Mathematical Foundations of the Relativistic Theory of Quantum Gravity

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 43 Explanation for red-shift anomalies 48 Superparticles (hypermassive Higgs bosons) and Big-Bang 49 Deduction of Reciprocal Fine Structure Constant and the Uncertainty Principle 51 Dark Matter, Dark Energy and Inflation Period 51 The Origin of the Universe 57 Solution for the Black Hole Information Paradox 59 A Creator’s need 61 The Origin of Gravity and Genesis of the Gravitational Energy 62 Explanation for the anomalous acceleration of Pioneer 10 64 New type of interaction 66 Appendix A 69 Allais effect explained 69 Appendix B 72 References 73

3 1. INTRODUCTION Quantum Gravity was originally studied, by Dirac and others, as the problem of quantizing General Relativity. This approach presents many difficulties, detailed by Isham [1]. In the 1970's, physicists tried an even more conventional approach: simplifying Einstein's equations by assuming that they are almost linear, and then applying the standard methods of quantum field theory to the thus oversimplified equations. But this method, too, failed. In the 1980's a very different approach, known as string theory, became popular. Thus far, there are many enthusiasts of string theory. But the mathematical difficulties in string theory are formidable, and it is far from clear that they will be resolved any time soon. At the end of 1997, Isham [2] pointed out several "Structural Problems Facing Quantum Gravity Theory". At the beginning of this new century, the problem of quantizing the gravitational field was still open. In this work, we propose a new approach to Quantum Gravity. Starting from the generalization of the action function we have derived a theoretical background that leads to the quantization of gravity. Einstein's General Relativity equations are deduced directly from this theory of Quantum Gravity. Also, this theory leads to a complete description of the Electromagnetic Field, providing a consistent unification of gravity with electromagnetism. 2. THEORY We start with the action for a free-particle that, as we know, is given by ∫−= ba dsS α where α is a quantity which characterizes the particle. In Relativistic Mechanics, the action can be written in the following form [3]: dtcVcLdtS t t t t∫ ∫ −−== 21 21 221α where 221 cVcL −−= α is the Lagrange's function. In Classical Mechanics, the Lagrange's function for a free-particle is, as we know, given by: where V is the speed of the particle and is a quantity hypothetically [ 2aVL = a 4] given by: 2ma = where is the mass of the particle. However, there is no distinction about the kind of mass (if gravitational mass, , or inertial mass ) neither about its sign m gm im( )± . The correlation between and a α can be established based on the fact that, on the limit , the relativistic expression for ∞→c L must be reduced to the classic expression .The result [2aVL = 5] is: cVL 22α= . Therefore, if mcac =2=α , we obtain . Now, we must decide if 2aVL = gmm = or imm = . We will see in this work that the definition of includes . Thus, the right option is , i.e., gm im gm .ma g 2= Consequently, cmg=α and the generalized expression for the action of a free-particle will have the following form: ( )1∫−= bag dscmS or

4 ( )21 2222 1 dtcVcmS t t g −−= ∫ where the Lagrange's function is ( )31 222 .cVcmL g −−= The integral dtcVcmS t t g 222 12 1 −= ∫ , preceded by the plus sign, cannot have a minimum. Thus, the integrand of Eq.(2) must be always positive. Therefore, if , then necessarily ; if , then . The possibility of is based on the well-known equation 0>gm 0>t 0<gm 0<t 0<t 22 0 1 cVtt −±= of Einstein's Theory. Thus if the gravitational mass of a particle is positive, then t is also positive and, therefore, given by 22 0 1 cVtt −+= . This leads to the well-known relativistic prediction that the particle goes to the future, if . However, if the gravitational mass of the particle is negative, then is negative and given by cV → t 22 0 1 cVtt −−= . In this case, the prediction is that the particle goes to the past, if . Consequently, is the necessary condition for the particle to go to the past. Further on, a correlation between the gravitational and the inertial masses will be derived, which contains the possibility of . cV → 0<gm 0<gm The Lorentz's transforms follow the same rule for and 0>gm 0<gm , i.e., the sign before 221 cV− will be when and if ( )+ 0>gm ( )− 0<gm . The momentum, as we know, is the vector VLp rr ∂∂= .Thus, from Eq.(3) we obtain VM cV Vm p g g r r r = −± = 221 The sign in the equation above will be used when and the ( )+ 0>gm ( )− sign if 0<gm . This means that will be always positive. Consequently, we will express the momentum gM p r in the following form ( )4 1 22 VM cV Vm p g g r r r = − = The derivative dtpd r is the inertial force which acts on the particle. If the force is perpendicular to the speed, we have iF ( )5 1 22 dt Vd cV m F gi rr − = However, if the force and the speed have the same direction, we find that ( ) ( )61 2322 dt Vd cV m F gi rr − = From Mechanics [6], we know that LVp −⋅ rr denotes the energy of the particle. Thus, we can write ( )7 1 2 22 2 cM cV cm LVpE g g g =− =−⋅= rr Note that is not null forgE 0=V , but that it has the finite value ( )8200 cmE gg = Equation (7) can be rewritten in the following form: ( ) ( )9 1 1 0 2 22 2 2 2 22 2 2 i i g Kii i g E i i i i g g g gg E m m EE m m cm cV cm cm m m cm cV cm cmE Ki =+= = ⎥⎥ ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎢ ⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − += =− − −= 444 3444 21 By analogy to Eq. (8), into the equation above, is the inertial energy at rest. Thus, 2 00 cmE ii = Kiii EEE += 0 is the total inertial

5 energy, where is the kinetic inertial energy. From Eqs. (7) and (9) we thus obtain KiE ( )10 1 2 22 2 0 .cM cV cm E i i i =− = For small velocities , we obtain ( cV << ) ( )1122120 VmcmE iii +≈ where we recognize the classical expression for the inertial kinetic energy of the particle. The expression for the gravitational kinetic energy, , is easily deduced by comparing Eq.(7) with Eq.(9). The result is KgE ( )12.E m m E Ki i g Kg = In the presented picture, we can say that the gravity, , in a gravitational field produced by a particle of gravitational mass , depends on the particle's gravitational energy, (given by Eq.(7)), because we can write g r gM gE ( )13222 2 22 r M G cr cM G cr E Gg ggg === Due to rg ∂Φ∂= , the expression of the relativistic gravitational potential, , is given by Φ 221 cV m r G r GM gg − −=−=Φ Then, it follows that 2222 11 cVcVr Gm r GM gg − = − −=−=Φ φ where rGmg−=φ . Then we get 22222 11 cVr Gm cVrr g − = −∂ ∂=∂ Φ∂ φ whence we conclude that 222 1 cVr Gm r g − =∂ Φ∂ By definition, the gravitational potential energy per unit of gravitational mass of a particle inside a gravitational field is equal to the gravitational potential of the field. Thus, we can write that Φ ( ) gm rU ′=Φ Then, it follows that ( ) 222 1 cVr mm G r m r rU F gggg − ′−=∂ Φ∂′−=∂ ∂−= If and0>gm 0<′gm , or and 0<gm 0>′gm the force will be repulsive; the force will never be null due to the existence of a minimum value for (see Eq. (24)). However, if gm 0<gm and 0<′gm , or and 0>gm 0>′gm the force will be attractive. Just for ig mm = and we obtain the Newton's attraction law. ig mm ′=′ On the other hand, as we know, the gravitational force is conservative. Thus, gravitational energy, in agreement with the energy conservation law, can be expressed by the decrease of the inertial energy, i.e., ( )14ig EE ΔΔ −= This equation expresses the fact that a decrease of gravitational energy corresponds to an increase of the inertial energy. Therefore, a variation iEΔ in yields a variation iE ig EE ΔΔ −= in . Thus gE iii EEE Δ+= 0 ; igggg EEEEE ΔΔ −=+= 00 and ( )1500 igig EEEE +=+ Comparison between (7) and (10) shows that 00 ig EE = , i.e., 00 ig mm = . Consequently, we have