the homogeneity j between acceleration and gravity is limited in space. exists only in the region of (61,08) and the inequality (61,07) and (61,10)).

However, when considering the entire space, the true gravitational field is the virtual gravitational field caused by acceleration. can be distinguished from those of In Newton's theory, this is Newton can be implemented by using boundary conditions on the potential of Ru. In Einstein's theory, distinguish between the true gravitational field and the virtual gravitational field The problem is most easily solved when using harmonic coordinates. Then weigh The components of the tensor are determined by the harmonic condition (53 , 13) and boundary conditions must be satisfied. [As shown in Section 93 In addition, the harmonic coordinates can be uniquely defined apart from the Lorentz transformations. Wear. Transformations of arbitrary coordinates that introduce a virtual gravitational field change the harmonic and boundary conditions. ignore. Therefore, the introduction of harmonic coordinates eliminates all virtual gravitational fields and can do. Suppose we are thus given the quadratic form (61,05) , the transition to harmonic coordinates lies in the transformation (61,02), probably accompanied by a Lorentz transformation. cormorant. The result of such a transition is a return to the quadratic form (61,01), whose form is Demonstrates the absence of a true gravitational field.

[In this section] [Above] In this paper, we do not use general tensor analysis. It was. Its application to (61,05) makes the fourth rank curvature tensor zero and thus would show that a true gravitational field does not in fact exist.

Let us return to the problem of using the equivalence principle to derive the gravity equation.

It is contradictory to interpret this principle as ``general relativity principle 1'' in a broader sense. We have made it clear that we will. However, this means that the equivalence principle is almost valid. Within certain limits, it does not preclude the use of the equivalence principle in a more restricted sense. virtue In the absence of a gravitational field, the reference frame is accelerated and in the presence of such a field. The analogy we have discussed between the inertial frame and the can be explained. Because converting the formula (61,01) to the format (61,05) It is possible that Newton's potential is accurate to this theory as a coefficient of dt. Because it gives us instructions to enter. However, the public theory of gravity Based on this idea of formulation, the approach is limited by its inherent limitations (i.e., equivalent Due to the local nature of the principle and the assumption that the field is homogeneous), [ It seems to us. Another drawback of this approach is that in accelerated motion The necessity of using a misdefined concept of a frame of reference. present invention [Our] aa-blow does not contain these drawbacks and has no inertial mass and no heavy weight. Regarding this principle, to the extent that the latter principle is valid in direct application to the force-mass law. , it can subsequently be obtained as a result of other assumptions. In this way, it means the hypothesis that space-time has Riemannian properties, and its mathematical expression is It is possible to introduce a geodesic coordinate system locally along the world line.

We emphasized the approximation properties of the equivalence principle. But Einstein's gravity From a theoretical point of view, the law of equivalence of inertial and gravitational masses is Since the concept of force-mass itself is approximate, it also has the property of approximation. these The concept is to the extent that Newton's laws of motion and gravity are valid and for any Mass is a quantity that characterizes a particular object independent of its position and the motion of other objects. To the extent that it can be defined, it can be applied. einstein's In gravitational theory, this is only approximately possible. Because the luck of material objects This is because dynamic laws have more complex properties. Nevertheless, the inertial mass and and the law of gravitational-mass equivalence are completely consistent with Einstein's theory of gravity. can be affirmed. This is because this law can be formulated in general. This theory follows very precisely.

On the other hand, Einstein's theory of gravity does not formulate the law of equality of masses. Yes: It involves essentially new physical principles. The first is ordinary relativity: space and already included in the union into a single four-dimensional manifold with an infinite metric of time.

This principle extends to the limiting nature of the speed of light and, closely related to this, means that For more precise identification, order of temporal events and also relationships by cause and effect. be kicked [(section 12)].

The second principle establishes the union of metric and gravity: it is Einstein's gravitational This is the very essence of theory.

These two principles form the basis of Einstein's theory of gravity. And, by guessing, perhaps as a result of local equivalence of acceleration and gravity, the relativity approximation It is not an extension of thought. However, they are curved spacetime or gravitational mass or inertia. Appendix III provides an atomic basis for quantitative equivalence. (The following text is from Nature, Vol. 336, pp. 71 1-712 (1988) and is incorporated herein by reference.

In accordance with the present invention, new developments have been made from this text. Additions are underlined. , and deletions are indicated by [in parentheses and initials. ) extends Hawking's results by assuming the existence and importance of [(Reference, drawing)]. Any thoughts on whether such fluctuations can occur? We do not have them, but when they occur and when their effects are relevant. , we can continue to analyze those effects that exist and persist. , should be emphasized. The first thing we know is that there is a continuous wave of distorted space-time. The hole, the filament must be very small, about 10-”cm, Quantum of gravity - natural magnitude for mechanical fluctuations. Thus, they can be directly observed. It would be impossible to measure it. Rather, as shown, their effects are natural It will be transmitted indirectly through the value of the constant.

The way wormholes affect our universe is that they lead to an increase in the number of universes. Rather than using a closely related variable α.

use. Wormholes affect the values of parameters in physical theories. Since the wormhole effect is dominated by the value of α1, all The constant of nature is a function of α1. This means that the mass of particles, the fine structure The structural constant, the gravitational constant, and, of course, the cosmological constant all depend on the topology of space. There are parameters that characterize the physical structure. [] Furthermore, if we can predict something about the distribution of α1 values, that we learn something about the values of physical parameters such as the cosmological constant Can be done. Since α1 is part of our description of the spatial geometry, their The probability distribution is determined by the wave function of the universe. Using Hawking's technique, the Ullmann states that the probability distribution for α1 is these parameters that make the cosmological constant zero. (small For a positive cosmological constant, it is proportional to 1/the exponent of the cosmological constant). instructor Therefore, the cosmological constant becomes zero. This is because the constant of nature is not zero, but a cosmological constant. Will the cosmological constant become even more infinite, just as the constant of nature takes the value that makes the number zero? It is et al.

[]29 Koulman-Hawking program [Ko adjusts the cosmological constant to zero avoid the major obstacles that have derailed other attempts to [. ] Our universe is Throughout most of its history, there has been some kind of small but indistinguishable cosmological constant. Filled with matter and radiation whose effects are completely obscured. Therefore, nothing how the mechanism determines the value of a constant and adjusts it to zero Could it be possible? The answer to Koulman's approach is that the universe is a wormhole. through the sky becomes the peak of the universe, and thus objects in our universe This avoids the problem of obscuring quality and radiation.

(The following text is from The Theory of Space, Time and Gravity.

ry of 5 pace and Gravitation), Chapter 111, pp , 114-135, and is incorporated herein by reference.

In accordance with the present invention, new developments have been made from this text. Additions are underlined. , and deletions are indicated by [in parentheses and initials. ) General tensor analysis 35. Our mathematical formulation of permissible transformational relativity for space and time coordinates As a basis, we use the wavefront equation %formula%) and the related equation for the square of the interval ds''=c2-dt'2-(dx''+d y2+dz") (35,02) ((ω)2 is the abbreviation for the differential equation on the left side of the wavefront equation It should be understood as ). Our normal variable x, = cl x , =x; x, =y; xs=z (35,03) and also the number When introducing eo=1; e+=ex=es=1' (35,04), Equations (ω)2 and ds2 = 0 It can be written as We understand that both these equations are under the Lorentz transformation. I know that it is unchanging. combined with the previous one by Lorentz transformation , new coordinates X'o=Ct':X1+X1:X'2=y';X'3=z'(35,07) When introducing k=0 is obtained.

Equations (ω)2 and ds2 have the form (35,05) and (35,06) or (3 0,08) and (30,09), for example (35,03) and (35,07) is called the Galilean coordinate, and this term is understood here to encompass time. be done.

Kokote X'0, x'l, x'2 and x'3 are as before i: (35,07) given, and therefore Galilean coordinates, the quantities x O% X 1, x2 and X3 is no longer equal to (35,03), but is instead of the form %expression %) With some auxiliary variable connected to x'0, x'l, x'2 and x'3 by the relationship , where f is an arbitrary function subject only to certain general conditions. Equation (35, 10) can be solved for X01X1, x2 and X3, so their Assume that the Jacobian must not be zero Further assume that the function f has a first cubic continuous derivative. Physical considerations There may be other conditions on fl that result from; these will be explained later .

If this change in variables is made, (ω)t becomes variable xOSxi, x2 and X3. has a homogeneous quadratic form in the first derivative. This format is written as Ru here Similarly, if the variable is changed in ds'', the result is here (35,13)o, and α=0 as easily estimated from (35,15) Therefore g is the determinant The quantity g'a will be the first subdeterminant of this determinant divided by g itself.

Using the rules for multiplication of determinants, is obtained. Here the second factor is equal to the Jacobian D of (35, 11) ,and eOele2e3 (35, 19) Therefore, the first factor is -D. After all (35, 20) xO has the property of time like x'0, but Xl, X2 and X3 are spatial coordinates. The variables xO1X1, X2 and x3 are controlled by conditions that ensure that they have the properties It is useful to limit These conditions must be formulated precisely. Before As mentioned above, the term "event" refers to thoughts at a particular moment in time. means a point in space: it is called a "point-instat" be able to. We define the spatial coordinate parameters X1, X2 and x3 have the same value, but different values xO* and for their time parameters. Two events with xO** and xO** are Requires thorough reading. We refer to temporal events [continuously] We know that the square of the interval is positive. This is especially true for infinitesimal intervals. Therefore, the difference XO* - XO** is assumed to be infinitesimal, and x*Q=xQ: x**0=xO+dxo (35, 21), It must be satisfied that ds2=g00dx”o (35, 22), where goo>O(35,23) It is. Furthermore, we have the same temporal parameter xO, but the spatial parameter - different values of -, i.e. (xLx2 and x3) and (X1+dX1,x2+ dx2, x3 + dx3) exist, the latter reaches the location of the sparkle Assume you don't.

Then, the following continuation < [for (12,01) and (12,02)] One could apply the equation: where rl, r2 and tl, tz are the positions and times of 1 and 2, respectively. It is. A pair of events for which inequality (12,05) is true is called quasi-simultaneous. Justified by the fact that: in one reference frame tz−t l > 0, and tz-1l can be found elsewhere. first The question of what happened to the sparkle has no unique answer here.

A quasi-simultaneous event is an invariant inequality following (12,05) It can be characterized by: Two relational expressions (12,05) and (12,0 6) are equivalent, so (12,05) is also unchanged. we are reality positive amount of R=4[(r2-r+)''-C2(tz-t+)2](12,07) two homogeneous It is called the space-like interval between time-events. For quasi-simultaneous events, ds2 is negative and Therefore, provided that all of the values of dXl, dx2 and dx3 are not Z, then whatever their values are. Quadratic form (35,24) is negative It must be negative-definite. . A necessary and sufficient condition for this is the inequality % expression %) It is a well-known algebraic fact that these inequalities are , incidentally, not all are independent. An independent set of conditions is For example, the first inequality of (35, 25), (35,'26) and the first inequality of (35, 27) 1 inequality.

If all these conditions are given to the coefficients g, , then they become (35°15) ds2 in the vicinity of any point, regardless of whether it is given by It can be expressed as the sum of squared terms, one with a positive sign and the remaining three one has a negative sign. The set of signs of the terms is a quadratic form of the sign (’signature) It is called. In our case the symbols are also written as (eOel and C3) can be written as (+−−1).

From the inequalities (35, 25) to (35, 27), the determinant g is always negative, and Also, a similar inequality involving g″′ with the above index can be applied. In the end, the result is g〉0　(35,28) and Here, C1, C2 and C3 are any three numbers, all of which are not zero.

We will not give proofs of these purely algebraic statements.

Thus, the parameter xO has the property of time, and the other three Xl, x2 In order for X3 to have the properties of spatial coordinates, goo must be positive and The quadratic form with coefficient g+h (Lk:l, 2.3) is said to be a negative definite sign. It is necessary and sufficient. Add limitations to quantities g10, g20 and g30 There is no need.

Next, consider the geometric meaning of the equations xQ=constant and xk=constant. conditions Let ω(x, y, z, t)=O(35,30) be the equation of a surface in motion. conditions that can be interpreted as follows. From this equation, we can see that the space and time loci The differential of the mark is coxdx + 9y + ωzdz + ω-t=0 (35, 31) where ωx1ωy1ω2 and ωt are x, y, z and t It holds that ω is the derivative of ω. Variations in the direction perpendicular to this surface Take the position (dx, dy, dz) and (35, 32) Assume that 1dnl is the absolute value of displacement. Substituting into (35, 31), we get l grad ωl dn + ωtdt = O (35, 33), therefore , the square of the displacement velocity teeth, is given by In this way, (35, 30) is interpreted as an equation for a certain surface. and each point on its surface moves with a velocity normally given by (35, 35) . However, this interpretation only applies as long as this speed does not exceed the speed of light. It is possible to According to (35,35) and (35°01), this is (ω)2≦0 (35, 36) (means not to be. The sign of equivalence is valid for motion at the speed of light.

On the other hand, (ω)”>0 (35, 37) It is. Equations (35, 30) can be solved in time (and have the form and here (gradρ2<□ (35, 39) Equations (35, 38) are such that all four-dimensional "point-instants" are quasi-simultaneous. In a method, everything at an instant of time belongs to the point. Such an equation can be called a "time-one equation." Ma Also, the equations arise [in Section 3] regarding the problem of properties of Maxwell's equations. [That's what we thought about the hyperspace in the original space-time manifold. space) can be considered as an equation. Next, such a hyperspace is It can be divided into two classes.

If (ω)2×0, one of the dimensions of hyperspace can be said to be temporal. (sometimes using the imprecise phrase ``surface is temporal''). (35,35) Therefore, this is an ordinary 2-dimensional surface* (*4-dimensional manifold) moving at a speed lower than the speed of light. , the hyperspace is three-dimensional, but in this case only these two are space Describe the

If (ω)2〉0, on the other hand, we say that the hyperspace is temporal. then It is the whole of infinite space, the various points, points (x, y, z) and use the time method, i.e., hypersurface represents the time t determined by the equation; assigned to any two points in space The times must be so close that the corresponding four-dimensional intervals are always spatial. No.

Using the fact that (ω)2 is invariant, we subsequently write ω=xO1ω=xl, ω =x2 and ω=x3. As a result, (VXO)2=gOO>O(35, 40) and (Vxl)2-gll<O; (vx2)2-g22<O; (Vx3)2-g ”<0(35,41) is given. Therefore the equation xO = constant is an equation of time, and the three The equation Xk = constant (k = 1.2.3) is Represents a surface that moves in a direction. These latter are thus the equations of the coordinate surface of moving space It is.

Also, from our conditions on the transformation of spatial and temporal coordinates, and a constant value of X3 has a velocity less than the speed of light in the inertial frame of reference This corresponds to the movement of a point.

In classical Newtonian mechanics, when interpreted as transitioning to a moving reference frame Transformations of interdependent coordinates are often used. Transformation of coordinates in Newtonian mechanics When we compare this with the transformation of time and space coordinates in relativity, we can realize the following. is required. First, in the general case of accelerated motion, the Newtonian force The very concept of an accelerated frame of reference in science is not the same as in relativity. stomach. Newton's concepts encompass the idea of an absolutely rigid body and instantaneous propagation of light.

In relativity, on the other hand, the concept of a rigid body has an absolute meaning, if not all (and also for unaccelerated motion in the absence of external forces) used and has auxiliary properties; the concept of reference frame is not based on it, but on the wavefront Based on the law of propagation (.Newton's prototype of the reference frame is a rigid scaffold , and the prototype of the relativity frame of reference is a station on a ladder.

Second, the class of transformations accepted in Newtonian mechanics is Newtonian mechanics is much broader than the above-mentioned No need to think about restrictions.

As an example, in Newtonian mechanics we can Consider a transformation that can be interpreted as moving. X', y', 2' and t' are customary If the coordinates of space and time in the sexual frame are Galilean coordinates, then Ru.

x’! x-Hat25 Y'-y; z'=z (35, 42) and also shall be. The variables xSy, z and t are new at some accelerated frame. can be interpreted as coordinates in space and time (in the sense of can. Substituting (35,42) and (35,43) into the equation for dS2 gives us , is obtained. Inequalities required for coefficients are applicable if the conditions are met. Wear. Additionally, it may require. As these inequalities show, the permutation ( 35゜42), (35,43) only in a part of space and at limited times Only lengths in between are accepted.

Another example is a transformation that corresponds to the introduction of a uniformly rotating frame.

ds2=[c2-Q)2(X2+y2)1 dt2-2G)(ydx- xdy) dt-dx2-dy2-dz2(35,48) is obtained. The conditions for the coefficients are , which is less than if the linear speed of rotation is equal to the speed of light Only the distance from the axis of rotation is satisfied.

Once again we emphasize that Newtonian mechanics is applicable to the examples given here. It has physical meaning only in the realm where it is possible [(see also section 61).

It is clear that the introduction of ordinary curve space coordinates is always a permitted transformation. conversion As long as does not include time, the transformation has the same geometrical meaning as in non-relativity. have Therefore, we refrain from discussing them.

36, General Tensor Analysis and Generalized Geometry In the previous section, we is the variable X1, x2, X3 instead of the spatial and temporal coordinates x, yXz and t By introducing call I thought about it. We assume that the variable XO characterizes the order of events in time and that the variable X1 , establish the conditions under which X2 and X3 can characterize its position in space did.

As such, the introduction of new variables cannot naturally affect the physical results of the theory. Can not. However, directly arbitrary without using Cartesian spatial coordinates and time Mathematical physics equations (e.g. equations of motion and field equations) by variables The development of mathematical expressions that allow the description of It is not only important, but also important in principle. The existence of such mathematical expressions is based on physics. It is possible to show a method for generalizing a theoretical theory.

If the equation is valid for any choice of independent variables, we The equation is commonly referred to as communion. A number that generally allows the description of coexisting tensor equations The scientific expression is called "general tensor analysis."

In general, the synergistic formula has already been used in Newtonian mechanics.

We refer to the Lagranche equation (of the second kind), which is a generalized describe the motion of a system of masses at a target and their generalization for continuous media . They state nothing new in physics compared to the problem in Cartesian coordinates. However, the Lagranche equation is nevertheless a subject of practical application and theoretical research. plays an important part in both.

In the theory of relativity, the analysis of general tensors has a similar purpose.

In general tensor analysis, the starting point is given by the square of the gradient and the square of the interval in four dimensions. The resulting pair of equations (36,01) and (36,02). These formulas are It can be said to characterize the metric of spacetime. The coefficients g″′ and gαβ entering the equation are variable. considered as a function of the numbers X01X1, X2 and X3.

Therefore, equations (36,01) and (36,02) become From (35,01) and (35,02) or (35,08) and ( 35,09), the coefficients g1 and gαβ are derived from the following four relationships: We assumed that fO1 can be represented by the numbers f1, f2 and f3. : (35, 16), gxj can be expressed by the same four functions. Ru.

However, do not assume that gαβ can be expressed in the form (36,03) The equations for general tensor analysis are rarely more complex, but their Instead they are simply given functions of the coordinates, i.e. the variables xO1X1, X2 and It is important to note that it is taken as a function of x3. This is a more general point of view corresponds to the introduction of non-Euclidean geometry and non-Euclidean metrics in spacetime do. Such a step goes beyond the limits of ordinary (so-called “special”) relativity. and new physical theories, such as Einstein's theory of gravity [this This relates to the formulation of ], which is given in Appendix VI. [Following chapters of this book are theory], but in this chapter] we apply a purely formal view. Then, by giving the metric, we can write about the assumption that gxβ is a function of known coordinates. We will develop general tensor analysis for all. Such a display has two advantages. No. 1, in order to be able to display gαβ in the format (36,03), gαβ is It is possible to discover a condition under which the generally gives a formalization of communism. Second, in this way, Einstein's gravity A mathematical device is obtained for formulating the theory.

Before giving a systematic explanation of general tensor analysis, it is important to understand that gαβ also has the form (36°03). Establish a connection between the expression (ω)2 and ds'', which exists regardless of whether the function ω If (xO, xi, X2, x3) satisfies (ω)2=0, then by the condition ω=constant The differential of the related coordinates shows that ds2=Q is satisfied.

shall be. (ω)2=0 in the form Rewrite with . The partial differential equation for ω is the Hamilton-Jacobi equation of classical mechanics. It has the same form as the equation of , and can be solved in the same way as the latter. That's ω Solve for 0 and write tsu=-H(.1..2..3) (36,07) , then the function H will be a Hamiltonian and Hamilton's equation will be. but And the first three equations of (36,05) have a differential dxα (α=1.2.3) We show that it is proportional to the partial derivative of G with respect to ω. The infinitesimal proportionality coefficient is *d Represented by p, is obtained. (36,04) and solve for ω, and the obvious relationship Person in charge is obtained, then You get what you ask for. In this way, if we describe the wavefront by the equation (ω)2=O, If we continue to think as ”=Q.

In the following, gxβ is considered to be a function of variables X01X1, X2 and x3. and they have continuous derivatives of all orders considered and mentioned in Section 35 Simply assume that the inequality is satisfied. In addition to gαβ, g a J, their connections is (36,04). gαβ can be expressed in the form (36,03) The conditions under which this is possible are established in Section 42 of the following reference, which is incorporated herein by reference. Trivia: The Theory of Space, Time and Gravity f 5 pace and Gravi tatton), V, Fock, th MacMilan Company, -- New York, 1964゜ In particular, section 20 of this reference gives the definition of a four-dimensional vector: section 21 gives the definition of a four-dimensional vector. describes the original tensor: clause 22 describes the false tensor; clause 37 describes the base of vectors and tensors by tensor algebra: Section 42 Transformation laws for Straffel symbols, and locally geodesic coordinate systems, constant ds'' into the form of coefficients; clause 43 describes the conditions for converting ds'' into the form of coefficients; sol; clause 44 describes the properties of the curvature tensor, and clause 3 1 describes the mass tensor. Also, clause 39 describes parallel transport to vectors. It is listed. And Section 40 describes the co-differentiation.

38. Geodesic equation Consider two times corresponding to two events in a time continuum, and The marks are represented by X, fl+ and X, +21, respectively. If X0=)(Q’ll Then, the coordinates of that space are xk=xk"', and xO=x032'. and the coordinates of that space are xk=xk''.

Since events X, during and X12' are assumed to be continuous in time, such time is possible at speeds lower than the speed of light. The corresponding time XO and spatial coordinate x k is here By doing so, it can be expressed parametrically using an auxiliary variable p. luck Since the moving speed is lower than the speed of light, the inequality % formula %) must be applicable for any infinitesimal interval along the path.

Here, the points represent the differentiation with respect to p. For a succession of times proportional to the interval of appropriate time τ The finite interval between events in is denoted by S=Cτ, and I is obtained.

Next, consider two weight events. Two points in space where an event occurs are connected by a certain curve. and note that the space-instant between the two is the inter-vehicle time. Therefore, it is possible to attribute a definite time to each point on this curve. You can write a "time equation" for a point. About curves and time equations The analytical expression of is restated in the form of equations (38,01) and (38,02) However, these equations can no longer be written as describing the motion of a point along a curve. and cannot be interpreted: they here give a stationary description of the curve as a whole. I can do it.

For any pair of infinitesimally separated midpoints 1.2. , Cape φ. φ pottery p2<O( 38,05) is obtained, and the spatial interval characterizes the length of the curve.

The time interval (38,04) between two events in a time sequence and the The problem arises of both extreme values of the spatial interval (38,06) between the temporal events. both The problem with these changes in is whether the interval is temporal or spatial. This leads to an equation of the same form. Geodetic equation of variation by analogy with surface theory It is called the equation of a line. However, the square of an infinitesimal distance is a positive finite number of coordinate derivatives. In the theory of surfaces, where there exists a quadratic form of , for sufficiently close terminal points), is the shortest line; and the position is different; the interval extremum is the largest for the temporal interval and the spatial Not minimum or maximum for spacing. This means that "ds" has the format (37,04). can be evaluated in the special case of the Galilean metric.

Since we can choose a frame of reference for events in the time continuum, The spatial coordinates of the first and final points are the same, and the time t as a parameter You can choose. Then, the following equation is obtained, (2) The solution to the variation problem in this case is given by constant values x1y and Z. So, it is v220. For other orbits, v2 is greater than zero at some point, so f♂　v　　　　,c, Therefore, for the spatial interval, t(2Lt(1); y(2)=, CI); z(2)=z(1) (3s, 1o) Here, select the reference frame so that X+21> x L l 1. can be done. If the coordinate X is a parameter, is obtained.

The solution to the variation problem is then given by constant values of y, z, and t, which is given. However, other curves y(x), z(x) or other times The equation t<x) is whether the square root in (38, 11) is greater than 1 on average, or or l>l,,t, or l<1.81, depending on whether it is smaller ．． It can be seen that it is.

Next, we derive the differential equation for the geodesic curve. The Lagrangian of the fluctuation problem is 1” Fuchi αβφ”φβ) (38, 13) Or, if we write X instead of φ6, we get 1" 38,・14).

integral l The condition for the extreme value of is the Euler-Lagrange equation% formula% That's the current selection (38, 16) The parameter p can be chosen to be equal to . These last equations is an integral , so condition (38, 19) is a result of (38, 21). About F Inserting a clear expression, we get from (38, 20) or the differential Implement.

Then, the differential equation of the geodesic line is gap “13” [β-y, ctl; <13; c, = g (38, 25 ).

Equations (38, 24) are called the first type of Christophel symbol. About the second derivative To solve equations (38, 25), multiply them by go and divide the sum by α. Ru. Then this new symbol (I) y, v) = gav[13zal (38゛ 26) using quality, +(βZ V) Xl3XY = 0 (38, 27) is obtained. Formula (38, 26) call the second type of Christophelic symbol, and often use another represents. Regarding uniformity, we also consider the first type of Christophel symbol We can introduce the corresponding conceptual form: [αβ, γi = “y+a( ’ (38, 29) and is obtained. In this concept, the geodesic equation takes the form: christolahue If the symbol corresponds to a metric tensor that can be written in the form (35,15), then Equations (38, 32) are the relational expressions for Galilean coordinates x’k be equivalent to. This is due to the communality of the equations and the Christophelian coordinates in Galilean coordinates. It is obtained from the fact that the sign becomes zero. In this case, therefore, the measurement The ground line equation leads to a linear dependence of the Galilean coordinates on the parameter p.

Evaluate that the expansion leading to (38, 32) remains valid no matter what the sign of F is. It's not difficult to value. If F>Q, the "geodesic curve" is and equations (38, 32) move at a speed lower than the speed of light. It can be interpreted as the equation of the mass point of freedom. The increment dp of p is the appropriate time T is proportional to the increment dr of and can be replaced by (38°. The length of the geodesic is , giving the appropriate time between the "departure" and "arrival" of the mass point. On the other hand, the measurement The ground line joins the two headway events and makes ap equal to the increment of space interval. can be done. Equation (38, 32) then appears as.

When F=O, it corresponds to a point moving along the line of the speed of light. In this case, Lagra (38,18) is zero, and the induction above the geodesic equation is no longer or not valid. However, equations (38, 32) themselves have their meanings taste, and since they have an integral (38°21), they satisfy the condition F =0 is compatible. In order to justify the equation in this case, in section 36 We can start from the Hamilton equation discussed. According to (36,08), , where the Hamiltonian H=−ω0 is the ωO formal solution (by which equation %formula%) If we use a function that is and express the derivative of H by the derivative of G, we have equation (36 , 36) can be written in symmetric form.

Here, ap is considered to be the differential of the independent variable p. The first four equations (38, 4 0) was already mentioned in Section 36. If we write the right side clearly, we get . these The equation is easily understood to be equal to (38,32), since is obtained and therefore therefore (38,44) Inserting (38,43) into (38,41) we get or (38 As a result of the first set of equations in ,41), if we exclude ω from these equations, we have , we finally get . These equations are identical to equations (38, 22), which An equation for a geodesic curve of the form (38, 32) was derived from the equation. (38,41 ) to (38, 47) is the passage from Hamilton's equation to Lagranche's equation. It's a normal progression.

Thus, the zero length geodesic curve is similarly determined by equation (38, 22) However, we have proven that the condition F=O is added.

Since F is a constant, we have to preserve its properties for the entire length of the geodesic curve. It should be noted: it always describes the motion of a point with a velocity less than the speed of light, or Or maybe it's a zero line, or finally it's spatial somewhere. Something can happen.

It can be thought of as the Hamilton-Jacobi equation for the function ω.

The Hamilton-Jacobi equation for the general case can also be easily obtained. can. Regarding finitude, we consider the case of a point moving at a speed lower than the speed of light. .

Select time t=xO as a parameter and dot the derivative with respect to it. In other words, the format %formula%) Problem *(*lags with opposite signs for equal conventions in mechanics It is convenient to introduce the Langian. After all, the sign of energy is Hamiltonian It is opposite to the sign of Ann. ) can be written. general The moments expressed are , and the Hamiltonian is by the usual rules It can be seen that the velocity X is expressed in moment pl by (38, 49).

Ldt=ds (38, 52) S is the length of the arc, and when observed, the four quantities ptpO can be written uniformly as Can be done. identity is the relational expression %formula%) This leads to the three Ayu from the four equations (38, 49) and (38, 51). , intersection 2 and intersection , can be seen as a result of eliminating.

The Hamiltonian H=-pO is obtained by solving (38,55) for pO. It will be done. The Hamilton-Jacobi equation can be expressed in spatial coordinates and time as By the usual convention of representing pl, p2, p3 and H as partial derivatives of S with respect to can get: These equations are also It can be written as Thus, the Hamilton-Jacobi form of the geodesic equation is It is. Hamilton-Jaco containing three arbitrary constants c1, c2 and c3 bi equation %formula%) If the complete integral of is known and we do not count the additional constant CO, then S with respect to the constant derivative of is also constant, as proven in mechanics. They are determined from the conditions of the problem. be done.

Comparison of (38, 58) and (38, 37) is done by zeroing the right side of (38, 58). Obtained by substitution. About spatial geodesics, Hamilton-Jacobi The right side of the equation is a negative constant that can be set equal to -1.

Appendix V (The following text is based on Theory of Space, Time and Gravity. e, Time and Gravitation), Chapter IV, pp, 178-1 82, and is incorporated herein by reference.

are shown, and deletions are indicated in [brackets] and by an initial letter. ) [,] Observations on the principle of relativity and the coexistence of equations [beginning of this book (section 6) In Appendix I, the formulation of the principle of relativity is obtained. , which together with the assumption that the speed of light has limiting properties forms the basis of the theory of relativity. can be done. Next, we say that the physics principle of relativity and the equations are co-operative. We will study the problem of connection with the requirements in more detail. We address this question in the introduction. I have already touched on the subject. ] Firstly, we have not yet made this concept more precise, and we are not familiar with the principle of relativity in general. An attempt is made to give a formalization of communism. In its most general form, the principle of relativity It shows precisely the equivalence of systems (or reference frames), and these coordinate systems are belongs to the class and has the form % expression %) This form is simply x'=f(x) (49 lines, o2 ) can be shown accurately.

In addition to the set of permissible transformations, the class of coordinate systems is characterized by certain auxiliary conditions. must be determined. Thus, for example, when considering the Lorentz transformation, this The linear transformation should not connect arbitrary coordinates of the two inertial reference frames, but It is obvious to connect only Galilean coordinates. any other (non-Galilei) It makes no sense to consider linear transformations between coordinates. Because Galilei's relativity This is because the principle theory is not valid for such artificial linear transformations. on the other hand If we introduce other variables instead of Galilean coordinates, the Lorentz transformation is can be clearly expressed in terms of these variables, but the expression for the transformation has a more complex form. will have.

Next, we will explain what is meant in the formulation of the principle of relativity by the equivalence of reference frames. Show more accurately. The two reference frames (X) and (X') are They are said to be physically equivalent if they proceed in the same way. as this means , a possible process is expressed as a function at the coordinate (X) ψ1 (x), ? 2(X le, , , ?n(X)　(49*,03) , the same function 9□(X'), (P□(,・),,, ,,,,? Other possible processes described by n(,・) (49*, 04) exists. Conversely, a process of the form (49,04) in the second system is corresponds to a possible process of the form (49*,03). [Corresponding like this The definition of the process is fully consistent with that given in Section 6. ]Thus relativity The principle concerns the existence of corresponding processes in the set of reference frames of a certain class. We accept systems of order 1 that are descriptions of this class as equivalent. From this definition it is clear As such, the principles of relativity itself and the equations of the two frames of reference are fundamental to physics. concepts, and the statement that one or both are valid is a finite physical assumption. It encompasses theories and is just not normal. Furthermore, the concept of the "principle of relativity" itself is only well defined when choosing a finite class of reference frames. become. In ordinary relativity, this class is that of inertial frames.

The function (49 lines, 03) or (49*, 04) that describes a physical process is It is called a field function or a state function. [As we have already shown in section 46 ] In the generally joint formulation of the equations that describe physical processes, the metric term The component g mu of the insor must be subsumed between the functions of the states. [we In this example] we obtain the following set of field functions: F,,(x),)v(X),g,,(x) (49 books, 05) Progressing relativity The requirements that go into the formulation of the equation apply equally to the metric tensor. Thus two things When comparing two corresponding phenomena in physically equivalent frames of reference, the old For the first phenomenon described in the coordinates, only the electromagnetic field and current density components , and the components of the metric tensor are the same in the new coordinates as with respect to the description. Must not have a single mathematical form.

We can further conclude that if we assume that the metric is fixed whether or not to consider phenomena that themselves affect the metric It will depend on. According to the [ordinary] sexual relativity theory [described in the previous section], , the metric is given once and for all and no physical process Assume that there is no process dependence.

The generally common formulation of the theory of relativity given by Risho [in this chapter] is based on this. It doesn't change anything. As long as this assumption remains valid, the properties of spacetime are and by introducing gμυ and achieving general communism for the first time, these The quantity depends only on the choice of coordinate system and not on the nature of the physical process discussed : They are functions of state only in a formal sense. On the other hand, in the theory of gravity , next chapter E] Appendices Vl and VII For the turned explanation, we make different assumptions about spacetime. Performed in terms of properties.

Here gμυ is a function of states other than formal states, but in fact: gμυ is This describes the physics fishing ground, that is, the gravitational field. However, the effect on heavy masses Also, when discussing small-scale processes that do not affect It can be assumed that Clear meaning in this environment In order to give to the principle of relativity, we need to understand not only the class of coordinate systems, but also the principle of It is essential to specify more precisely the nature of the physical process that has been formulated.

First, the assumption is that the metric remains fixed (“rigid”), or the metric is There are some things that can be thought of as fixed for a certain class of physical processes. We start from this assumption. on corresponding phenomena in two physically equivalent coordinate systems Returning to the definition and following it, all the following equations containing the components of the metric tensor are , for the first process described at the old coordinates, or for the first process described at the new coordinates. must have the same mathematical form for the second process. gμυ For these quantities, the first and There is no need to make a distinction between the second process and to consider the transformation of the coordinates. It is only necessary. However, the amount gμυ(X) and g’ u v (x’) (49 lines, 06) are tensor transformations. 1 and must have the same mathematical form. The requirement of the principle of relativity that there is no (for infinitesimal coordinate transformations) is the equation δgμυ= Transformed to 0. [Discussed in Sections 48 and 49. ] Most general classes of transformations that satisfy these equations have a parameter of 1o. and is possible only in a uniform space-time, where RIAV, Qβ=K ’ (gvagLLI3-ggagvl (49 tree, o7) [(see, equation 4 9.12)] is valid. When K is zero in these relations, space-time is Galilean, and the transformation in question is a Lorentzian transformation, although perhaps They exclude that they can be written in other (non-Galilean) coordinates.

Thus, using the rigidity assumption about the metric, the principle of relativity asserts the uniformity of space and time. If =O can be applied to the additional condition, then the Galilean metric at appropriate coordinates is obtained. Next, we transform the general form of the relativity principle into Galilean's relativity principle. Ru. For the condition =Q, it yields an additional homogeneity of spacetime: of the Galilean coordinates When changing the scale, the element spacing scale changes by the same proportion. []This property Subsequently, unlike the absolute scale that exists for speed according to the speed of light, there is a means that there is no absolute measure: there is no absolute measure of space and time. This leads to the equation =0.

Next, when discussing phenomena that themselves have an effect on metrics, the principle of relativity can be explained under certain conditions. We must evaluate the possibility that it is also valid in nonuniform spaces.

For this to be the case, the motion of the mass that produces the non-uniformity is relevant to the description of the phenomenon. It is necessary to do so.

In fact, [as shown at the end of this book] spacetime is uniform at infinity. Under the assumption (here it must be Galilean) that an inertial frame is similar to Show that we can choose a class of coordinate systems that are similar and are defined up to Lorentz transformations. can be done. For this class of coordinate systems, the principle of relativity states that the mass has a finite distance Despite the fact that spacetime is nonuniform in Applicable in format. However, the boundary that requires uniformity at infinity We must keep in mind the essential role that conditions play.

The maximum possible homogeneity is represented by the Lorentz transformation, so that There cannot be a more general principle of relativity than that discussed in .

General relativity as a physical principle that can be applied with respect to any frame of reference Even more so, it cannot exist.

To make this fact clear, the existence of corresponding phenomena in different reference frames A physical principle that assumes that the equations are It is essential to make a sharp distinction between the simple requirement of being communist. principle of relativity It is clear that the converse is not true, although it implies the coexistence of differential equations: Coexistence of equations is also possible when the principle of relativity is not satisfied. all Quite apart from the fact that laws are not transformed into differential equations, Even fields described by differential equations have their conditions as well as these equations. and also requires all kinds of initial boundary conditions and other conditions. these The condition is not communism. Therefore, the conservation of those physical constants is in their mathematical form changes in the formulas and, conversely, the preservation of their mathematical form requires changes in their physical constants. It means a change in number. However, the feasibility of a process with new physical constants is This is an independent problem that cannot be solved quantitatively. within the class of a given reference. If a physical process is possible, the principle of relativity can be applied. If the opposite is true , it is impossible. However, such a physical process The representation of models, and in particular the representation of such models in metrology, requires at most a narrow reference system. It is clear that the class is possible and certainly not limited.

As this reasoning once again shows (without relying on the concept of uniformity) A general theory of relativity that can be applied with respect to any reference frame is not possible.

[In the general case, the expression for ds2 is always a homogeneous function of the derivatives of the coordinates, but Also, the coordinates can depend on themselves in a non-homogeneous manner. Ko But general relativity is also unnecessary as a motivating factor for the co-representation requirement of the equations. Ru. The requirement of communality can be justified independently. The coordinate system is not fixed on the front side. In all cases, equations written in different coordinate systems are mathematically equivalent. , etc. is a trivial and purely logical requirement. The equation is communism In the case of essential transformations, this corresponds to the class of coordinate system considered. In this way, Laure When dealing with inertial frames that are related by 3D transformations and using Galilean coordinates In that case, it is sufficient to require coexistence with respect to the Lorentz transformation. [(child (as carried out in Children's Books I and II). ]However, any coordinate When using , it is necessary to require general covariance [(Chapter ■V)].

If and when the principle of relativity exists for the class of reference frames used It should be noted that the covariance of coordinate systems acquires a clear physical meaning only . Such things are commensurate with respect to Lorentz transformations. This concept is a physical law It is very effective in formulating rules. Because it is a concrete time-geometrical contains elements (linearity and uniformity of motion) and dynamical elements (mechanical and This is from the concept of inertia in the electromagnetic sense [: Section 5]). For this reason, it is Concerning the principle of physical relativity and becoming concrete and physical in itself. However However, when discussing any transformation instead of the Lorentz transformation, the relative Stop selecting the class of coordinate systems in which the gender principle exists, and implement this thereby destroying the connection between physics and the concept of communion.

Purity to the concept of coexistence as a constant requirement to equations written in different coordinate systems A logical perspective remains. Naturally, this requirement is necessary, but it must always be satisfied. It is a kimono.

Deals with a class of reference frames that are more general than relativity to which the principle of relativity applies Sometimes the need arises to replace a clear formulation of a principle with some other description. Akira A clear formulation consists in showing a class of physically equivalent reference frames: the new The formulation must express the properties of space and time for which the principle of relativity is possible. . Using the stiffness metric assumption, this introduces an additional equation (49*,07) This is achieved by

As we have shown [in this chapter], the additional assumption of the absence of a measure of the universe Using , these equations are part of the theory of relativity without changing their physical content. generally leading to the formalization of communism. Next, Galileo Lorentz's principle of relativity is completed. Maintained in full range.

In general, the very possibility of formulating ordinary relativity in the form of a common theory is It shows particularly clearly the difference between the principle of relativity and logical requirements. moreover , such a formulation opens the way to a generalization based on relaxing the rigid metric assumption. The relaxation of makes the auxiliary condition (49*, 07) better reflect the nature of space and time. offers the possibility of replacing it with something else. This tells us that Einstein leading to the theory of gravity, which is presented in Appendices Vl and VII [which will be discussed in the next chapter]. and discussed on all space-time scales.

Appendix Vl (The following text is based on Theory of Space, Time and Gravity. eSTime and Gravitation), Chapter V, pp, 183-20 9 (1965), and is incorporated herein by citation, and deleted. They are shown in [brackets] and initials. ) principles of gravity theory 50. Generalization of Galileo's law What is the most essential property of the gravitational field that makes it different from all other fields known to physics? The nature is revealed in the effect of the field on the motion of a freely moving object. gravity field , all otherwise free objects have initial conditions of their motion, i.e. If their initial positions and velocities are the same, then they move in the same way (.Basically Galileo's law states that in the absence of resistance, all objects fall equally fast. It can be thought of as a generalization of the law.

At this point, it is appropriate to recall the definitions of inertial mass and gravitational mass. Ru. Inertial mass is a measure of an object's ability to resist acceleration: for a given force, Speed is inversely proportional to inertial mass. A gravitational mass generates a gravitational field and the production of such a field is a measure of the ability of an object to undergo a force: in a given field, the force experienced by an object is Force is proportional to mass.

Using these definitions, the above generalization of Galileo's law It can be formulated as a statement that gravitational mass and gravitational mass are equal.

The atomic basis for Galileo's law is that all matter is confined to three-dimensional space-time. This means that it is composed of elementary particles of matter with a mass of . normal The two-dimensional spatial manifold of matter is the shell of a sphere. equal this curvature discontinuity It is solved by this. The resulting gravitational equation is also It will be demonstrated that it contains body mass.・Einstein's field equation Equations are derived based on this isodominance, and they represent the boundary line in the Mills orbit. According to Newton, the gravitational field is the gravitational potential U (x, yS z). isolated at a point external to itself The gravitational potential generated by a spherically symmetric mass M is, where r is the mass It is the distance from the center. The quantity γ is Newton's gravitational constant (in cgs), which This has a value. Thus U has the dimension of the square of the velocity. we immediately acknowledge In all cases encountered in nature, the Sun or a superdense star Even on the surface, the quantity U is very small compared to the square of the speed of light U<<c2 (50, 03) In the general case of any mass distribution, Newton's potential U is Poisson equation %formula%) , where ρ is the mass density. Newton's potential is as follows is completely determined by the Poisson equation with continuity and boundary conditions: The function U and its first derivative are not spatial over a finite, single value series. must be, and there must be no tendency to zero at infinity.

Assume that Newton's potential U is given. Gravitational field of potential U The force experienced by an object (mass point) with gravitational mass (m), is F=(m),,g radU (50,05). On the other hand, due to Newton's laws of motion, , (m),,w=F (50,06) is obtained. therefore, (m) +++W= (m) g, g r a dU (50,07) Galilei By the generalization of the law, the motion of an object in a given gravitational field depends on its mass . Therefore, the ratio of inertial mass f: m) tx/force weight mass) 8 is must be the same for the field; thus it is a universal constant and its value is It can depend on the choice of units for the two masses. very generally accepted In the units shown, we simply obtain (m), , = (m), , (50,08). Therefore, the inertial mass and gravitational mass are equal.

The equivalence of inertial mass and gravitational mass is a well-known fact and is usually and accept it. However, matter is not so simple: their identical Equality is another very important law of nature and is closely related to the generalization of Galileo's law. related.

As a result of the equivalence of inertial and gravitational masses, the equation of motion (% formula %) has universal properties and thus formally represents a generalization of Galileo's law.

The equation of motion (50,09) can be obtained from the principle of variation. This fact is gravity This is a guideline when constructing a theory.

51. The phenomenon of gravity in an interval square universe in Newton's approximation is The Theory of Intervals [This was the subject of the previous chapter. ] Expand the composition. This extension requires The importance of this will become clear from the following considerations.

format From the wavefront propagation equation, which can be precisely shown as Recognize. But light has energy, and the law of proportion of mass and energy By law, all energy is inseparably related to mass. death Therefore, light must have mass.

On the other hand, according to the law of gravity in the universe, a mass located in a gravitational field will must experience an action, so generally the motion is not in a straight line. There will be (heavy hair edict and decaying pouch 31 frog history 2 哩蒋μ±坤if 杆龆MEYU.Wavelength is a two-dimensional manifold, a Mills orbit (see Appendix IV). that Therefore, in a gravitational field, the law of wavefront propagation has a somewhat different form than the one described above. I know it's indispensable. However, the equation of wavefront propagation is is a fundamental characteristic of the nature of

Therefore, the existence of a gravitational field must affect the properties of space and time; It turns out that their metrics are rigid. This is true that we are here This is the conclusion reached in the theory of gravity that begins to be constructed.

[As shown in Chapter I, the equation for the propagation of this two-p wavefront can be modified with some additional assumptions. Using the equation, we derive the following equation for the square of the interval: ds2− c”dt2− (dx2 + c1y2 + dz2) (51,02) Properties of space and time The influence of the gravitational field on the wavefront propagation equation and the coefficient in the spacing squared equation are (5 1,01) and (51,02).

Second, Newton's theory relies on a generalization of Galileo's law to guide us. We have to find an approximate form for the square of the interval in the gravitational field of the tensile U. No. The laws of motion for objects moving freely in a gravitational field do not depend on the properties of the objects. The basic fact is that the relationship between the laws of motion and the metrics of space and time is Enabling discovery.

The equation of a geodesic curve in the air with a given metric [is studied in Section 38 and is in the Appendix] IV. These equations then apply to a free body in a given gravitational field. We will try to discover a metric that roughly matches Newton's equation of motion. this If the attempt is successful, the free object (mass point) is geodesic in spacetime with a given metric. One could introduce the hypothesis that it moves along a line: in this way, A connection between the laws of motion and metrology will be established.

As we know, the equation of a geodesic curve derives from the principle of variation, If the interval has the form (50,02), then ds==/(c2-v2)dt (51, 04) Or for small speeds, is obtained.

When (51,04) or (51,05) is inserted into (51,03), constant speed Given the equation describing , which is in fact free motion in the absence of a gravitational field . Then, for small velocities and weak gravitational fields (U<<c2), the interval expression is of the form ds=! (c2-2U-v2) dt (5i, 06) or I take the. The theory of deflection of light in a ten-gravity field [is described in section 59 below. It is described in the record. Also, the explanation that the wavefront is a two-dimensional manifold, a Mills orbit, See Appendix IV. instead of (51,04) or (51°05). Additional Since neither constants nor constant multiples are important in the Lagrangian, we The principle of fluctuation (51,03) uses ds taken from (51°07) to calculate the fluctuation. This gives the same result as the principle, which was formulated at the end of section 50, but this In fact, we have described the free motion of an object in a gravitational field.

Since the additional constants and multiple factors in the Lagrangian are not important, Equation (51,08) can be expressed as (51,03) and (51,06) by any constant C. It is true that we can obtain using a sufficiently large value of: In the absence of , when U=O, the equation (51,06) for the interval becomes ■ This means that whatever the value of 2, it will be in Galilean form (51,04). It must require . This requirement makes the constant C in (51,06) equal to the speed of light. Fix it tightly.

These inferences are based on the condition Later, the square of the interval is of the form %formula%) gives excellent reasons for the assumption that there is little difference between Here, the coefficient of dt2 The relative error in will certainly be of higher order than the term involved 2U/C''. Regarding the coefficient of the purely spatial part, it differs from 1 by an amount of 2U/C2. There is. In fact, the theory of gravity to be developed in the next section is based on a more precise formula Under the condition (51,09), the difference between (51,10) and (51,11) is As expected, it is negligible.

In principle, the value found for the coefficient of dt'' allows an experimental evaluation.

At a certain point (X+, yl, 2+) where the gravitational potential is U, an appropriate period Assume that there is a radiator with radiation between T0. The waves radiated by it are of the form , where T is T. is not equal to dt, but it means that dt is dτ, It relates to it in the same way as it relates to the proper time derivative of the projectile. for brevity Assuming that the radiator is in the selected reference frame, approximately is obtained and therefore is obtained.

In this problem, the dependence of the gravitational potential on time can be ignored Therefore, the gravitational field can be treated as stationary. Then propagates from the radiator The wave will retain its time dependence (51, 12) throughout space.

Furthermore, the dependence of the gravitational potential on time is ignored, so the gravitational field is different At some other point (X2, y2, Z2) with the value U2, a second radiator, e.g. For example, assume that other atoms of the same element exist. The waves radiated by it are space through format has a time dependence of , where Thus, they are emitted by the same source but originate from locations with different gravitational potentials. The two waves are only have different durations. U2 is the potential on the sun, and Ul is the potential on the earth. If U2>U, and the factor T0 in (51,17) The value of is approximately be equivalent to. Thus the wavelengths of spectral lines originating from the Sun are generated on Earth. Displaced by 2 parts (1 million) towards the red end of the spectrum with respect to the corresponding line I have to.

However, the radiation of spectral lines on the Sun is subject to different physical conditions than on Earth. The change in period due to the difference in gravitational potential is to a large extent It must be noted that this may be masked by other corrections.

However, there are some super dense stars, such as the Sirius binary, which are It has a density 10,000 times that of water. On those surfaces, the value of the gravitational potential is 20 times larger in the case of a Sirian binary star significantly larger than on the surface of the Sun - For such stars the correction for the difference in 9 gravitational potential is very obvious. Chassis effects that can be determined and experimentally detected are determined by the mechanisms described in Appendix I. successfully revealed in the boundary conditions by using the Sbauer phenomenon Ta. []52, Einstein's equation of gravity Einstein's theory of gravity, in its restricted, non-cosmological form, is based on Have fundamental ideas.

The geometric properties of real physical space and time correspond to Euclidean geometry. Although it is not, it corresponds to Riemannian geometry. discussed the basic assumptions]. The derivation of their geometrical properties from Euclidean is Or, more precisely, the pseudo-Euclidean form appears in nature as a gravitational field. Geometry The physical properties are inseparably connected to the distribution and motion of matter that is worth considering. this relationship is reciprocal. On the one hand, the derivation of geometric properties from Euclidean is On the other hand, the motion of the mass is determined by these inductions. determined. Simply put, mass determines the geometry of space and time, and These properties determine the movement of the mass.

Next, we will try to formulate these ideas mathematically.

In the previous section, we showed that for practical purposes there is a certain In the species coordinate system, the Newtonian gravitational potential U is The coefficient of dt2 in the formula, that is, the coefficient g of the general formula. Enters 0. On the other hand, In Newton's approximation, the gravitational potential U is Poisson's equation %formula%) Satisfy. The required generalization of Newton's theory of gravity concerns the transformation of arbitrary coordinates. It must be communist. Therefore, the coefficient g. . or terms in this entire coefficient cannot be regarded as a generalization of Newton's potential. That cost Therefore, the entire set of coefficients g1o must be considered, and Newton's appears as a generalization of the The fundamental metric tensor is generally a co-equal equation , and in Newton's approximation, one of them is the potential We have to proceed to Poisson's equation for U. The total number of equations is one In general, the number of unknown functions is equal to the number of components of the tensor gμυ, which is 10. Yes.

On the left side of Poisson's equation, there is a second-order differential operator and Laplace operator that act on U. Exists. Therefore, the simplest generally communal generalization on this left side is linearly measurable. It will be a tensor encompassing the second derivative of the quantity tensor gμυ.

Such tensors are curvature tensors (of second or fourth rank). Fourth The rank curvature tensor R1u+11a is inappropriate. Because its ingredients This is because it does not contain an expression that can be a generalization of the Laplace operator acting on U. be. Also, since it has too many components, the number is 20, there are twice as many unknown functions. Even the excessive number of equations for the 2°[2 tensor Rmn, ab Comparability can be proved as in the case for spaces of constant curvature ( (49, 12)), but in that case the equation makes the metric purely locally , that is, it can be determined without using boundary conditions. Therefore, they are It has different characteristics from the Poisson equation, which requires boundary conditions. ] Therefore, A second rank curvature tensor remains with exactly the correct number of components.

On the right side of Poisson's equation, the mass density ρ appears. Required tensor properties The generalization of mass with has a mass tensor T' whose invariant is equal to the invariant mass density °.

Thus, the required generalization of Poisson's equation is the second rank curvature tensor R~ This leads to the conclusion that there must be a relationship between

[In the previous chapter, we saw that] in the absence of a gravitational field, the tensile The divergence of Sol T#'' must be zero VvTllV=O(5 2,03) [Keeping this equation for the general case and solving problems related to it (the energy of the gravitational field) ``V=''-1fV up to Chapter VII. R(52,04) is known as Einstein's tensor or conserved tensor, and its data It has the remarkable property that the divergence is exactly zero.

vVGl, Lvso 0 (52,05) Here, assuming that X is constant, the equation for the mass tensor (52,03 ) is the result of (52,06).

As we know, the metric tensor g□ itself also has (52°05) Mitas: Therefore, the left storage tensor of (52,06) has the form 1 gmn, Here, λ is constant, without ignoring (52,03), add the tensor of You will be able to do so.

[As we saw in Section 31] there is a purely local condition on the mass tensor ( Conditions that depend on the field components or other functions of the state, but do not depend on the coordinates; field equations whose divergence is zero), it means that the two constants Determined only apart from the numbers. More precisely, the tensor T” satisfies the conditions stated , these are also expressed by Tl1V=aTI−LV+pgμV (52,07) It will be filled. Here the constant a depends on the choice of energy units, and the constant β is also infinite depends on the conditions. At replenishment of local conditions, the field becomes zero, infinitely , the mass tensor is also zero, β is zero, and the mass tensor is uniquely determined.

The replacement of the tensor T″′ by its own linear function (52,07) is the gravity due to This corresponds to the substitution of equation (52,06). The constant λ is called the cosmological constant. these As is clear from the observation that the problem of the value of λ uniquely determines the tensor T#” acquires a clear meaning after the conditions for it are formulated. Such conditions are inevitable will have non-local properties; therefore, they will have non-local properties as a whole; It can only be formulated by starting from clear assumptions about .

At the beginning of this section, we note that Einstein in its restricted (non-cosmological) formulation The basic assumptions of Stein's theory of gravity were precisely demonstrated. According to these, the infinity of space The concept of retains its meaning, and spacetime at infinity is quasi-Euclidean (Galilean). It is. Deviations from Euclidean properties occur at finite distances from large objects. observed only. However, in this case, the mass tensor is When Euclid is assumed, continue to follow the requirements stated for that case. Can be done. At infinity, where the field becomes zero, the mass tensor must also become zero. required. Then, it has a uniquely defined meaning for calculating the value of the cosmological constant. do. According to our basic assumption, the absence of a gravitational field means that spacetime from Euclidean implies the absence of bias in the geometry of, and therefore also the curvature tensor R'' and This means that the invariant R becomes zero. On the other hand, how does the tensor T#” If this is also zero, there is no gravitational field. Therefore, the equation T0=0 and R”=O must indeed be comparable, and this means that the equation for T1 is the term This is possible only if it does not contain λg0 (ie, when λ=0).

Thus, the assumptions we made at the beginning of this section are given and the mass tensor Given our definition of , the Poisson equation for the potential A suitable generalization is just equation (52,06).

Regarding equations (52,08), if the problem is stated cosmologically, they can be used should be used, and in this case the concept of spatial infinity is inapplicable, and The tensor T#'' contains an unknown constant β even after the units are fixed. The value of the so-called cosmological constant λ must be chosen according to the value of the constant; Obviously related to β. For a given normalization of To, the choice of a certain value is special. , which must be introduced explicitly; this also applies to the value λ=0. It is true.

be. This model integrates gravity on atomic and cosmological scales. In Appendix VII , the Mills orbit integrates atomic theory and general relativity at the atomic level, Completely integrated gravitational non-cosmological cases and equations for all scales of spacetime Returning to equation (52,05), the conditions stated (Poisson equation, general coexistence, guv The linearity of the second derivative of, vanishing identically to the left of (52°05) and the quality (corresponding to the Euclidean property of the absence of quantities), these equations are uniquely It is.

Equations (52,06) are called Einstein's gravity equations; they are Demonstrate the basics of force theory. We will discuss them in the next section.

53. Properties of Einstein's equations. Propagation speed of gravity.

Einstein's equations of gravity with derivatives of the linear equations with respect to their properties We begin our discussion of. From a physical point of view, the characteristic equation is the wavefront of a gravitational wave. represents the propagation law for .

Multiplying (53,01) by guv and summing gives us the invariant of the curvature tensor and Relational expression that combines the invariants of the mass tensor (% expression %) is obtained. Using this relational expression, the gravity equation can be expressed as ”) can be written as (53,03).The contrainflectional curvature tensor R# is expressed as It can be created here, αβ is created. To raise the subscript from β Therefore, the amount obtained is n-', (1 = gCL9gβ'5nl, (53, 05) Therefore, the last term in (53,04) does not contain a second derivative, but p is a homogeneous quadratic form in tβ and therefore also the first derivative of the metric tensor be.

The second derivative appears in the first term and also in n, but the dependence of the latter is on the quantity derivative rv-fl’r”αβ(53-06) exists only through [this was introduced in section 41]. we do the following Recall] The function slatted d'Alemberthan is of the form % expression % can be written as, here and also ra　,,-口xCL(53,10) r　Mu is the symmetric contravariant derivative of the vector from F′: or in detail is obtained by rules that are formally identical to the rules forming . Of course, r′ is rl is not a tensor because it is not a vector. This situation is Einstein's It proves very useful when simplifying equations.

Einstein's equations are generally commensurate, so any four functions Enables transformation of containing coordinates. The equations are solved in any system with coordinates. Assume that Next, take the four solutions of equation entry ψ=0 as independent variables. You can proceed to other coordinates by

These solutions satisfy the inequalities that guv must obey according to Appendix IV. They can also be selected in two ways, subject to certain additional conditions. I can do it. However, each of the coordinates XO%X1, X2 and X3 is equation entry X, =0 When satisfying ``'=O(53,13)'' in that system, and therefore also r””=O(53,14) is obtained. We call such a coordinate system harmonic. At this moment, We are interested in the problem of uniqueness of the sum coordinate system or additional conditions that can guarantee uniqueness. It has no taste. Here, equation (53, 13) is compatible with Einstein's equation. compatibility, and they impose no essential restrictions on the latter solution, and the allowed coordinate systems It is important to note that this serves to narrow the class of

Under the condition (53, 13), the formula for R'' can be simplified to become. Therefore, the quadratic The derivative is created on the left by a single quantity gμυ with the same index as a certain R”. In the d'Alembert operator used, it only appears in conjunction.

The form of the characteristic equation for a given system of equations is the derivative of the order that occurs at the highest It depends only on the term containing . For the systems (53,01) and (53,13) , these terms are just the terms that include Dharamberthan.

Therefore, the system of gravity equations has the same properties as d'Alembert's equations. Deaf, mouth W=O(53,16) ω0αx1. x2Jx3) = constant (5'3.18) is the wavefront equation, i.e. , no discontinuities in the gravitational field exist on it (the surface equation is .

The equations (53, 17) for the propagation of gravitational wavefronts are based on the complete theory of space and time. (Identical to the corresponding equation for the wavefront of a light wave in empty space. When we derive the law from Maxwell's equations in Section 3, we assumed that spacetime is Euclidean.

However, following the observations at the end of Appendix E, the Maxwellian equation described in Section 46 Starting from the generally commensurate form of the equation, one can obtain the same result without using this assumption. can be done. ] Simply put, we can say that gravity propagates at the speed of light. .

In Einstein's theory of gravity, the basic meaning of propagating at the speed of light is It is a fact that The assumed form of the gravitational equation is consistent with the general assumptions of the theory of relativity. Then, following this assumption, there is a limiting speed for all types of action propagation, i.e. There exists the speed of light in free space. The existence of a finite propagation velocity for gravity is , a characteristic of Newton's theory of gravity that admits instantaneous action at a certain distance. Eliminate contradictions. , ,54, Comparison with the problem statement in Newton's theory. border boundary condition.

In Newton's theory of gravity, the gravitational potential is expressed by equation 6%) , and tends to zero at infinity, so here M is a problem. The sum of the masses of all objects in the system is and is equal to . gravity equation Einstein's theory based on Newton's theory is, in the first approximation, must give the same result. Newton's theory covers all space. The total mass given by the integral (54,03) remains finite. It can be applied to mass distributions such as .

This condition is especially true for mass distributions with an insular character. It will be done. We use this term to mean that all the masses of the system we study are located within a finite volume. concentrated and this volume is much larger than all the other masses that form part of the system. Used to describe the case where the two are separated by a large distance. If these other masses are ten Their influence on a system of given masses can be ignored when they are far apart in minutes. This given system can then be treated as isolated.

When formulating Einstein's theory, the assumption is that the mass distribution is isolated Start in the same way. This assumption creates a clear limiting condition at infinity. ton's theory, and thus treat mathematical problems as determined. can be done.

Theoretically, other assumptions are also permissible. For example, the mass fraction that is uniform throughout the mean space Cloth can be assumed. Such a perspective is very large and nebula in comparison It is suitable for studying distances where even the distance between them is considered to be very small. Such a big Very little is known about the mass distribution over long distances; Therefore theories dealing with them are necessarily less reliable and smaller scales. There is less possibility of experimental evaluation than theory of degree celestial phenomena.

[The bulk of this book will be devoted to the case of isolated distributions of masses.

The assumption of a uniform distribution is only considered in sections 94 and 95, where we introduce this assumption. gives Friedman-Rabaczewski's theory of space. In this way, next, that spacetime is primarily Euclidean, or rather pseudo-Euclidean, and The deviation of the geometry of spacetime from Euclidean geometry is a result of the presence of a gravitational field. Assume that In the absence of a gravitational field, the geometry must be Euclidean. No. For an isolated distribution of masses, the gravitational field tends to zero at infinity. , and therefore, at points far away from the mass, the spacetime is Euclidean become a target. However, when the geometry is Euclidean, the Galilean coordinate x , y, z and t, by which the square of the interval is of the form % expression %) has. As experiments have shown, the geometry of spacetime is now significantly different from Euclidean geometry. Since the square of the interval is biased from (54,05) in the entire space, the bias is It can be expected that there are variables for which the In the following, this We give a more precise definition of these pseudo-Galilean coordinates.

Newton's theory is exactly in Galilean coordinates, i.e. in the inertial reference frame. It is recognized that formalization is the simplest. After all, Einstein's theory is its -filmization, and compare it using coordinates with similar properties as much as possible. .

Newton's theory is non-relative, and progresses from relativity to non-relativity When doing so, we specifically choose the speed of light as the large parameter.

Therefore, we no longer introduce quantities into the parameters: (35,03 ) instead of then Write x, = t; x, = x; x2 = y; x3 = z (54, 06). Thus now After that, the variable xO simply means the time t and not ct as before.

Then, the equation (54,05) for the square of the interval is ds2=c2dxo”−(dx l"+dx2"+dx3") (54,07). This is must be valid over long distances, where the geometry is Euclidean It is.

general formula , g- is a metric tensor that at infinity must have the following limit: The corresponding limiting value of the contravariant component of Le will be. These are then the metric tensors should be considered as a boundary condition for

However, the number of boundary conditions mentioned so far is insufficient: large distances from the mass Some additions that characterize the asymptotic behavior of the difference g, ,−(g,,)− boundary conditions must be added.

As we showed in the previous section, at least ``If °=0, then a Instein's equation is a type of wave equation. Because their main This is because the term is a d'Alembert operator. Outside the mass distribution, the tensor T″′ becomes zero, and the equation has the form Ra″=O(54,11) , where r''=0 and the tensor R'' has the form. to a large distance , the difference g''-(g') and their first and second derivatives are 1/ r tends to be zero, where r　w+　V(xt2+　x22+　x 32). (This assumption will be justified in the following.) So, at large distances, the second term of (54, 12) is uniform in the −th derivative. It is a highly quadratic form and tends to be zero as 1/r2. d'Alemberthan For terms containing, the coefficients therein are replaced by their limit values to the same approximation be able to. After these simplifications, is obtained. As can be shown [a more complete study of the asymptotic behavior of g′° Section 87. As it shows] the asymptotic form of g## is (54, 1 Although it is affected by the order 1/r2 term omitted from 3), quantitatively the difference g "-(g') The behavior is the same as that of the function W that satisfies the wave equation, where △ is It is a normal Euclidean Laplace operator.

We are interested in the solution of the wave equation (54, 14), which is infinitely It corresponds to a departing wave that disappears. They have an asymptotic form, where n minutes is a unit vector with , and f is an arbitrary function. all its independent variables The function f and its derivative with respect to (argument) are assumed to be finite. Germany The standing variable n gives the dependence of f on the direction along which a point retreats infinitely far. .

Other possible solutions of the wave equation must not be discarded for physical reasons.

Because in our description of this problem, we consider this system to be isolated. This is because that. This means that the wave does not collide with it from the outside and all Waves have as their source the objects of the system, and in isolated types of systems they all Since all objects are concentrated in a finite region, all waves originate from this region. and therefore has an asymptotic form (54, 15) at large distances from this region. Ru.

The condition that the solution to the wave equation has the form shown at large distances is the differential form It can be stated in the formula This condition holds that all to=t+r/C at r-4oo and any fixed interval. cannot be applied for all values. It is called the outward radiation condition. be able to. It is the function W and its first derivative with respect to X% Y% Z and t We say that the function is constrained everywhere and vanishes at infinity as 1/r. It guarantees the uniqueness of the solution as far as it is related to the requirements.

As we note, the above considerations do not apply to Einstein's equations. Instead, we will strictly discuss the ordinary wave equation (54, 14). Therefore, the difference asymptotic form of %formula%) is, in fact, somewhat different from (54, 15). However, the differential form (54, 17 The slightly modified condition for outward radiation written in ) is for (54°18) It would be effective.

To summarize, in our description of this problem, the metric tensor is It does not satisfy the conditions of being Euclidean and also the conditions of outward radiation. I can say that it is not.

55. Solution of Einstein's equation of gravity in first approximation and determination of constants In order to compare Einstein's and Newton's theories of gravity, first We must determine the constant X that enters Instein's gravitational equation. this constant The value of can be obtained by approximately solving Einstein's equations. It can be found by comparing.

For the mass tensor on the right side of (55,01), the Euclidean metric elastic body The case is clearly given.

In the equation, Xo simply means time and does not mean ct: therefore , the concept TO'' is now equal to 02T00, and To' in equation (32, 34) is now equal to cT'° and TIk is unchanged.

Thus X. = t, equations (32, 34) become (27ik = pvi Vk-pik becomes. In this approximation, the term corresponding to density and current energy −Umov Do not ignore the pseudoscalars and vectors of (and simply c"T00=p: C"T"=ρVl (55,03) is valid. To the same precision, we set the invariant of the mass tensor to the value T20 (55,04) can be replaced with By equations (55,03) and (55,04), format %formula%) Calculate the approximate value of the tensor component on the right side of Einstein's equation written as This format was described in (53,03).

Using Galilei's value of g##, was gotten.

On the other hand, according to (54,13) and using harmonic coordinates, approximately, is obtained, where Δ is the Euclidean Labrasian operator. Interested in quasi-stationary solutions For taste, terms containing second-order time derivatives can be discarded. (55 ,07) and (!5.5.06) into (55,Q5), we get Δgik Tomi-Xρδ swimming is obtained. Next, mention the formula for squared intervals in Newton's approximation do. According to (51,10), go. = c”-2U (5!5.09) is obtained. , where U is Newton's potential. In this approximation, the metric tensor Replace the remaining components of Le with Galilean values. The relational expression and the product g,,g'' are 1 and the ratio Using the fact that it is very small compared to can be taken. However, Newton's potential is equation 6%) Show off. By comparing this with the first equation in (55,08) , Einstein's gravitational constant X is related to Newton's constant g When related by , it can be seen that the two match. newton's potential Le U is a solution of (55, 13) that satisfies appropriate boundary conditions at infinity. The solution is the body It is well known that Newton's potential can be written in integral integral form. In parallel, we introduce a function, which is ΔUi, 4. ．． )vi, (55, 18) and also looked at the conditions at infinity vinegar. Like the corresponding electromagnetic quantities, these functions represent the gravitational vector potential can be called. Next, the solution to (55,08) is It can be written as (55,15) was used to exclude X.

As we note, U is the square dimension of velocity; It is a cubic dimension. When estimating the order of the magnitude of a quantity, let U have order q2. can be taken to have order q3, where q is much smaller than the speed of light. This is the speed at which

Next, from the contravariant component of the metric tensor, we can obtain its co-component, its determinant, etc. It's purely algebraic.

To simplify the algebraic operations, we introduce a system of quantities, where 1, k=1.2.3 It is.

It is easy to evaluate. The set of quantities ash can be interpreted as a metric in three-dimensional space. , but only its algebraic properties are important to us here. .

awDetaHL (55, 22) and therefore g=-agoo (55, 24) is obtained. Directly from the definition (55, 20) m=1 and also xl becomes. If g'° has the value (55, 19), and therefore becomes. Go. If we accept the form of gooaik, c28yo(55,29) is obtained with an error of higher order than U/c2. Ru.

Therefore, with the same relative error, giot(2glO(55,30) is obtained. Using these results, for the covariant components of the metric tensor is obtained. Knowing the approximate values of g61 and g61, we can then use (55, 11). The accuracy with which use is justified can be assessed. (52,26) is U, is degree q 3, the above equation differs from 1 by an amount of order q2/c2.

Therefore, (55, 11) is not only this, but also the next higher approximation Can be used in U/c2 or v2/c2. (52, 2 6) is lead to. Go here. is positive, and the double sum is a positive definite quadratic form. Therefore, always and very strictly, gOOgoO≦1(55,34) is obtained, but as we understand, the deviation from 1 on the left is extremely small. stomach.

Next, the values of the determinant g and the covariant component of the %JC-g)×metric tensor will be described.

gllV-J(-g), g#Lv(55,35) writing. Then, -g-c2+4U (55, 36) and therefore is obtained. Therefore, using (55, 35), we have the second order in the −th derivative: We must estimate the magnitude of the ignored term in a certain R''.

Use an approximate form of the just-induced metric tensor to calculate the low value You will be able to do it. However, we will not perform these calculations here [why If so, the quadratic term will be determined in detail in Chapter V■, and we will now use the following approximation. and solve the gravity equation]. Here we only need the order of magnitude of the quadratic term. . The terms in Rae and R'' are of order 6 and those in R1k are 1/c It is fourth order. In our approximation these terms have no effect.

Conditions that ensure coordinates are harmonic It remains to evaluate whether or not it satisfies the required approximation. First, applicable clarify what kind of precision is required to make it work. Formula for R″′ Do not omit the term in r in (53,04), but instead If we keep them to an accuracy comparable to (55,07), then the substitute for (55,07) Instead, you get Term of type Δg# in which the previously ignored term in 0 is retained G, is higher in smallness in C than hal/c', so that it is actually smaller compared to It is necessary that G' has an order higher than 1/C2. these The condition is actually satisfied. This is because G' is quartic in 1/c This is because it is directly clear from (55, 38). Regarding Go, 4 of them The next term is It is. These must not be zero, so the equation cannot be applied. Must-have. As is clear from the definition of U, U' (differential method with boundary conditions (by equation or by volume integration) This equation is a factual expression This relation expresses the conservation of mass in Newton's approximation.

Thus, the expression derived for the metric tensor is equivalent to the gravity equation for the first approximation: Not only that, but it also satisfies the ``condition of harmony.'' Furthermore, they are bounded at infinity. satisfies the boundary conditions. The formula for the square of the element spacing corresponding to the derived formula is +, 2 (Uldxl + U2dx2 + U3dx3) dt (55, 44 ) It is.

Usually, the term containing the product d□d has no significance. If you omit them, the expression %formula%) is obtained, which contains only Newton's potential. This formula is in clause 51, equation It has already been quoted without proof in equations (51, 11).

56, the gravitational equation metric tensor in the stationary case and in conformal space has its components but As in, the time coordinate X. = independent of t, and furthermore, goH-0( If i=1.2.3) (56,02), it is called stationary.

If selective masses exist, they are not in motion (which is not the case in physics) The problem of motion of a system of 4° [4 masses, which is clear from scientific considerations, is discussed in detail in Chapter Vl. Therefore, the stationary metric tensor is the same in the case of a single mass. It can happen once. Notwithstanding the limited applicability, in the case of stationary He has an interest in physics. Firstly, it is based on Newton's theory of gravity (which It also allows a deeper insight into the similarities with the theory of rest) and Also, in the stationary case, it is easy to discuss the problem of uniqueness of solutions. Ru. Also, it is possible to find exact solutions to Einstein's equations in this case. This is because it can be done.

Under conditions (56,01) and (56,'02), the time coordinates are uniquely determined. spatial coordinates allow a group of transformations between them. therefore, In this problem, we will use the apparatus of three-dimensional tensor analysis and It is natural to write the gravity equation accordingly.

Three-dimensional tensor analysis can be applied to the spatial part of ds or also to a certain factor. can be applied directly to this spatial portion multiplied by . Einstein's Recalling the form of the approximation of (55.29) obtained from Eq. By introducing a factor inversely proportional to the vector into the spatial part, shall be. Consider the quantity V2 to be a three-dimensional scalar and use the quadratic form % expression %) Let be the square of the length of an arc in some auxiliary space. Comparing 3D tensors As can be understood by Assume Clidian and relate the quantity v2 to the Newtonian potential U can do.

thus, goo　　C2v21goi　-0/and also is obtained. Here, the quantity h1. and h +i is hi3hkJ = hki(h ki=δkl)' (56,07), and this relational expression is Similar to (37, 18).

If we express the determinant of hlk by h, we can easily get is obtained. therefore, ? (−g)・gik=−cJh−hjk is obtained, and applied to the function ψ d'Alembert operator [(41, 11)] is where (△v) is the Laplace operator in the conformal space be: Therefore, the spatial coordinates that are harmonic in the four-dimensional sense are also the conformal spaces in the three-dimensional sense. It can be seen that there is harmony between them.

The four-dimensional Christophel symbol (formed from the metric tensor g2.) is defined as (Cp, , ), and the three-dimensional Christophel symbol (from the metric tensor h1. ) is expressed as (G, k). Similarly, the subscripts g and h are metric Attach to a quantity that is a tensor with respect to (g#,) and (h, k).

The 4-dimensional Christophel symbol and curvature tensor are expressed by the corresponding 3-dimensional quantities. can be expressed.

These equations include the derivative of the three-dimensional scalar ■, which is expressed as . attached to V' Subscripts are sometimes omitted for simplicity. A purely spatial indexed The strike sophel symbol would be. one or all indexes are zero If , the Christophel symbol is zero: (rooo)g=0. compensation k), -0 (rkto) g=o (56” 4) last If two of the indices are zero, then we get general formula %formula%) can be used to convert a 4-dimensional curvature tensor into a 3-dimensional tensor and a 3-dimensional scalar It can be expressed by the co-derivative of -■. element, but rather a lengthy plan. Without considering the calculation, for a component with four spatial indices, ("i, mk) g - (R11, mk) h + htrtl $Vlk) h-hi k’$V’m)h is obtained, where (V,,)ゎ is the value of V with reference to the metric (h). It is the second co-derivative: and (V’h)h are the same derivatives with raised subscripts: (V’k)h − h ■ (vik) h (56, 19) If only one index is zero, then ( ROi, mk) g - 0 ("o, mk) g - 0 (R'l, .k) g - 0 (56 , 20) is obtained. If the two indices are zero, then (Roo, mk) , -0(56,21) and also (56.22) and finally is obtained.

Next, we can form a reduced curvature tensor that enters Einstein's gravitational equation. It's easy.

formula %formula%) Using (56,17) and (56,22) to (56,25) is obtained, where (AV)h is the Laplace operator applied to V: The expression %formula%) From, using (56, 23), (RoO) g-c2V2 (V・(aV)h −(V, Vt)h) (56・ 28) is obtained.

For the mixed components of R, , they are zero-four-dimensional as a result of (56,02) According to the relational expression (56°06) between the metric tensor and the three-dimensional metric tensor, Equations (56,25), (56,28) and (56°29) are invariants (R), leads to the following formula: (R)g = Vl(R)h+2V (AV)h-4 In the (V, Vl) h(5” 30) conformal space, “□ preserves the Einstein Tensor saved by intensormo and Hlk represents. The latter invariant is and the result is (Rlk)h=HHk−hHkH (56, 34) is obtained. save einsta For the spatial part of the intensor, a simple formula is obtained, which means that it does not contain the second derivative of the three-dimensional scalar ■ It is noticeable in the fruit.

The mixture component of the conservation tensor becomes zero, G1゜=O(56,36) is the component G6゜ Goo = c2V2 (-V2H2V (AV) h+ 3 (VtVbh) (56°37). Proceed to formulate the gravity equation. We As just mentioned, the quantity G1. does not contain the second derivative of ■. the other smell As shown by (,56,28), Ro. does not include the second derivative of hlk.

Therefore, the gravity equation is Ro. When written in the form solved for and Glk, ■ The second derivatives of and will be separated from each other. Communism of (53,03) A general equation of the form ”oo--X(”00-%ooT) (56,39) is obtained, where T is invariant T = (T), = (7o0 + 711 + TZ2 + T33) g (5” 40) . Therefore, by (56,05) we get and Ro. The value of (56,2 8) using (56,42) is obtained.

The spatial component equation is is obtained. Equations for mixture components (% formula %) , where the left side is zero due to (56,36) and the right side is also zero due to (56,36). Since B, the flow of mass is zero, so they are filled equally.

The equations so obtained are Acquire a pictorial form.

shall be. The quantity m is also of the form %formula%) It can be written as As can be understood by comparing this with the approximate formula (55,02) , the quantity m represents a certain mass density: it can be interpreted as a mass density in the conformal space. Recognize.

Furthermore, if we replace ■ with the quantity φ, we get the equation %formula%) Accordingly, the relation between the space-time metric and the metric in conformal space is of the form ds2-c2e-2 φdt2-e2φd, 2. , (5, 6, 47), so is obtained. Therefore, the gravity equations (56,42) and (56,43) are HHk=-2φiφk-hik (φ1φ])h-xTik(56,50) You can write it.

The first of these equations is essentially Poissonian for Newton's potential U. It is an equation. In fact, the symbol △ is a generalization of the Laplace operator, and m is the mass density. And by (55, 15), the constant X is obtained by.

Then, the quantity U is the equation %formula%) This is the equation for Newton's potential (53°13) and different. Also, where g is the component of gravitational acceleration.

Next, we will clarify the physical meaning of (56, 50). Apart from the factors, The term containing φ1 can be interpreted as gravitational stress.

If 2φiφk - hilc (φjφj) h=xT'1k (56, 55) , (56,50) It can be equally replaced with =-x(TJik+%) (56,56). ten We understand that the three-dimensional divergence of the sol T, , is related to the metric (h, ,) , and (56,49) gives 1 , vk, -, , h=-, φi' (56, 58) are obtained.

On the other hand, since Hl is a conserved tensor in conformal space, its diver- -gence is zero. Therefore, apart from its sign, the tensor of the gravitational stress The divergence of Tlk is the tensor of elastic and other resting stresses. be equivalent to. In this way, (vkTlk)h, , >, (56, 59) is obtained. Ru. These equations are similar to the usual equations in statics for elastic bodies in a gravitational field. It represents a single membrane.

The equation for statics in conformal space, written in the form (56,56), is It holds true in the same way as Einstein's equations in space and time.

In both sets of equations, the left side contains the conservation tensor, while the right side contains the stress tensor. or its four-dimensional generalization. Here, the gravitational stress is expressed in an unambiguous form as It appears after space splits from time and after progression into conformal space.

A conformal space would be almost Euclidean. In fact, (56,54) and ten As understood from the estimation formula (55,02) for sol T, k, (56,5 The right side of 6) will have order g12/C2. This is their Euclidean value This leads to the result that the deviation of hl, from will have order U2/c'. This conclusion The result is consistent with the approximation formula (55, 45), which was just the basis for the introduction of conformal space. .

For empty space, when T − z = O and μ = 0, the equation (56°4 9) is the result of (56,50). Hl using (56,57).

It is easy to understand this by making the divergence of Ru.

If the mass tensor T, , is zero, then the entire space has no singularity and no boundary line. The only static solution to Einstein's equations that satisfies the Euclidean space This would be a solution corresponding to a pseudo-Euclidean space. This can be demonstrated in the following way Wear. In the case of empty space, equation (56, 49) gives (△φ), = 0. Ru. This is an elliptical type equation for f and represents a generalization of the Laplace operator. vinegar. The function f and its derivative f, are finite and continuous everywhere, and At spatial infinity there must be a tendency to zero. However, these provisions The only solution to the Labrace equation that satisfies the condition is the solution φ=0. But then the derivative φ will also be zero and therefore also equation (56,55). Further Tl Since k=0, we also obtain H, ,=0. Therefore, the curvature of the conformal space Insor is zero, and space itself is Euclidean.

Appendix VII (The following text is based on Theory of Space, Time and Gravity. e, Time and Gravitation), Chapter V, pp, 209-21 5, and is incorporated herein by reference.

In accordance with the present invention, new developments have been made from this text. Additions are underlined. , and deletions are indicated by [in parentheses and initials. ) and a Gravitating Mass) [Concentrated Mass] A strict solution of the gravity equation for In the case of [enriched] masses as Mills orbits, the strict sphere pair of the gravity equation can find the solution of the term. Since we are dealing with the stationary case, Using the results from the previous section and converting ds2 to ds2-c2V2dt2-5(F2( 57,01) d　a2-h,dx;dxk(57,02).

If Xl, X2 and X3 are harmonic coordinates, “1” r” Sinθc ostpX2 w r’ SFnθsin” (57,03) x3 w r ”　cos　e By doing so, we can introduce spherical coordinates for them.

The assumption of spherical symmetry means that the expression for dσ2 is of the form da" = F2df2 + p2 (d 62+Sin26dg2) (57・04), where F and and ρ are functions only of γ*. The coefficient V also depends only on r* (and Must not be.

First one Fdr’=dr (57,05) Then, the general formula (57,04) can be transformed to F=1, so da2 = d r　2　+p2(d62+　5in26dq+”)　(57-06) Implement this When we In the sense, it means that the harmonic coordinates x1, X2 and and x3. Equation (57,03) with r* replaced by r Forming a Laplace operator for , we later always get a "harmonic" radius vector r. Ru.

Regarding weighing (57,06), is obtained and therefore hxo, hq+rxo, hr6=0 is obtained. therefore 4h-p25in6 (57,09) And the Laplace operator in the conformal space can be written as (where △*W is empty This is the Laplace operator between: As a result of (56,10), the harmonic coordinates represent the Lavrace equation in conformal space. I have to see it. For the quantity (57,03), △*)(,=-2x+ (57, 12) is obtained. Therefore, the condition for harmonic coordinates △x,=o (57,13) transforms into This is used when proceeding from the initial radius vector r to the harmonic one r*. This is the equation that should be used. The general formula is metric tensor (57,07) and (57,0 8), we derive the following equation for 18 Christoszff construction symbols: can: Here, the prime sign means differentiation with respect to r. Christophel symbol and All tensor quantities used in this section refer only to conformal spaces: Therefore, there is no need to add a subscript as was done in section 56.

The fourth run in three dimensions using the Christophel symbols tabulated in (57, 15). form the curvature tensor of Form the curvature of the ranks. In the equation for off-diagonal components, the first lower The term whose index is equal to the index above is omitted: the 4th rank curvature Due to the symmetries of the insoles, these terms are zero when the coordinate systems are orthogonal. one In the general formula (56, 16), leave only the terms that are different from zero, and therefore it is likely that Inserting the value of the Christophel symbol from (57, 15) is obtained. for calculation From this, it is obtained that rq+r has the same value. therefore Continuing the calculation, is obtained, therefore Inserting (57, 22) and (57, 24) into (57,, 16) results in In particular, the formula %formula%) is found. Similarly, calculations show that, as expected for reasons of spherical symmetry, Rq, (P=s+n26R66,,,'(57,26) is obtained.Second The off-diagonal elements of the rank curvature tensor are proven to be zero: The invariant of the three-dimensional curvature tensor is expressed as can be calculated from, and would be given by By applying equation (56, 32) Hre= OH God=OH%-0 (In a somewhat simpler way, we can use the fourth rank of common ten in three-dimensional space. These expressions can be obtained using the relational expressions connecting the sol and the conservation tensor. It will be possible. This relationship is discussed in Appendix G.

In the concepts of this section, the equation (G, '13) in this appendix is of the form hH'' = R 64,,66; hHθe = Rrφ, rφ; hHHm” Rre, re ;hH’>wr”$r, re ;h) (4’r xm Rr13.Q ; hHrew+”e$, $r; (57, 31) It can be written as These equations are derived from equations (57, 30) as found earlier. It is easy to understand that According to the equation we derived, Einstein can write the gravitational equation of gravity in clear form. As we saw in the previous section, format ds2 %formula%) where has the value (57,04), then the gravity equation is Hik “−2 φiφk −)’ik (φiφJ)h−xm, , (57, 34) where [quality The quantity density μ is given by (56,44) and (56,45), and appears as Ru. Radius r. Proceeding in this case of a shell with concentrated mass at [point] and , the Laplace operator has the form (57,10), but the quantity H1k is (57,3C1 ), using spherical coordinates, given by is obtained. The equation for Hφφ is the equation for Hθθ and the factor s Although the difference is in''θ, the remaining equations (57, 34) are satisfied the same.

Integrating (57, 35), we get ρ2φ'=-α (57, 37) is obtained, where is a constant. Equation (57,35) has positive μ (57 , 33), the constant α should be positive. In fact, first of all (5 7, 33) and -(57, 38) Te. (57, 37) is applicable everywhere for r≧ro, given by where m is the mass of the Mills orbit. where μ is Newton's gravitational constant and for X by (56,51). In this way, in this way, (57,,7 The constant α in 3) is made to match the boundary condition of the discontinuity of space-time curvature for r>re. It can be solved by The principle of superposition can be applied: thus the total song The rate is given as the sum of the curvatures produced by the Mills orbit, and α(57, The mass in the relational expression for 39A) is the total mass; all of the objects that are attracted by gravity All Mils [4 bowi, 2 o-M (57, 38) Here, the integral is extended over the entire range where μ differs from zero, and (57, 30) is can be applied outside the domain of , where α is given by. Here γ is Newton is the gravitational constant of , and is related to X by (56°51). As is clear, M is the mass of the object attracted by the gravitational force:] When proceeding in the case of concentrated mass , this quantity, and with it α, stop at Arukuma and positive. The dimension of α is the length That's why we call it the gravitational radius of the mass. φ from (57, 37) ′ into the first culm equation of (57,36), we get and infinitely If we take the square root to satisfy the requirement that ρ′+1, pp””’J<p2+α2) (57, 41) is obtained. Express this formula in terms of r. When differentiated, ρρ′+ρ′2=1 (57, 42) is obtained, which shows that the second equation (57, 36) is also satisfied.

The differential equation (57, 41) can be easily solved by the product method; After zeroing 1 , Yoyo 2 + a2) (57, 43) p,,,% 2-a2) (57, 44) Thus, finally da2 = dr2 + (r2- a2Xd6” + sin”6 dqJ) ("7.4") Due to its physical properties, ρ must be positive and Therefore, the range of the variable r is r≧α (57, 46) Next, we will discuss the harmonization conditions. The value of ρ from (57, 44) is (57°14) It can be seen that the harmonic radius vector r* satisfies . as is obvious , this equation has a solution r*=r ('57.48) has. This is uniquely true in the entire domain (57, 46) for finite r. finitely and differs from r by no more than a finite amount at infinity. can be easily demonstrated. Therefore, the variable r in our equation is itself tuned. and instead of (57,03), simply X, mr sin 1llcos epXz, r sinθ5vncp (57,49)X3 z r CoSθ can be written.

We have not found the quantity φ yet. Integrate (57, 37) and consider boundary conditions and from (57,44) Equation (57,01) or (57,32) for ds2 takes the form and Inserting the value of ds2 from equation (57,45) gives .

The strict solution so obtained is that the conformal space is almost Euclidean. where φ is approximately equal to U/c2, where U is Newton's potion which can be corroborating the conclusion that

In fact, as equations (57, 45) show, the components of the metric tensor of ds2 are of order with relative deviations from their Euclidean values, and let φ be placed by U/c2 The converting errors will have the same order. [As the estimates presented in the next section show, , the quantity (57, 76) is extremely small. This close relationship with Newton's theory It should be noted that a close match can only be obtained when using harmonic coordinates. .

[(57,54) Equivalent form but enriched quality in arbitrary anharmonic coordinates The solution to the quantity problem was first derived by Schwarzschild [18] and then It bears his name. ] Modifications and substitutions made by those skilled in the art are within the scope of the invention. and the invention is not limited except by the scope of the following claims.

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Claims (1)

[Claims] 1. Process: prepare the material elements; forming the element of material to a second curvature opposite to the first curvature, and having a second curvature; Energy from an energy source is applied to the material element, where it is attracted by gravitational force. A repulsive force is created away from the mass, and the mass attracted by the attractive force and the applied said energy from said material element in response to a force exerted by said energy receiving said repulsive force on the source; gives a repulsive force from a mass attracted by an attractive force having a space-time manifold of first curvature consisting of Method. 2. The step of preparing the material element comprises the step of donating electrons, as described in item 1 above. How to put it on. 3. The method according to item 1 above, wherein the first curvature is a positive curvature. 4. The process of forming is It consists of applying one of a quadrupole electrostatic field, a quadrupole magnetic field, and a quadrupole electromagnetic field. , and further, said selection of a quadrupole electrostatic field, a quadrupole magnetic field, and a quadrupole electromagnetic field. 4. The method of claim 3, further comprising the step of moving the electrons through one of the channels. 5. The method according to item 4 above, wherein the step of moving includes the step of containing the electrons. 6. The process of adding energy involves the use of static electricity, magnetism, and electromagnetic energy. 2. The method of item 1 above, comprising the step of adding at least one. 7. A structure that can move the received repulsive force with respect to the mass attracted by the attractive force. 7. The method of claim 6, further comprising the step of adding to. 8. Rotate the structure around the axis to reduce the force of gravity due to the mass pulled by the gravitational force. The vector of angular momentum of said circularly rotating structure parallel to the center vector of the force 8. The method of claim 7, further comprising the step of applying a force having a torque. 9. By changing the direction of the angular momentum vector, create a table of the mass attracted by the gravitational force. The method further comprises accelerating the structure through a trajectory parallel to a plane. The method described in Section 8. 10. Process: providing an element of material having a second curvature opposite to the first curvature; Energy from an energy source is applied to the element of the material mentioned above, where it is attracted by gravitational force. A repulsive force is created that separates the mass attracted by the attractive force and the applied mass. said energy from said material element in response to a force exerted by said energy. - receiving said repulsive force on the source, gives a repulsive force from a mass attracted by an attractive force having a space-time manifold of first curvature consisting of Method. 11. Elements of matter; means for forming said material elements to have a curvature opposite to that of the object to which they are attracted by gravity; , means for applying energy to said oppositely curved material element, wherein A repulsive force is applied to said oppositely bent material element in response to the energy applied to said The energy is generated by the gravitational force in the direction away from the object that is attracted by the gravitational force having the space-time manifold of the first curvature, which is applied to the means that applies the A device that applies repulsive force from an object. 12. The device according to item 11 above, wherein the material element consists of electrons. 13. The forming means includes: means for applying a negatively curved field to an electron; and a means for passing the electron through the negatively curved field. means for moving the electron to take a negative curvature; 13. The device according to claim 12, comprising: 14. The means for providing the negatively curved field includes a quadrupole electrostatic field, a quadrupole magnetic field, and a quadrupole magnetic field. 14. The apparatus of claim 13, comprising means for providing one of the heavy pole electromagnetic fields. 15. The moving means is means for accelerating said electrons along a selected path; and means for accelerating said electrons along a selected path. 14. The apparatus of claim 13, comprising at least one means for containing. 16. The accelerating means may cause at least one of an electric field gradient, a magnetic field, and an electromagnetic field to 16. The apparatus of claim 15, comprising means for feeding along a selected path. 17. said containing means is spatially and temporally coherent along said selected path; at least a means for providing strong light, a means for providing an electric field, and a means for providing a magnetic field. 16. The apparatus of claim 15, comprising: 18.Furthermore, A circularly rotatable structure with a moment of inertia, and said repulsive force circularly rotatable means to add to the possible structure, where The vector of angular momentum of the circularly rotatable structure is the object attracted by the gravitational force. parallel to the central vector of the gravitational force generated by The device according to item 11 above. 19. The accelerating force is a structure that can rotate circularly around the object that is attracted by the gravitational force. Item 18 above, which is selectively added to give a trajectory parallel to the surface of The device described. 20. The accelerating force is a structure that can rotate circularly around the object that is attracted by the gravitational force. 19. The apparatus of claim 18, wherein the device is selectively added to provide a hyperbolic trajectory of . 21. Elements of matter that have a curvature opposite to that of the object that is attracted by gravity, and Beauty means for applying energy to said oppositely curved material element, wherein A repulsive force is applied to said oppositely bent material element in response to the energy applied to said The energy is generated by the gravitational force in the direction away from the object that is attracted by the gravitational force having the space-time manifold of the first curvature, which is applied to the means that applies the A device that applies repulsive force from an object.