Mathematical Foundations of the Relativistic Theory of Quantum Gravity Fran De Aquino Maranhao State University, Physics Department, S.Luis/MA, Brazil. Copyright © 2008-2011 by Fran De Aquino. All Rights Reserved Abstract: Starting from the action function, we have derived a theoretical background that leads to the quantization of gravity and the deduction of a correlation between the gravitational and the inertial masses, which depends on the kinetic momentum of the particle. We show that the strong equivalence principle is reaffirmed and, consequently, Einstein's equations are preserved. In fact, such equations are deduced here directly from this new approach to Gravitation. Moreover, we have obtained a generalized equation for the inertial forces, which incorporates the Mach's principle into Gravitation. Also, we have deduced the equation of Entropy; the Hamiltonian for a particle in an electromagnetic field and the reciprocal fine structure constant directly from this new approach. It was also possible to deduce the expression of the Casimir force and to explain the Inflation Period and the Missing Matter, without assuming existence of vacuum fluctuations. This new approach to Gravitation will allow us to understand some crucial matters in Cosmology. Key words: Quantum Gravity, Quantum Cosmology, Unified Field. PACs: 04.60.-m; 98.80.Qc; 04.50. +h Contents 1. Introduction 3 2. Theory 3 Generalization of Relativistic Time 4 Quantization of Space, Mass and Gravity 6 Quantization of Velocity 7 Quantization of Time 7 Correlation Between Gravitational and Inertial Masses 8 Generalization of Lorentz's Force 12 Gravity Control by means of the Angular Velocity 13 Gravitoelectromagnetic fields and gravitational shielding effect 14 Gravitational Effects produced by ELF radiation upon electric current 26 Magnetic Fields affect gravitational mass and the momentum 27 Gravitational Motor 28 Gravitational mass and Earthquakes 28 The Strong Equivalence Principle 30 2 Incorporation of the Mach's Principle into Gravitation Theory 30 Deduction of the Equations of General Relativity 30 Gravitons: Gravitational Forces are also Gauge forces 31 Deduction of Entropy Equation starting from the Gravity Theory 31 Unification of the Electromagnetic and Gravitational Fields 32 Elementary Quantum of Matter and Continuous Universal Fluid 34 The Casimir Force is a gravitational effect related to the Uncertainty Principle 35 The Shape of the Universe and Maximum speed of Tachyons 36 The expanding Universe is accelerating and not slowing down 38 Gravitational and Inertial Masses of the Photon 39 What causes the fundamental particles to have masses? 40 Electron’s Imaginary Masses 41 Transitions to the Imaginary space-time 44 Explanation for red-shift anomalies 50 Superparticles (hypermassive Higgs bosons) and Big-Bang 51 Deduction of Reciprocal Fine Structure Constant and the Uncertainty Principle 53 Dark Matter, Dark Energy and Inflation Period 53 The Origin of the Universe 59 Solution for the Black Hole Information Paradox 61 A Creator’s need 63 The Origin of Gravity and Genesis of the Gravitational Energy 64 Explanation for the anomalous acceleration of Pioneer 10 66 New type of interaction 68 Appendix A 71 Allais effect explained 71 Appendix B 74 References 75 3 1. INTRODUCTION b S= −α∫ ds Quantum Gravity was originally a studied, by Dirac and others, as the where α is a quantity which problem of quantizing General characterizes the particle. Relativity. This approach presents In Relativistic Mechanics, the many difficulties, detailed by Isham action can be written in the following [1]. In the 1970's, physicists tried an form [3]: t t even more conventional approach: S=2 Ldt = −2 αc1 − V2 c 2 dt ∫∫t t simplifying Einstein's equations by 1 1 assuming that they are almost linear, where and then applying the standard L= −α c1 − V2 c 2 methods of quantum field theory to is the Lagrange's function. the thus oversimplified equations. But In Classical Mechanics, the this method, too, failed. In the 1980's Lagrange's function for a free-particle a very different approach, known as is, as we know, given by: L= aV 2 string theory, became popular. Thus where V is the speed of the particle far, there are many enthusiasts of and a is a quantity hypothetically [4] string theory. But the mathematical given by: difficulties in string theory are formidable, and it is far from clear that a= m 2 they will be resolved any time soon. where m is the mass of the particle. At the end of 1997, Isham [2] pointed However, there is no distinction about out several "Structural Problems the kind of mass (if gravitational Facing Quantum Gravity Theory". At mass, mg , or inertial mass mi ) neither the beginning of this new century, about its sign (±). the problem of quantizing the The correlation between a and gravitational field was still open. α can be established based on the In this work, we propose a new fact that, on the limit c → ∞ , the approach to Quantum Gravity. relativistic expression for L must be Starting from the generalization of the reduced to the classic expression action function we have derived a 2 2 theoretical background that leads to L= aV .The result [5] is: L= α V2 c . the quantization of gravity. Einstein's Therefore, if α = 2ac= mc , we obtain General Relativity equations are L= aV 2 . Now, we must decide if deduced directly from this theory of m= mg or m= mi . We will see in this Quantum Gravity. Also, this theory work that the definition of m includes leads to a complete description of the g Electromagnetic Field, providing a mi . Thus, the right option is mg , i.e., consistent unification of gravity with a= mg 2 . electromagnetism. Consequently, α = m c and the g 2. THEORY generalized expression for the action of a free-particle will have the We start with the action for a following form: free-particle that, as we know, is b S= − mg c ds ()1 given by ∫a or 4 t2 2 2 2 sign if mg < 0 . Consequently, we will S= − mg c1− V c dt ()2 ∫t 1 express the momentum pr in the where the Lagrange's function is following form 2 2 2 r L= − mg c1 − V c . ()3 m V r g r t2 p = = MVg ()4 2 2 2 2 2 The integral S= mg c1− V c dt , ∫t1 1−V c preceded by the plus sign, cannot The derivative dpr dt is the have a minimum. Thus, the integrand inertial force F which acts on the of Eq.(2) must be always positive. i Therefore, if m > 0 , then necessarily particle. If the force is perpendicular g to the speed, we have t > 0; if m < 0 , then t < 0 . The r g r mg dV F = 5 possibility of t < 0 is based on the i () 1−V2 c 2 dt well-known equation t= ± t1 − V2 c 2 0 However, if the force and the speed of Einstein's Theory. have the same direction, we find that Thus if the gravitational mass r r mg dV of a particle is positive, then t is also Fi = 3 ()6 2 dt positive and, therefore, given by ()1−V2 c 2 t=+ t1 − V2 c 2 . This leads to the From Mechanics [6], we know that 0 r well-known relativistic prediction that pr ⋅ V − L denotes the energy of the the particle goes to the future, if particle. Thus, we can write V→ c . However, if the gravitational m c2 r r g 2 mass of the particle is negative, then Eg = p ⋅ V − L = = Mg c ()7 2 2 t is negative and given by 1−V c 2 2 Note that E is not null forV=0, but t= − t0 1 − V c . In this case, the g prediction is that the particle goes to that it has the finite value the past, if V→ c . Consequently, 2 mg < 0 is the necessary condition for Eg0 = mg0 c (8) the particle to go to the past. Further Equation (7) can be rewritten in on, a correlation between the the following form: gravitational and the inertial masses 2 2 mg c 2 will be derived, which contains the Eg= m g c − −mg c = 1−V2 c 2 possibility of mg < 0 . The Lorentz's transforms follow ⎡ ⎤ ⎢ ⎥ the same rule for m > 0 and m < 0 , m ⎛ 2 ⎞ g g g ⎢ 2 ⎜ mi c 2 ⎟⎥ = mi c + − mi c = 2 2 m ⎢ ⎜ 2 2 ⎟⎥ i.e., the sign before 1−V c will be i ⎝ 1−V c ⎠ ⎢ 14244 4344 ⎥ E ()+ when mg > 0 and ()− if mg < 0 . ⎣⎢ Ki ⎦⎥ The momentum, as we know, mg mg r r =()EEi0 + Ki = Ei ()9 is the vector p=∂ L ∂V.Thus, from mi mi Eq.(3) we obtain By analogy to Eq. (8), E= m c2 into r i0 i 0 r mg V r the equation above, is the p = = MVg ±1 −V2 c 2 inertial energy at rest. Thus, EEE= + is the total inertial The ()+ sign in the equation above i i0 Ki energy, where EKi is the kinetic will be used when mg > 0 and the (−) 5 inertial energy. From Eqs. (7) and (9) ∂Φ Gmg = we thus obtain ∂r 2 2 2 2 r1 − V c mi0 c 2 Ei = = Mi c ()10 By definition, the gravitational 2 2 . 1 − V c potential energy per unit of For small velocities (V<< c), we gravitational mass of a particle inside obtain a gravitational field is equal to the E≈ m c2 + 1 mV 2 (11) gravitational potential Φ of the field. i i0 2 i Thus, we can write that where we recognize the classical U() r expression for the inertial kinetic Φ = m′ energy of the particle.