Mathematical Foundations of the Relativistic Theory of Quantum Gravity

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

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

1. INTRODUCTION b −= α∫ dsS 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: 2 LdtS 2 α 1−−== 22 dtcVc ∫∫t t simplifying Einstein's equations by 1 1 assuming that they are almost linear, where and then applying the standard α 1 −−= cVcL 22 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: = aVL 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 = ma 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 = aVL .The result [5] is: = α 2cVL . the quantization of gravity. Einstein's Therefore, if α = 2 = mcac , we obtain General Relativity equations are = aVL 2 . Now, we must decide if deduced directly from this theory of = mm g or = mm i . 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 = g 2.ma electromagnetism. Consequently, α = cm 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 −= g dscmS ()1 given by ∫a or

t2 2 22 sign if mg < 0 . Consequently, we will −= g 1− dtcVcmS ()2 ∫t 1 express the momentum pr in the where the Lagrange's function is following form 2 22 r g 1−−= .cVcmL ()3 Vm r g r t2 p = = gVM ()4 2 22 22 The integral = g 1− dtcVcmS , ∫t1 1− cV preceded by the plus sign, cannot The derivative r dtpd 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 Vd F = 5 possibility of t < 0 is based on the i () 1− cV 22 dt well-known equation 1−±= cVtt 22 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 Vd of a particle is positive, then t is also Fi = 3 ()6 2 dt positive and, therefore, given by ()1− cV 22 1−+= cVtt 22 . This leads to the From Mechanics [6], we know that 0 r well-known relativistic prediction that r −⋅ LVp denotes the energy of the the particle goes to the future, if particle. Thus, we can write → cV . However, if the gravitational cm 2 r r g 2 mass of the particle is negative, then g LVpE =−⋅= = g cM ()7 22 t is negative and given by 1− cV 22 Note that E is not null forV=0, but 0 1−−= cVtt . In this case, the g prediction is that the particle goes to that it has the finite value the past, if → cV . Consequently, 2 mg < 0 is the necessary condition for g0 = g0cmE (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 g cm 2 will be derived, which contains the gg cmE −= g cm =− 1− cV 22 possibility of mg < 0 . The Lorentz's transforms follow ⎡ ⎤ ⎢ ⎥ the same rule for m > 0 and m < 0 , m ⎛ 2 ⎞ g g g ⎢ 2 ⎜ icm 2 ⎟⎥ = icm + − icm = 22 m ⎢ ⎜ 22 ⎟⎥ i.e., the sign before 1− cV will be i ⎝ 1− cV ⎠ ⎢ 14244 4344 ⎥ E ()+ when mg > 0 and ()− if mg < 0 . ⎣⎢ Ki ⎦⎥

The momentum, as we know, mg mg r r ()0 EE Kii =+= Ei ()9 is the vector Lp ∂∂= V.Thus, from mi mi Eq.(3) we obtain By analogy to Eq. (8), = cmE 2 into r ii 00 r gVm r the equation above, is the p = = gVM 1−± cV 22 inertial energy at rest. Thus, = + EEE is the total inertial The ()+ sign in the equation above 0 Kiii 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 222 2 1 − cVr i0cm 2 Ei = = icM ()10 By definition, the gravitational 22 . 1 − cV potential energy per unit of For small velocities (<

mg E = .E ()12 Kg Ki ′ mi If mg > 0 and mg < 0 , or mg < 0 and In the presented picture, we m′ > 0 the force will be repulsive; the r g can say that the gravity, g , in a force will never be null due to the gravitational field produced by a existence of a minimum value for mg particle of gravitational mass M g , (see Eq. (24)). However, if mg < 0 depends on the particle's gravitational and m′g < 0 , or mg > 0 and m′g > 0 energy, E g (given by Eq.(7)), because the force will be attractive. Just we can write for = mm and ′ = mm ′ we obtain E cM 2 ig ig Gg g G g 13 the Newton's attraction law. 22 −=−= 22 () cr cr On the other hand, as we Due to ∂Φ∂= rg , the expression of know, the gravitational force is the relativistic gravitational potential, conservative. Thus, gravitational Φ , is given by energy, in agreement with the energy

GM g Gmg conservation law, can be expressed −=−=Φ by the decrease of the inertial energy, r 22 1− cVr i.e., Then, it follows that Δ g −= ΔEE i (14)

This equation expresses the fact that GM g Gmg φ −=−=Φ = a decrease of gravitational energy r 1− 22 1− cVcVr 22 corresponds to an increase of the inertial energy. Therefore, a variation ΔEi in whereφ −= g rGm . E yields a variation in . Then we get i Δ −= ΔEE ig E g

Φ∂ ∂φ Gm Thus = 0 +ΔEEE iii ; 0 Δ =+= 0 −ΔEEEEE igggg = = g ∂r 1−∂ cVr 222 1− cVr 22 and + = + EEEE 15 whence we conclude that igig 00 ( ) Comparison between (7) and (10)

shows that = EE ig 00 , i.e., = mm ig 00 . Consequently, we have 6

The minimum energy of a particle is obviously its inertial energy at rest =+ + igig 00 = 2EEEEE i0 (16) 2 2 g = i cmcm . Therefore we can write However ii 0 += EEE Ki.Thus, (16) becomes hn 22 0 −= Kiig .EEE (17) 2 2 = g cm Note the symmetry in the equations of 8 Lm maxg

Ei and E g .Substitution of 0 = −EEE Kiii Then from the equation above it follows into (17) yields that nh − gi = 2EEE Ki (18) mg ±= ()23 Squaring the Eqs.(4) and (7) and cLmax 8 comparing the result, we find the whence we see that there is a minimum following correlation between value for mg given by gravitational energy and momentum : 2 h E m ()ming ±= ()24 g 2 += 2 2 .cmp ()19 cL 8 c 2 g max The relativistic gravitational mass The energy expressed as a function of 1 22 − 2 the momentum is, as we know, called gg (1−= cVmM ) , defined in the Hamiltonian or Hamilton's function: Eqs.(4), shows that 2 2 2 g += g .cmpcH ()20 g()min = mM g ()min (25) Let us now consider the problem The box normalization leads to the of quantization of gravity. Clearly there is conclusion that the propagation number r something unsatisfactory about the kk == 2 λπ is restricted to the whole notion of quantization. It is important to bear in mind that the values = 2π Lnk . This is deduced quantization process is a series of rules- assuming an arbitrarily large but finite of-thumb rather than a well-defined 3 cubical box of volume L [8]. Thus, we algorithm, and contains many ambiguities. In fact, for electromagnetism have we find that there are (at least) two = nL λ different approaches to quantization and From this equation, we conclude that that while they appear to give the same Lmax theory they may lead us to very different nmax = quantum theories of gravity. Here we will λmin follow a new theoretical strategy: It is and known that starting from the Schrödinger min = nL λminmin = λmin equation we may obtain the well-known Since n =1. Therefore, we can write expression for the energy of a particle in min periodic motion inside a cubical box of that edge length L [ 7 ]. The result now is max = LnL minmax (26) hn 22 From this equation, we thus conclude E = n = 21,...3,2,1 n 2 () that 8 g Lm = nLL min (27) Note that the term 2 2 (energy) 8 g Lmh or will be minimum for = LL where L L max max L = max ()28 is the maximum edge length of a cubical n box whose maximum diameter Multiplying (27) and (28) by 3 and = Ld 3 (22) max max reminding that = Ld 3 , we obtain is equal to the maximum length scale of the Universe. 7

From this equation one concludes dmax = min dorndd = ()29 that we can have or n = VV max

Equations above show that the length = VV max 2 , but there is nothing in (and therefore the space) is quantized. between. This shows clearly that V By analogy to (23) we can also max conclude that cannot be equal to c (speed of light in vacuum). Thus, it follows that maxhn M g()max = ()30 n = 1 = VV cL 8 max min n = 2 = VV 2 since the relativistic gravitational mass, max − 1 22 2 n = 3 = VV max 3 Tachyons gg (1−= cVmM ), is just a ...... multiple of m . g 1 ()nVVnn −=−= 1 Equation (26) tells us that x max x −−−−−−−−−−−−−−−−−−−−−−−− min = max nLL max . Thus, Eq.(30) can be rewritten as follows = x max x cnVVnn ←== 2 hn 1 ()nVVnn +=+= 1 Tardyons M = max ()31 x max x g()max 2 nVVnn +=+= 2 cLmax 8 x max ()x Comparison of (31) with (24) shows that ...... = 2 mnM (32) g ()max max g ()min x bigaisnwhere number. which leads to following conclusion that Then c is the speed upper limit of 2 g = mnM g()min (33) the Tardyons and also the speed lower This equation shows that the limit of the Tachyons. Obviously, this limit gravitational mass is quantized. is always the same in all inertial frames. Substitution of (33) into (13) leads Therefore c can be used as a reference to quantization of gravity, i.e., speed, to which we may compare any speed V , as occurs for the relativistic GM g 2 ⎛ Gm g ()min ⎞ g == n ⎜ ⎟ = factor 1− cV 22 . Thus, in this factor, c r 2 ⎜ ()nr 2 ⎟ ⎝ max ⎠ does not refer to maximum propagation = 4 gn 34 min () speed of the interactions such as some From the Hubble's law, it follows that authors suggest; c is just a speed limit ~ ~ Vmax lH max == ()dH max 2 which remains the same in any inertial ~ ~ frame. V lH == ()dH 2 min min min The temporal coordinate x 0 of whence 0 0 space-time is now = maxtVx ( = ctx Vmax d max = is then obtained when →cV ). V d max min min ~ Equations (29) tell us that Substitution of max == nnVV (lH ) into this 0 ~ 0 max min = ndd max . Thus the equation equation yields max == ()1 nVxt ( lxH ). above gives ~ On the other hand, since V = lH and V V = max ()35 = nVV we can write that min n max max ~ −1 0 ~ ~ which leads to following conclusion =Vl max nH .Thus ( lx ) ()ntH == tH max . Therefore, we can finally write V max V = ()36 n ~ 0 this equation shows that velocity is also = (1 nt )( ) = max ntlxH (37) quantized. which shows the quantization of time. 8

From Eqs. (27) and (37) we can ( Δ− ) Vmm easily conclude that the spacetime is not p =Δ g g continuous it is quantized. 1 − ()cV 2 Now, let us go back to Eq. (20) From the Eq.(16) we obtain: which will be called the gravitational = − = 22 Δ =+− − ΔEEEEEEEE Hamiltonian to distinguish it from the 0 ()00 0 iiiiiiig However, Eq.(14) tells us that −Δ =ΔEE ; inertial Hamiltonian Hi : gi

2 2 2 what leads to = 0 +ΔEEE gig or = 0 +Δmmm gig . i += i0 cmpcH . ()38 Thus, in the expression of Δ p we Consequently, Eq. (18) can be rewritten in the following form: can replace ( g − Δmm g ) for mi0 , i.e,

− gi = 2ΔHHH i (39) Vm p =Δ i 0 where Δ H is the variation on the 2 i 1 − ()cV inertial Hamiltonian or inertial kinetic We can therefore write energy. A momentum variation Δp yields Δp cV = ()42 a variation Δ H i given by: 2 i0cm 1 − ()cV 2 2 2 224 2 4 i () i0 +−+Δ+=Δ i0 cmcpcmcppH ()40 By substitution of the expression above By considering that the particle is into Eq. (41), we thus obtain: initially at rest (p = 0). Then, Eqs. (20), −1 ⎡ 22 2 ⎤ 2 mm (12 ) −−−= 1 mcV ()43 (38) and (39) give respectively: = gg cmH , ig 0 ⎣⎢ ⎦⎥ i0 2 = ii 0cmH and For V =0 we obtain = mm .Then, ⎡ 2 ⎤ ig 0 ⎛ Δp ⎞ 2 H ⎢ 1 +=Δ ⎜ ⎟ −1⎥ cm g()= mm i0 () i ⎢ ⎜ cm ⎟ ⎥ i0 min min ⎝ i0 ⎠ Substitution of into the quantized ⎣ ⎦ mg (min )

By substituting H g , H i and ΔH i into expression of M g (Eq. (33)) gives Eq.(39), we get 2 g = mnM i0()min ⎡ 2 ⎤ ⎛ Δp ⎞ ⎢ ⎜ ⎟ ⎥ mm ig 0 12 +−= −1 mi0. ()41 where is the elementary ⎜ ⎟ mi0(min ) ⎢ ⎝ i0cm ⎠ ⎥ ⎣ ⎦ quantum of inertial mass to be determined. This is the general expression of the ForV = 0 , the relativistic correlation between the gravitational and 22 inertial mass. Note that expression gg 1−= cVmM becomes for >Δ i0 cmp ( 25 ), the value of mg = =mMM ggg 00 . However, Eq. (43) shows becomes negative. that g 0 = mm i0 . Thus, the quantized

Equation (41) shows that mg expression of M g reduces to decreases of Δmg for an increase of 2 i0 = mnm i0()min Δp . Thus, starting from (4) we In order to define the inertial quantum obtain number, we will change n in the ()Δ− Vmm expression above for ni . Thus we have pp =Δ+ g g 2 1 − ()cV 2 0 = mnm iii ()min0 (44) By considering that the particle is initially at rest (p = 0), the equation above gives 9 which shows the quantization of accuracy of one part in 107 . inertial mass; ni is the inertial Posteriorly, Eötvos experiment has quantum number. been repeated with accuracy of one We will change n in the quantized part in 109 . In 1963, the experiment expression of M g for ng in order to was repeated with an even greater define the gravitational quantum accuracy, one part in 1011 . The result number. Thus, we have was the same previously obtained. In all these experiments, the 2 22 −17 = mnM igg 0()min (44a) ratio 2cV is less than10 , which is much smaller than the accuracy of −11 Finally, by substituting mg 10 obtained in the previous more given by Eq. (43) into the relativistic precise experiment. expression of M g , we readily obtain Then, we arrive at the conclusion that all these experiments m g say nothing in regard to the relativistic M g = = 1 − cV 22 behavior of masses in relative motion. − 1 M ⎡()12 −−= 22 2 − ⎤McV ()451 Let us now consider a planet in i ⎣⎢ ⎦⎥ i the Sun’s gravitational field to which, in the absence of external forces, we By expanding in power series apply Lagrange’s equations. We and neglecting infinitesimals, we arrive at the well-known equation: arrive at: 2 2 ⎛ dr ⎞ 2 ⎛ dϕ ⎞ 2GM i ⎜ ⎟ + r ⎜ ⎟ =− E ⎝ dt ⎠ ⎝ dt ⎠ r ⎛ V 2 ⎞ M ⎜1−= ⎟M ()46 2 dϕ g ⎜ 2 ⎟ i r =h ⎝ c ⎠ dt

where M i is the inertial mass of the Thus, the well-known expression for Sun. The term E −= i aGM , as we the simple pendulum know, is called the energy constant; period, =2π ()gi ( glMMT ) , can be rewritten a is the semiaxis major of the Kepler- in the following form: ellipse described by the planet l ⎛ V 2 ⎞ around the Sun. T = π ⎜12 + ⎟ << cVfor By replacing M into the g ⎜ 2c 2 ⎟ i ⎝ ⎠ differential equation above for the Now, it is possible to learn why expression given by Eq. (46), and Newton’s experiments using simple expanding in power series, neglecting penduli do not find any difference infinitesimals, we arrive, at: between M g and M i . The reason is due to the fact that, in the case of 2 2 2 ⎛ dr ⎞ 2⎛ dϕ ⎞ 2GM g 2GM g ⎛V ⎞ 22 ⎜ ⎟ + r ⎜ ⎟ E +=− ⎜ ⎟ penduli, the ratio 2cV is less than ⎜ 2 ⎟ ⎝ dt ⎠ ⎝ dt ⎠ r r ⎝ c ⎠ 10−17 , which is much smaller than the accuracy of the mentioned Since = ω = ( ϕ dtdrrV ) , we get experiments. Newton’s experiments have 2 2 2 ⎛ dr ⎞ ⎛ dϕ ⎞ 2GMg 2 grGM ⎛ dϕ ⎞ been improved upon (one part in ⎜ ⎟ + r2⎜ ⎟ E +=− ⎜ ⎟ dt dt r c2 dt 60,000) by Friedrich Wilhelm Bessel ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ which is the Einsteinian equation of the (1784–1846). In 1890, Eötvos planetary motion. confirmed Newton’s results with 10

Multiplying this equation by Gmg (ddt ϕ)2 and remembering that g −= 2 1− cVr 22 ()ϕ 2 = rddt h 24 , we obtain 2 3 ⎛ dr ⎞ ⎛ r4 ⎞ 22 rGMrGM ⎜ ⎟ r2 =+ E⎜ ⎟ g ++ g where V is the velocity of the mass ⎜ ⎟ ⎜ 2 ⎟ 2 2 ⎝ dϕ ⎠ ⎝ ⎠ hh c m , in respect to the observer. V is Making =1 ur and multiplying both g also the velocity with which the members of the equation by u 4 , we get observer moves away from mg . If 2 3 ⎛ du ⎞ E g 22 guGMuGM the observer is inside the gravitational ⎜ ⎟ u2 ++=+ ⎜ ⎟ 22 2 dϕ ⎠⎝ hh c field produced by mg , then, V is the This leads to the following expression velocity with which the observer 2ud GM ⎛ 3 u h 22 ⎞ u g ⎜1+=+ ⎟ 2 2 ⎜ 2 ⎟ dϕ h ⎝ c ⎠ escapes from mg (or the escape In the absence of term 3h cu 222 , the velocity from the gravitational field of integration of the equation should be mg ). Since the gravitational field is immediate, leading to 2π period. In created by a particle with non-null order to obtain the value of the gravitational mass, then obviously, perturbation we can use any of the < cV . If << cV the escape velocity is well-known methods, which lead to given by an angleϕ , for two successive perihelions, given by Mm ′ 22 1 2 gg 6 MG g g′ = GVM 2π + 2 22 r c h Calculating per century, in the case of whence we obtain Mercury, we arrive at an angle of 43” for the perihelion advance. 2Gmg This result is the best V 2 = theoretical proof of the accuracy of r Eq. (45). By substituting this expression into Now consider a relativistic g particle inside a gravitational field. the equation of , above obtained, The condition for it to escape from the the result is gravitational field is that its inertial kinetic energy becomes equal to the Φ∂ Gmg absolute value of the gravitational g = = ∂r 2 2 energy of the field, which is given −21 g rcGmr MGM ′ ()rU −= gg = r whence we recognize the Schwzarzschilds’ equation. Note in MGm ′ this equation the presence of m , −= gg g 1− cVr 22 whose value, according to Eq. (41) can be reduced or made negative. In Since =Φ ()MrU g′ and = ∂Φ ∂rg then, we get

11 this case*, the singularity g → ∞ , frequency of a wave, measured in units produced by Schwzarzschilds’ radius of universal time, remains constant 2 during its propagation, and that it can be =2 g cGmr , ( = mm ig ), obviously expressed by does not occur. Consequently, Black ∂ψ ω = Hole does not exist. 0 ∂t For << cV we get 22 +≅− 211 cVcV 22 . where ψ dtd is the derivative of the 2 eikonal ψ with respect to the time. Since =2 g rGmV , then we can write that On the other hand, the frequency of the wave measured in units of own 22 Gmg φ cV 2 111 +=+≅− 2 time is given by crc ∂ψ ω = Substitution of 22 11 +=− φ ccV 2 into the ∂T well-known expression below Thus, we conclude that ω ∂t 1 −= cVtT 22 = ω0 ∂T which expresses the relativistic whence we obtain correlation between own time (T) and ω t 1 universal time (t), gives == ω T ⎛ φ ⎞ ⎛ φ ⎞ 0 ⎜1 + ⎟ tT ⎜1+= ⎟ 2 ⎝ c 2 ⎠ ⎝ c ⎠ It is known from the Optics that the By expanding in power series, neglecting infinitesimals, we arrive at: ⎛ φ ⎞ * ωω ⎜1 −= ⎟ This can occur, for example, in a stage of 0 2 gravitational contraction of a neutron star (mass > ⎝ c ⎠ In this way, if a light ray with a frequency 2.4M~), when the gravitational masses of the neutrons, in the core of star, are progressively ω0 is emitted from a point where the turned negative, as a consequence of the increase gravitational potential isφ , it will have a of the density of magnetic energy inside the 1 neutrons, 1 2 , reciprocally produced by frequencyω1 . Upon reaching a point n = 2 μ0HW n the spin magnetic fields of the own neutrons, where the gravitational potential is φ2 its rr r 1 3 3 frequency will beω2 . Then, according to n = 2 0( 2 ) = ( 4πγπμ rmSerMH nnnnnn ) due to the equation above, it follows that decrease of the neutrons radii, r , along the very n ⎛ φ ⎞ ⎛ φ ⎞ ωω ⎜1 −= 1 ⎟ and ωω ⎜1 −= 2 ⎟ strong compression at which they are subjected. 01 2 02 2 −6 −3 ⎝ c ⎠ ⎝ c ⎠ Since ∝rW , and ρ ∝ r , then W nn nn n Thus, from point 1 to point 2 the increases much more rapidly – with the decrease frequency will be shifted in the of r – than ρ . Consequently, the ratio n n interval Δω = ω − ω21 , given by W ρ increases progressively with the nn ⎛ − φφ 12 ⎞ =Δ ωω 0 ⎜ ⎟ compression of the neutrons star. According to ⎝ c 2 ⎠ Eq. (41), the gravitational masses of the neutrons can be turned negative at given stage of the If Δω < 0 , (φ > φ21 ) , the shift occurs in the compression. Thus, due to the difference of direction of the decreasing frequencies (red-shift). If Δω > 0 , φ < φ the blue- pressure, the value of W ρ nn in the crust is (21 ) smaller than the value in the core. This means shift occurs. that, the gravitational mass of the core becomes Let us now consider another negative before of the gravitational mass of the consequence of the existence of crust. This makes the gravitational contraction correlation between M g and M i . culminates with an explosion, due to the repulsive gravitational forces between the core and the Lorentz's force is usually written in crust. Therefore, the contraction has a limit and, the following form: consequently, the singularity does not occur.

r r r r p ×+= BVqEqdtd surface dA )is equal to the energy dU r absorbed per unit volume ()ddU .i.e., where r 1−= cVVm 22 . However, V p i0 dU dU dU 22 dP === 47 Eq.(4) tells us that r 1−= cVVmp . () g dV dxdydz dAdz Therefore, the expressions above must Substitution of = vdtdz ( v is the speed be corrected by multiplying its members of radiation) into the equation above by mm ig 0 ,i.e., gives r m m r Vm dU ( dAdtdU ) dD r g g i0Vm g r dP == = ()48 p = = = p d v v m m 22 22 V i0 i0 1 − cV 1 − cV Since dPdA = dF we can write: and dU r dFdt = ()49 pd d ⎛ mg ⎞ mg ⎜ r ⎟ rrr v = ⎜p ⎟ ()×+= BVqEq dt dt ⎝ mi0 ⎠ mi0 However we know that = dtdpdF , then That is now the general expression for dU dp = ()50 Lorentz's force. Note that it depends on v mg . From Eq. (48), it follows that When the force is perpendicular to VdDd dPddU V == ()51 the speed, Eq. (5) gives v r r 22 = g()1− cVdtVdmdtpd .By comparing Substitution into (50) yields dDd with Eq.(46), we thus obtain dp = V ()52 rr r r 2 1 − 22 ×+= BVqEqdtVdcVm v ( i0 )( ) or Note that this equation is the expression Δp 1 D V of an inertial force. dp = dDd ∫0 2 ∫∫00 V Starting from this equation, well- v known experiments have been carried whence out in order to verify the relativistic VD Δp = 2 ()53 expression: 1 − cVm 22 . In general, v i This expression is general for all types of the momentum variation Δp is waves including non-electromagnetic expressed by Δ = ΔtFp where F is the waves such as sound waves. In this applied force during a time interval Δt . case, v in Eq.(53), will be the speed of Note that there is no restriction sound in the medium and D the intensity concerning the nature of the force F , of the sound radiation. i.e., it can be mechanical, In the case of electromagnetic electromagnetic, etc. waves, the Electrodynamics tells us that For example, we can look on the v will be given by momentum variation Δp as due to dz ω c v === absorption or emission of dt κ r με rr ⎛ 2 ⎞ ⎜ 1 ()ωεσ ++ 1⎟ electromagnetic energy by the particle 2 ⎝ ⎠ (by means of radiation and/or by means of Lorentz's force upon the charge of the where kr is the real part of the r r particle). propagation vector k ; +== ikkkk ; In the case of radiation (any ir type), Δp can be obtained as follows. It ε , μ and σ, are the electromagnetic is known that the radiation pressure , characteristics of the medium in which the incident (or emitted) radiation is dP , upon an area = dxdydA of a propagating ( ε = ε ε where ε is the volume dV = dxdydz of a particle( the r 0 r relative dielectric permittivity and incident radiation normal to the −12 ε0 ×= /10854.8 mF ;μ = μ r μ 0 where 13

μ is the relative magnetic permeability ⎧ ⎡ 44 ⎤⎫ r ⎪ −13 R ω ⎪ −7 m ()diskg ⎨ ⎢ ×+−≅ 1012.1121 −1⎥⎬m 0()diski and πμ ×= 104 m/H ; σ is the ⎢ f ⎥ 0 ⎩⎪ ⎣ ⎦⎭⎪ electrical conductivity). For an atom Note that the effect of the inside a body, the incident (or emitted) electromagnetic radiation applied upon radiation on this atom will be propagating the disk is highly relevant, because in the inside the body, and consequently, absence of this radiation the index of σ=σbody, ε=εbody, μ=μbody. refraction, present in equations above, It is then evident that the index of becomes equal to 1. Under these refraction r = vcn will be given by circumstances, the possibility of strongly c με reducing the gravitational mass of the rr ⎛ 2 ⎞ disk practically disappears. In addition, nr == ⎜ 1 ()ωεσ ++ ⎟ ()541 v 2 ⎝ ⎠ the equation above shows that, in On the other hand, from Eq. (50) follows practice, the frequency f of the that radiation cannot be high, and that U ⎛ c ⎞ U extremely-low frequencies (ELF) are Δp = ⎜ ⎟ = nr v ⎝ c ⎠ c most appropriated. Thus, if the frequency of the electromagnetic radiation applied Substitution into Eq. (41) yields upon the disk is = 1.0 Hzf (See Fig. I (a)) ⎧ ⎡ 2 ⎤⎫ ⎪ ⎢ ⎛ U ⎞ ⎥⎪ and the radius of the disk is = 15.0 mR , mg ⎨ 121 +−= ⎜ r ⎟ −1 ⎬mn i0 ()55 ⎢ ⎜ 2 ⎟ ⎥ and its angular speed ⎪ ⎝ i0cm ⎠ ⎪ ⎩ ⎣ ⎦⎭ ω ×= 4 srad ( 000,100~/1005.1 rpm), the If the body is also rotating, with an result is angular speed ω around its central axis, m ()diskg ≅ − 6.2 m 0 ()diski then it acquires an additional energy This shows that the gravitational mass of equal to its rotational energy a body can also be controlled by means 1 2 ( k = 2 IE ω ). Since this is an increase of its angular velocity. in the internal energy of the body, and In order to satisfy the condition this energy is basically electromagnetic, << ΔUU , we must have << Δ dtUddtdU , we can assume that E , such as U , k where r = dtdUP is the radiation power. corresponds to an amount of By integrating this expression, we electromagnetic energy absorbed by the get = r 2 fPU . Thus we can conclude body. Thus, we can consider E as an k that, for << ΔUU , we must have increase =Δ EU in the electromagnetic k 2 << 1 IfP ω 2 , i.e., energy U absorbed by the body. r 2 2 Consequently, in this case, we must r << ω fIP replace U in Eq. (55) for ()+ ΔUU . If By dividing both members of the Δ<< UU , the Eq. (55) reduces to expression above by the area = 4πrS 2 , ⎧ ⎡ 2 ⎤⎫ we obtain ⎪ ⎛ ω 2nI ⎞ ⎪ m ⎢ 121 +−≅ ⎜ r ⎟ −1⎥ m ω 2 fI g ⎨ ⎢ ⎜ 2 ⎟ ⎥⎬ i0 D << ⎪ ⎝ 2 i0cm ⎠ ⎪ r 2 ⎩ ⎣⎢ ⎦⎥⎭ 4πr For σ << ωε , Eq. (54) shows that Therefore, this is the necessary condition in order to obtain Δ<< UU . In the case 2 r vcn == με rr and r = 4πμσ fcn in of the Mumetal disk, we must have the case of σ >> ωε . In this case, if the 25 2 Dr << 10 r ( / mwatts ) body is a Mumetal disk From Electrodynamics, we know −17 (μr = gaussat σ ×= .101.2;100000,105 mS ) that a radiation with frequency f 2 propagating within a material with with radius R , ( = 1 RmI ), the equation 2 i0 electromagnetic characteristics ε, μ and above shows that the gravitational mass σ has the amplitudes of its waves of the disk is 14

−1 attenuated by e =0.37 (37%) when it .DG −=∇ ρ penetrates a distance z, given by † ∂B 1 E −=×∇ G z = G ∂t ⎛ 2 ⎞ 1 ⎜ 1 + ()ωεσεμω − 1⎟ 2 ⎝ ⎠ .BG =∇ 0

Forσ >>ωε , equation above reduces to ∂DG 1 jH GG +−=×∇ z = ∂t fσπμ where D = 4ε ε 0 EGGrGG is the In the case of the Mumetal subjected to gravitodisplacement field ( ε rG is the an ELF radiation with frequency gravitoelectric relative permittivity of the = 1.0 Hzf , the value is = 07.1 mmz . medium; ε is the gravitoelectric Obviously, the thickness of the Mumetal 0G disk must be less than this value. permittivity for free space and G = gE Equation (55) is general for all is the gravitoelectric field intensity); ρ is types of electromagnetic fields including the density of local rest mass in the local gravitoelectromagnetic fields (See Fig. I rest frame of the matter; (b)). B = μ μ0 H GGrGG is the gravitomagnetic

Transmitter field ( μrG is the gravitomagnetic relative ELF electromagnetic radiation

permeability, μ0G is the gravitomagnetic

permeability for free space and H G is

Mumetal disk the gravitomagnetic field intensity;

= −σ Ej is the local rest-mass Motor G GG

current density in this frame (σ G is the

gravitoelectric conductivity of the Balance medium). Then, for free space we can write (a) that ⎛ GM ⎞ D gE === 444 εεε ⎜ ⎟ 0 0GGGG 0G 2 ⎝ r ⎠ Gravitoelectric Field But from the electrodynamics we know

that q εED == Acceleration 2 4πr

By analogy we can write that

M g D = G 2 Gravitomagnetic 4πr Field By comparing this expression with the

(b) previous expression of D , we get G Fig. I – (a) Experimental set-up in order to measure the 1 −− 2128 gravitational mass decreasing in the rotating Mumetal ε 0G ×== ..1098.2 mNkg disk. A sample connected to a dynamometer can measure 16πG the decreasing of gravity above the disk. (b) which is the expression of the Gravitoelectromagnetic Field. gravitoelectric permittivity for free space. The Maxwell-like equations for The gravitomagnetic permeability weak gravitational fields are [9] for free space [10,11] is 16πG −26 † μ0G . ×== 10733 kgm Quevedo, C. P. (1978) Eletromagnetismo c 2 McGraw-Hill, p.269-270. We then convert Maxwell-like equations

15 for weak gravity into a wave equation for ⎧ ⎡ 2 ⎤⎫ ⎛ ⎞ free space in the standard way. We ⎪ ⎢ ⎜ WG ⎟ ⎥⎪ mg ⎨ 121 +−= −1 ⎬mi0 conclude that the speed of Gravitational ⎢ ⎜ ρ c2 ⎟ ⎥ ⎪ ⎢ ⎝ ⎠ ⎥⎪ Waves in free space is ⎩ ⎣ ⎦⎭ 1 v = = c ⎧ ⎡ 2 ⎤⎫ ⎛ 2 ⎞ με 00 GG ⎪ ⎢ BG ⎥⎪ ⎨ 121 +−= ⎜ ⎟ − ⎬ i0 ()551 am This means that both electromagnetic ⎢ ⎜ ρμ c2 ⎟ ⎥ ⎪ ⎢ ⎝ 0G ⎠ ⎥⎪ and gravitational plane waves propagate ⎩ ⎣ ⎦⎭ at the free space with the same speed. where ρ = mi0 V . Thus, the impedance for free space is This equation shows how the E 16πG gravitational mass of a particle is altered Z G == μεμ c == G H 000 GGG c by a gravitomagnetic field. G A gravitomagnetic field, according and the Poynting-like vector is r r r to Einstein's theory of general relativity, ×= HES GG arises from moving matter (matter For a plane wave propagating in the current) just as an ordinary magnetic field arises from moving charges. The vacuum, we have = HZE GGG . Then, it Earth rotation is the source of a very follows that 2 22 weak gravitomagnetic field given by r 1 r 2 ω r 2 c ω 2 S = EG = h = h0i μ0G ⎛ Mω ⎞ −14 −1 2Z 2Z 32πG B ,EarthG −= ⎜ ⎟ ≈ 10 .srad G G 16π ⎝ r ⎠ which is the power per unit area of a Earth harmonic plane wave of angular Perhaps ultra-fast rotating stars can frequencyω . generate very strong gravitomagnetic In classical electrodynamics the fields, which can make the gravitational density of energy in an electromagnetic mass of particles inside and near the star negative. According to (55a) this will field,We , has the following expression occur if G > .061 cB 0G ρμ . Usually, W = 1 E 2 + 1 μμεε H 2 2 re 0 2 r 0 however, gravitomagnetic fields In analogy with this expression we produced by normal matter are very define the energy density in a weak. gravitoelectromagnetic field, WG , as Recently Tajmar, M. et al., [12] follows have proposed that in addition to 1 2 1 2 the London moment, BL , WG = 2 εε 0 EGGrG + 2 μμ 0 H GGrG ** −11 * For free space we obtain ( L −= ( emB )ω . ×≅ 10112 ω ; m and * μ = ε rGrG =1 e are the Cooper-pair mass and charge 2 respectively), a rotating superconductor 0G =1 με 0G c should exhibit also a large = μ0GGG cHE gravitomagnetic field, B , to explain an and G = μ HB apparent mass increase of Niobium 0 GGG Cooper-pairs discovered by Tate et Thus, we can rewrite the equation of WG al[13,14]. According to Tajmar and Matos as follows [15], in the case of coherent matter, BG 2 ⎛ 1 ⎞ ⎛ B ⎞ B2 is given by: B −= 2 λμωρ 2 where ρ W = 1 ⎜ ⎟ Bc 22 + 1 μ ⎜ G ⎟ = G G 0 grGc c G 2 ⎜ 2 ⎟ G 2 0G ⎜ ⎟ μμ ⎝ μ0Gc ⎠ ⎝ 0G ⎠ 0G is the mass density of coherent matter and λgr is the graviphoton wavelength. Since , ( is the volume of the = WU GG V V By choosing λgr proportional to the local particle) and n = 1 for free space we r density of coherent matter, ρc . i.e., can write (55) in the following form 16

By means of (55c) it is possible to 1 ⎛ gr cm ⎞ = ⎜ ⎟ = ρμ check the changes in the gravitational 2 ⎜ ⎟ 0 cG λgr ⎝ h ⎠ mass of the coherent part of a given we obtain material (e.g. the Cooper-pair fluid). Thus ⎛ 1 ⎞ for the electrons of the Cooper-pairs we 2 ⎜ ⎟ have BG −= 2 0 grGc −= 2 μωρλμωρ 0Gc ⎜ ⎟ = ⎝ 0 ρμ cG ⎠ ⎡ 2 ⎤ −= 2ω ⎛ 2 ⎞ ⎢ BG ⎥ and the graviphoton mass, m , is mm iege 112 −−+= ⎜ ⎟ mie = gr ⎢ ⎜ ρμ c 2 ⎟ ⎥ ⎢ ⎝ 0 eG ⎠ ⎥ m gr = μ 0 ρ cG h c ⎣ ⎦ 2 Note that if we take the case of no ⎡ 2 ⎤ ⎢ ⎛ 4ω ⎞ ⎥ local sources of coherent matter ()ρc = 0 , m 112 −−+= ⎜ ⎟ m = ie ⎢ ⎜ 2 ⎟ ⎥ ie the graviphoton mass will be zero. ⎝ 0 ρμ eG c ⎠ ⎣⎢ ⎦⎥ However, graviphoton will have non-zero += χ mm ieeie mass inside coherent matter ()ρc ≠ 0 . This can be interpreted as a consequence of the graviphoton gaining where ρe is the mass density of the mass inside the superconductor via the electrons. Higgs mechanism due to the breaking of In order to check the changes in gauge symmetry. the gravitational mass of neutrons and It is important to note that the protons (non-coherent part) inside the minus sign in the expression for BG superconductor, we must use Eq. (55a) 2 can be understood as due to the and BG −= 2 0 λωρμ grG [Tajmar and change from the normal to the 2 coherent state of matter, i.e., a switch Matos, op.cit.]. Due to 0 λρμ grcG = 1, between real and imaginary values that expression of BG can be rewritten in for the particles inside the material the following form when going from the normal to the 2 coherent state of matter. Consequently, BG −= 2 0 grG −= 2 ()ρρωλωρμ c in this case the variable U in (55) Thus we have must be replaced by iU and not by U ⎡ 2 ⎤ G G ⎛ B2 ⎞ ⎢ ⎜ G ⎟ ⎥ only. Thus we obtain mm ingn 12 +−= −1 min = ⎢ ⎜ ρμ c2 ⎟ ⎥ ⎧ ⎡ 2 ⎤⎫ ⎢ ⎝ 0 nG ⎠ ⎥ ⎪ ⎛ U ⎞ ⎪ ⎣ ⎦ m ⎢ 121 −−= ⎜ G ⎟ − ⎥ ()551 bmn g ⎨ ⎜ 2 r ⎟ ⎬ i0 2 ⎢ cm ⎥ ⎡ 2 2 ⎤ ⎪ ⎝ i0 ⎠ ⎪ ⎛4 ()ρρω ⎞ ⎩ ⎣ ⎦⎭ m ⎢ 12 +−= ⎜ cn ⎟ −1⎥m = in ⎢ ⎜ 2 ⎟ ⎥ in Since =WU GG V , we can write (55b) ⎝ 0 ρμ nG c ⎠ ⎣⎢ ⎦⎥ for nr = 1, in the following form −= χ mm innin ⎧ ⎡ 2 ⎤⎫ ⎪ ⎛ W ⎞ ⎪ ⎡ 2 ⎤ ⎢ ⎜ G ⎟ ⎥ 2 mg ⎨ 121 −−= −1 ⎬mi0 ⎛ B ⎞ ⎢ ⎜ ρ c 2 ⎟ ⎥ ⎢ ⎜ G ⎟ ⎥ ⎪ ⎝ c ⎠ ⎪ mm ipgp 12 +−= −1 mip = ⎩ ⎣ ⎦⎭ ⎢ ⎜ ρμ c2 ⎟ ⎥ ⎢ ⎝ 0 pG ⎠ ⎥ ⎧ ⎡ 2 ⎤⎫ ⎣ ⎦ ⎪ ⎛ B 2 ⎞ ⎪ ⎢ G ⎥ 2 ⎨ 121 −−= ⎜ ⎟ − ⎬ i0 ()551 cm ⎡ ⎤ ⎢ ⎜ 2 ⎟ ⎥ ⎛ 2 2 ⎞ ⎪ ⎝ 0 ρμ cG c ⎠ ⎪ ⎢ 4 ()ρρω cp ⎥ ⎩ ⎣ ⎦⎭ m 12 +−= ⎜ ⎟ −1 m = ip ⎢ ⎜ 2 ⎟ ⎥ ip where ρ = m V is the local density ⎢ ⎝ 0 ρμ pG c ⎠ ⎥ ic 0 ⎣ ⎦ of coherent matter. −= χ mm Note the different sign (inside ippip the square root) with respect to (55a). 17

where ρn and ρ p are the mass density Δ = − mmm igg 0 −= χ mip 0 of neutrons and protons respectively. In Tajmar’s experiment, induced By comparing this expression with accelerations fields outside the −4 mg ×≈Δ 101 mi which has been superconductor in the order of 100μg , at obtained from Tajmar’s experiment, we angular velocities of about 500 .srad −1 conclude that at angular velocities were observed. ω ≈ 500 .srad −1 we have Starting from = Gmg ()initialg r we −4 χ p ×≈ 101 can write that +Δ ( ()initialg += Δ g ) rmmGgg . From the expression of mgp we get Then we get Δ=Δ rmGg . For g 2 ⎡ 2 ⎤ ⎢ ⎛ B ⎞ ⎥ η ==Δ η ()initialg rGmgg it follows that χ 12 += ⎜ G ⎟ − 1 = p ⎢ ⎜ 2 ⎟ ⎥ . Therefore a ⎝ 0 ρμ pG c ⎠ g =Δ ηmm ()initialg = ηm i ⎣⎢ ⎦⎥ variation of =Δ ηgg corresponds to a ⎡ 2 ⎤ ⎛ 2 2 ⎞ ⎢ 4 ()ρρω cp ⎥ gravitational mass variation Δ =ηmm ig 0. 12 += ⎜ ⎟ − 1 ⎢ ⎜ 2 ⎟ ⎥ −4 ⎢ ⎝ 0 ρμ pG c ⎠ ⎥ Thus μ ×=≈Δ 101100 ggg corresponds ⎣ ⎦ to where ρ = m pp V is the mass density m ×≈Δ 101 −4 m p g i0 of the protons. On the other hand, the total In order to calculate V we need gravitational mass of a particle can be p expressed by to know the type of space (metric) inside the proton. It is known that there are just cENmNmNmNm 2 =Δ+++= pgeegppgnng 3 types of space: the space of positive

()χ ()χ mmNmmN ippippinninn +−+− curvature, the space of negative 2 curvature and the space of null ()χ pieeiee cENmmN =Δ+−+ curvature. The negative type is 2 obviously excluded since the volume of ()pieeippinn cENmNmNmN −Δ+++= the proton is finite. On the other hand, 2 ()χχχ pieeeipppinnn cENmNmNmN =Δ+++− the space of null curvature is also (χχχ )Δ+++−= cENmNmNmNm 2 excluded since the space inside the pieeeipppinnni proton is strongly curved by its enormous where ΔE is the interaction mass density. Thus we can conclude that energy; N n ,,N p N e are the number of inside the proton the space has positive neutrons, protons and electrons curvature. Consequently, the volume of the proton, , will be expressed by the respectively. Since ≅ mm ipin and Vp ρ ≅ ρ it follows that χ ≅ χ and 3-dimensional space that corresponds to pn pn a hypersphere in a 4-dimentional space, consequently the expression of mg i.e., will be the space of positive Vp reduces to curvature the volume of which is [16] 2ππ π (2 χχ ) Δ++−≅ 2 (55dcENmNmNmm ) = r 3 2 sinsin = 2πφθχθχ rddd 32 ig 0 pieeeippp V p ∫∫∫ p p 000 In the case of Earth, for example, Assuming that χ <<2 χ mNmN ipppieee and ρ Earth << ρ p . Consequently the 2 <<Δ 2 χ mNcEN Eq. (55d) reduces to p ippp curvature of the space inside the Earth is approximately null (space approximately 4 3 2 χ −=−≅ χ (55emmmNmm ) V ≅ πr . 0 ipiipppig flat). Then Earth 3 Earth or −15 For rp . ×= 1041 m we then get 18

mp ⎡ 2 ⎤ ρ ×≅= 103 16 / mkg 3 ⎛ 4ω 2 ⎞ p χ* ⎢ 112 −−= ⎜ ⎟ ⎥ . ×= 10840 −4 Vp ⎢ ⎜ * 2 ⎟ ⎥ ⎝ 0G ρμ c ⎠ Starting from the London moment ⎣⎢ ⎦⎥ it is easy to see that by precisely From this equation we then obtain measuring the magnetic field and the angular velocity of the ρ * ×≅ 103 16 / mkg 3 superconductor, one can calculate the * mass of the Cooper-pairs. This has Note that p ≅ ρρ . been done for both classical and high-Tc Now we can calculate the superconductors [17-20]. In the graviphoton mass, m , inside the experiment with the highest precision to gr date, Tate et al, op.cit., reported a Cooper-pairs fluid (coherent part of the disagreement between the theoretically superconductor) as predicted Cooper-pair mass in Niobium * * −52 of e = 02 .mm 999992 and its m = 0Ggr ρμ h c ×≅ 104 kg experimental value of . (210000841 ), Outside the coherent matter ()ρc = 0 the where me is the electron mass. This graviphoton mass will be zero anomaly was actively discussed in the literature without any apparent solution (m gr = μ 0 ρ cG h c = 0). [21-24]. Substitution ofρ , =ρρ *andω ≈ 500 .srad −1 If we consider that the apparent p c mass increase from Tate’s into the expression of χ p gives measurements results from an increase in the gravitational mass m* of the −4 g χ p ×≈ 101

Cooper-pairs due to BG , then we can write Compare this value with that one obtained from the Tajmar experiment. m m** g g == .0000841 Therefore, the decrease in the * 2me mi gravitational mass of the superconductor, *** ** expressed by (55e), is −=Δ mmm ()initialggg mm ig =−= , ≅ mm , − χ m ,SCipSCiSCg ** = .0000841 mm ii =− −4 m ,SCi −≅ 10 m ,SCi −4 *** . 10840 i =×+= χ mm i This corresponds to a decrease of the * −4 where χ . ×= 10840 . order of 10−2%in respect to the initial From (55c) we can write that gravitational mass of the superconductor. However, we must also consider the 2 gravitational shielding effect, produced ⎡ 2 ⎤ ⎛ 4ω ⎞ −2 mm ** ⎢ 112 −−+= ⎜ ⎟ ⎥m* = by this decrease of ≈ 10 % in the ig ⎢ ⎜ * 2 ⎟ ⎥ i ⎝ 0G ρμ c ⎠ gravitational mass of the particles inside ⎣⎢ ⎦⎥ the superconductor (see Fig. II). += χ mm *** Therefore, the total weight decrease in i i the superconductor will be much greater

−2 * than10 %. According to Podkletnov where ρ is the Cooper-pair mass experiment [25] it can reach up to 1% of density. the total weight of the superconductor Consequently we can write −1 at 6523 .. srad (5000rpm). In this experiment a slight decrease (up to ≈ 1% ) in the weight of samples hung above the disk (rotating at 5000rpm) was 19 observed. A smaller effect on the order acceleration, g′ , upon the particle m′g , of 10 %. has been observed when the GM disk is not rotating. The percentage of is r′ g . This means that gg +=−= 2 ˆμ weight decrease is the same for samples R of different masses and chemical in this case, the gravitational flux,φ g′ , compounds. The effect does not seem to through the particle m′ will be given by diminish with increases in elevation g above the disk. There appears to be a φ g′ = ′ = − gSSg = −φ g , i.e., it will be “shielding cylinder” over the disk that symmetric in respect to the flux when extends upwards for at least 3 meters. ′ = mm ′ (third case). In the second case No weight reduction has been observed ig 0 under the disk. (m′g ≅ 0), the intensity of the It is easy to see that the decrease gravitational force between m′ and M in the weight of samples hung above the g g disk (inside the “shielding cylinder” over will be very close to zero. This is the disk) in the Podkletnov experiment, equivalent to say that the gravity is also a consequence of the acceleration upon the particle with mass

Gravitational Shielding Effect showed in m′g will be g′ ≅ 0 . Consequently we can Fig. II. write that φ ′ = ′Sg ≅ 0 . It is easy to see In order to explain the g Gravitational Shielding Effect, we start that there is a correlation between with the gravitational field, ′ mm ig ′0 and φ′ φ gg , i.e.,

GM g r , produced by a particle g −= 2 ˆμ R _ If ′ mm ig ′0 = −1 ⇒ φ′ φ gg = −1 with gravitational mass, M g . The

_ If ′ mm ig ′0 = 1 ⇒ φ′ φ gg = 1 gravitational flux,φ g , through a spherical surface, with area S and radius R , _ If ′ mm ig ′0 ≅ 0 ⇒ φ′ φ gg ≅ 0 concentric with the mass M g , is given by r r Just a simple algebraic form contains the φ g SgdSgSdg ==== S ∫∫ S requisites mentioned above, the correlation GM g 2 = ()R = 44 ππ GM φg′ m′g R 2 g = φg mi′0 Note that the flux φ g does not depend on By making ′ mm ′ = χ we get the radius R of the surface S , i.e., it is ig 0 the same through any surface concentric ′ with the mass M g . g = φχφ g Now consider a particle with This is the expression of the gravitational gravitational mass, m′g , placed into the flux through m′ . It explains the gravitational field produced by M . g g Gravitational Shielding Effect presented According to Eq. (41), we can ‡ in Fig. II. have mm ig ′′ 0 −= 1, ′ mm ig ′0 ≅ 0 , ′ mm ig ′0 =1, As φg = gS andφ g′ = ′Sg , we obtain etc. In the first case, the gravity ′ = χ gg ‡ The quantization of the gravitational mass (Eq.(33)) shows that for n = 1 the gravitational This is the gravity acceleration inside m′g . mass is not zero but equal to mg(min).Although the Figure II (b) shows the gravitational gravitational mass of a particle is never null, shielding effect produced by two particles Eq.(41) shows that it can be turned very close to at the same direction. In this case, the zero. 20 gravity acceleration inside and above the flux on the surface α returns from O′ to 2 second particle will be χ g if g = mm i12 . O and is detected by the These particles are representative galvanometer G . That is, there is no of any material particles or material deflection for the cathodic rays. Then it substance (solid, liquid, gas, plasma, r r follows that = eEeVB y since = FF EB . electrons flux, etc.), whose gravitational mass have been reduced by the Then, we get factor χ . Thus, above the substance, the E y V = gravity acceleration g′ is reduced at the B same proportion χ = mm , and, This gives a measure of the velocity of ig 0 the electrons. consequently, ′ = χ gg , where g is the Thus, by means of the gravity acceleration below the substance. experimental set-up, shown in Fig. III, we Figure III shows an experimental can easily obtain the velocity V of the set-up in order to check the factor χ electrons below the body β , in order to above a high-speed electrons flux. As we calculate the theoretical value of χ . The have shown (Eq. 43), the gravitational experimental value of χ can be obtained mass of a particle decreases with the by dividing the weight, P′ = m g′ of increase of the velocity V of the particle. β gβ Since the theory says that the the body β for a voltage drop V~ across factor χ is given by the correlation the anode and cathode, by its mm then, in the case of an electrons ig 0 weight, Pβ = mgβ g , when the voltage flux, we will have that χ = mm iege where V~ is zero, i.e., m as function of the velocity V is P′ g′ ge χ = β = given by Eq. (43). Thus, we can write Pβ g that According to Eq. (4), the gravitational

mass, M , is defined by ⎧ ⎡ ⎤⎫ g mge ⎪ 1 ⎪ χ ⎨ −== 21 ⎢ − 1⎥⎬ m m 22 g ie ⎢ 1 − cV ⎥ M g = ⎩⎪ ⎣ ⎦⎭⎪ 22 1 − cV Therefore, if we know the velocity V of While Eq. (43) defines mg by means of the electrons we can calculate χ . ( mie is the electron mass at rest). the following expression When an electron penetrates the ⎪⎧ ⎡ 1 ⎤⎪⎫ electric field E (see Fig. III) an electric mg ⎨ −= 21 ⎢ − 1⎥⎬mi0 y 22 r r ⎩⎪ ⎣⎢ 1 − cV ⎦⎥⎭⎪ force, −= EeF , will act upon the E y In order to check the gravitational mass r electron. The direction of FE will be of the electrons it is necessary to know r the pressure P produced by the contrary to the direction of E . The y electrons flux. Thus, we have put a r piezoelectric sensor in the bottom of the magnetic force FB which acts upon the r glass tube as shown in Fig. III. The electron, due to the magnetic field B , is r r electrons flux radiated from the cathode B = eVBF ˆμ and will be opposite to FE is accelerated by the anode1 and strikes because the electron charge is negative. on the piezoelectric sensor yielding a By adjusting conveniently B we pressure P which is measured by r r means of the sensor. can make = FF . Under these B E circumstances in which the total force is zero, the spot produced by the electrons 21

g g′ < g g′ < 0

mg = mi mg < mi mg < 0

g g g

(a)

Particle 2 P2 = mg 2 g′ = mg 2 ( x g ) mg 2 g′ < g due to the gravitational g′ shielding effect produced by mg 1

mg 1 = x mi 1 ; x < 1 Particle 1

mg 1 P1 = mg 1 g = x mi 1 g g

(b)

Fig. I I – The Gravitational Shielding effect.

Let us now deduce the correlation piezoelectric sensor is the resultant of between P and M ge . all the forces Fφ produced by each When the electrons flux strikes electrons flux that passes through the sensor, the electrons transfer to it each hole of area Sφ in the grid of the a momentum == geeee VMnqnQ . anode 1, and is given by

Since =Δ= 2FdtFQ V, we conclude that ⎛αnSφ ⎞ 3 ()PSnnFF === ⎜ ⎟ VVM ~ 2 2d ⎛ F ⎞ φ φ ⎜ 2ed 2 ⎟ ge M = ⎜ ⎟ ⎝ ⎠ ge 2 ⎜ ⎟ V ⎝ ne ⎠ where n is the number of holes in the The amount of electrons, n , is given grid. By means of the piezoelectric e sensor we can measure F and by e = ρSdn where ρ is the amount of consequently obtain M ge . electrons per unit of volume We can use the equation (electrons/m3); S is the cross-section above to evaluate the magnitude of of the electrons flux and d the the force F to be measured by the distance between cathode and piezoelectric sensor. First, we will find anode. the expression of V as a function of In order to calculate n we will e V~ since the electrons speed V start from the Langmuir-Child law and ~ the Ohm vectorial law, respectively depends on the voltage V . given by We will start from Eq. (46) 3 which is the general expression for V~ 2 J = α and = ρ VJ , ()ρ = ρ e Lorentz’s force, i.e., d c c r pd rrr mg where J is the thermoionic current ()×+= BVqEq dt m − 3 i0 −− 16 2 density; α ×= 10332 ... VmA is When the force and the speed the called Child’s constant; V~ is the have the same direction Eq. (6) gives r voltage drop across the anode and pdr m Vd g cathode electrodes, and V is the = 3 dt 22 2 dt velocity of the electrons. ()1− cV By comparing the Langmuir- By comparing these expressions we Child law with the Ohm vectorial law obtain r we obtain mi0 Vd rrr 3 3 ×+= BVqEq ~ 2 αV 22 2 dt ρ = ()1 − cV ed 2V In the case of electrons accelerated Thus, we can write that by a sole electric field (B = 0), the 3 α~ 2 SV equation above gives n = e edV r r ~ r Vd Ee 22 2 Ve and a ()1 −== cV dt mie mie ⎛ 2ed 2 ⎞ M = ⎜ ⎟P Therefore, the velocity V of the ge ⎜ 3 ⎟ ⎝α VV~ 2 ⎠ electrons in the experimental set-up Where = SFP , is the pressure to be is measured by the piezoelectric 3 2 Ve~ 12 −== cVadV 22 4 sensor. () mie In the experimental set-up the From Eq. (43) we conclude that total force F acting on the 23

Dynamometer (D)

d ~ Collimators g’= χ g ↓g + Vy Grid d Collimators B O’ + α γ F y e V Ey O⋅ e

Anode 1 Filaments Cathode Anode 2 − Piezoelectric G ↓iG sensor - +

+ −

Fig. III – Experimental set-up in order to check the factor χ above a high-speed electrons flux. The set-up may also check the velocities and the gravitational masses of the electrons.

m ≅ 0 when ≅ 7450 cV . Substitution 2 ge . If we have nSφ ≅ .160 m and of this value of V into equation above = .080 md in the experimental set-up ~ givesV ≅ .1479 KV . This is the then it follows that 3 voltage drop necessary to be applied 14 ~ 2 F . ×= 10821 ge VVM across the anode and cathode ~ electrodes in order to obtain m ≅ 0. By varying V from 10KV up to 500KV ge we note that the maximum value for Since the equation above can ~ be used to evaluate the velocity V of F occurs when V ≅ .7344 KV . Under these circumstances, ≅ 70 cV and the electrons flux for a givenV~ , then . M ≅ 280 m . Thus the maximum we can use the obtained value of V to ge . ie r evaluate the intensity of B in order to value for F is F ≅ ≅ 19091 gfN produce = eEeVB in the max . y ~ experimental set-up. Then by Consequently, for Vmax = 500KV , the adjusting B we can check when the piezoelectric sensor must satisfy the electrons flux is detected by the following characteristics: galvanometer G . In this case, as we have already seen, = eEeVB y , and − Capacity 200gf the velocity of the electrons flux is − Readability 0.001gf calculated by means of the Let us now return to the expression = y BEV . Substitution of explanation for the findings of V into the expressions of m and ge Podkletnov’s experiment. Next, we M ge , respectively given by will explain the decrease of 0.1% in ⎧ ⎡ ⎤⎫ the weight of the superconductor ⎪ 1 ⎪ when the disk is only levitating but not mge ⎨ −= 21 ⎢ − 1⎥⎬mie 22 ⎩⎪ ⎣⎢ 1 − cV ⎦⎥⎭⎪ rotating. Equation (55) shows how the and gravitational mass is altered by electromagnetic fields. mge M ge = The expression of nr for 22 1− cV σ >> ωε can be obtained from (54), yields the corresponding values of in the form 2 mge and M ge which can be compared c μσc n == ()56 with the values obtained in the r v 4πf experimental set-up: Substitution of (56) into (55) leads to

⎧ 2 ⎫ = χ = ( ′ )mPPmm ⎡ ⎤ ge ie ββ ie ⎪ ⎢ μσ ⎛ U ⎞ ⎥⎪ mg ⎨ 121 +−= ⎜ ⎟ −1 ⎬mi0 2 ⎢ ⎜ ⎟ ⎥ F ⎛ 2ed ⎞ ⎪ 4πf ⎝ i cm ⎠ ⎪ M = ⎜ ⎟ ⎩ ⎣ ⎦⎭ ge ~ 3 ⎜ αnS ⎟ VV 2 ⎝ φ ⎠ This equation shows that atoms of ferromagnetic materials with very- where Pβ′ and Pβ are measured by high μ can have gravitational the dynamometer D and F is masses strongly reduced by means measured by the piezoelectric of Extremely Low Frequency (ELF) sensor. electromagnetic radiation. It also shows that atoms of superconducting 25 materials (due to very-high σ ) can other hand the transition temperature, also have its gravitational masses for high critical temperature (HTC) strongly reduced by means of ELF superconducting materials, is in the electromagnetic radiation. order of102 K . Thus (58a) gives Alternatively, we may put ⎧ ⎡ −9 ⎤⎫ Eq.(55) as a function of the power ⎪ 10~ ⎪ m ,CPfluidg ⎨ ⎢ 121 −+−= 1⎥⎬m,CPfluidi ()58b density ( or intensity ), D , of the ⎢ ρ2 ⎥ ⎩⎪ ⎣ CPfluid ⎦⎭⎪ radiation. The integration of (51) Assuming that the number of Copper- gives = vDU . Thus, we can write V pairs per unit volume is ≈ 10 mN −326 (55) in the following form: [27] we can write that ⎧ ⎡ 2 ⎤⎫ * −4 3 ⎪ ⎛ 2Dn ⎞ ⎪ ρ Nm ≈= 10 / mkg m ⎢ 121 +−= ⎜ r ⎟ −1⎥ m 57 CPfluid g ⎨ ⎜ 3 ⎟ ⎬ i0 () ⎪ ⎢ ⎝ ρc ⎠ ⎥⎪ Substitution of this value into (58b) ⎩ ⎣ ⎦⎭ yields where ρ = mi0 V . m = m − 1.0 m Forσ >> ωε , nr will be given by ,CPfluidg ,CPfluidi ,CPfluidi (56) and consequently (57) becomes ⎧ ⎡ 2 ⎤⎫ This means that the gravitational ⎪ ⎢ ⎛ μσD ⎞ ⎥⎪ masses of the electrons are mg ⎨ 121 +−= ⎜ ⎟ − ⎬mi0 ()581 ⎢ ⎝ 4 ρπ cf ⎠ ⎥ decreased of ~10%. This ⎪ ⎣ ⎦⎪ ⎩ ⎭ corresponds to a decrease in the In the case of Thermal gravitational mass of the radiation, it is common to relate superconductor given by the energy of photons to 2 m Δ+++ cEmmmN temperature, T, through the ,SCg ( ge gp gn ) = 2 = relation, m ,SCi ()inipie Δ+++ cEmmmN ≈ κThf ⎛ Δ+++ cEmmm 2 ⎞ = ⎜ ge gp gn ⎟ = where κ . 10381 −23 / °×= KJ is the ⎜ 2 ⎟ ⎝ inipie Δ+++ cEmmm ⎠ Boltzmann’s constant. On the other 2 hand it is known that ⎛ 9.0 inipie Δ+++ cEmmm ⎞ = ⎜ ⎟ = 4 ⎜ 2 ⎟ = σ BTD ⎝ inipie Δ+++ cEmmm ⎠ −8 42 where σ B . ×= 10675 / °Kmwatts = 999976.0 is the Stefan-Boltzmann’s constant. Where ΔE is the interaction energy. Thus we can rewrite (58) in the Therefore, a decrease of following form ( − . ) ≈1099997601 −5 , i.e., approximately ⎧ ⎡ 2 ⎤⎫ −3 ⎪ ⎛ μσσ hT3 ⎞ ⎪ 10 % in respect to the initial m ⎢ 121 +−= ⎜ B ⎟ − ⎥ ()581 am g ⎨ ⎢ ⎜ ⎟ ⎥⎬ i0 gravitational mass of the ⎪ ⎝ 4πκρc ⎠ ⎪ ⎩ ⎣⎢ ⎦⎥⎭ superconductor, due to the local Starting from this equation, we can thermal radiation only. However, here evaluate the effect of the thermal we must also consider the radiation upon the gravitational mass gravitational shielding effect produced, in this case, by the of the Copper-pair fluid, m ,CPfluidg . −3 Below the transition temperature,T , decrease of ≈ 10 % in the c gravitational mass of the particles (TT c < .50 ) the conductivity of the inside the superconductor (see Fig. superconducting materials is usually II). Therefore the total weight larger than 1022 / mS [26]. On the decrease in the superconductor will 26

−3 r r be much greater than ≈ 10 % . This Δ= VJ de can explain the smaller effect on the where Δ is the density of the order of 10 %. observed in the e free electric charges ( For cooper Podkletnov measurements when the conductors Δ ×= 1031 10 m/C. 3 ). disk is not rotating. e

Let us now consider an electric Therefore increasing Vd produces an current I through a conductor increase in the electric current I . subjected to electromagnetic Thus if mge is reduced 10 times radiation with power density D and ≈ 10 m.m the drift velocity V is frequency f . ( ge e ) d Under these circumstances the increased 10 times as well as the gravitational mass m of the electric current. Thus we conclude ge that strong fluxes of ELF radiation electrons of the conductor, according upon electric/electronic circuits can to Eq. (58), is given by suddenly increase the electric ⎧ ⎡ 2 ⎤⎫ currents and consequently damage ⎪ ⎢ ⎛ μσD ⎞ ⎥⎪ mge ⎨ 121 +−= ⎜ ⎟ −1 ⎬me these circuits. ⎪ ⎢ ⎝ 4 ρπ cf ⎠ ⎥⎪ ⎩ ⎣ ⎦⎭ Since the orbital electrons moment of inertia is given where ×= 10119 −31 kg.m . e by = Σ ( ) rmI 2 , where m refers to Note that if the radiation upon i i j j i the conductor has extremely-low inertial mass and not to gravitational mass, then the momentum = IL iω of frequency (ELF radiation) then mge can be strongly reduced. For the conductor orbital electrons are not affected by the ELF radiation. example, if ≈ 10−6 Hzf , ≈ 105 m/WD 2 Consequently, this radiation just and the conductor is made of copper affects the conductor’s free electrons 7 ( μ ≅ μ0;σ ×= 1085 m/S. and velocities. Similarly, in the case of ρ =8900 m/kg 3 ) then superconducting materials, the momentum, = IL ω , of the orbital ⎛ μσD ⎞ i ⎜ ⎟ ≈ 1 electrons are not affected by the ⎝ 4 ρπ cf ⎠ gravitomagnetic fields. and consequently ge ≈ 10 m.m e . r r The vector = ()V vUD , which we According to Eq. (6) the force may define from (48), has the same upon each free electron is given by r direction of the propagation vector k r and evidently corresponds to the r mge Vd r r Fe = 3 = Ee Poynting vector. Then D can be 1− cV 22 2 dt r r () replaced by E×H.Thus we can write 1 1 ( μ) === 1 []()= 1 ()1 μμ EvvEEBEEHD 2. where E is the applied electric field. 2 2 2 2 For σ >> ωε Eq. (54) tells us that Therefore, the decrease of mge = 4 fv μσπ . Consequently, we obtain produces an increase in the velocity

V of the free electrons and 1 2 σ = 2 ED consequently the drift velocity Vd is 4 fμπ also increased. It is known that the This expression refers to the density of electric current J through instantaneous values of D and E . a conductor [28] is given by The average value for E 2 is equal to 1 2 2 Em because E varies sinusoidaly 27

( Em is the maximum value for E ). penetrates the magnetic field, its Substitution of the expression of D negative inertial mass increases, but into (58) gives its total inertial mass decreases, i.e., although there is an increase of ⎧ ⎡ 3 2 ⎤⎫ ⎪ ⎛ σμ ⎞ E ⎪ inertial mass, the total inertial mass m ⎢ 121 +−= ⎜ ⎟ − ⎥ ()591 am g ⎨ 2 ⎜ ⎟ 2 ⎬ i0 (which is equivalent to gravitational ⎪ ⎢ 4c ⎝4πf ⎠ ρ ⎥⎪ ⎩ ⎣ ⎦⎭ mass) will be reduced. 2 1 2 Since rms = EE m 2 and = 2 EE m we On the other hand, Eq.(4) can write the equation above in the shows that the velocity of the body following form must increase as consequence of the gravitational mass decreasing ⎧ ⎡ 3 2 ⎤⎫ ⎪ μ ⎛ σ ⎞ E ⎪ since the momentum is conserved. m ⎢ 121 +−= ⎜ ⎟ rms − ⎥ 591 am g ⎨ 2 ⎜ ⎟ 2 ⎬ i0 () ⎢ 4c ⎝4πf ⎠ ρ ⎥ Consider for example a spacecraft ⎩⎪ ⎣ ⎦⎭⎪ with velocity Vs and gravitational Note that for extremely-low mass M . If M is reduced to m frequencies the value of f − 3 in this g g g then the velocity becomes equation becomes highly expressive. ′ = VmMV Since = vBE equation (59a) s ( ) sgg can also be put as a function of B , In addition, Eqs. 5 and 6 tell us that i.e., the inertial forces depend on mg . ⎪⎧ ⎡ ⎛ σ ⎞ B4 ⎤⎪⎫ Only in the particular case of m ⎢ 121 +−= ⎜ ⎟ − ⎥ 591 bm g ⎨ ⎜ 2 ⎟ 2 ⎬ i0 () = mm the expressions (5) and ⎪ ⎢ ⎝ 4 cf ⎠ ρμπ ⎥⎪ g i 0 ⎩ ⎣ ⎦⎭ (6) reduce to the well-known For conducting materials with Newtonian expression = amF . σ ≈ 107 m/S ; μ = 1; ρ ≈103 m/kg 3 i0 r Consequently, one can conclude that the expression (59b) gives the inertial effects on the spacecraft ⎪⎧ ⎡ ⎛ ≈ 10−12 ⎞ ⎤⎪⎫ will also be reduced due to the ⎢ ⎜ ⎟ 4 ⎥ mg ⎨ 121 +−= ⎜ ⎟ −1 ⎬mB i0 decreasing of its gravitational mass. ⎢ ⎝ f ⎠ ⎥ ⎩⎪ ⎣ ⎦⎭⎪ Obviously this leads to a new This equation shows that the concept of aerospace flight. decreasing in the gravitational mass Now consider an electric of these conductors can become current = 0 2πftsinii through a experimentally detectable for conductor. Since the current density, example, starting from 100Teslas at r r r r 10mHz. J , is expressed by == σESddiJ , One can then conclude that an then we can write that interesting situation arises when a = σ = ( 0 σ ) 2πftsinSiSiE . Substitution body penetrates a magnetic field in of this equation into (59a) gives the direction of its center. The ⎧ ⎡ 4 ⎤⎫ gravitational mass of the body ⎪ i0μ 4 ⎪ mg ⎨ ⎢ 121 +−= sin π − ⎥⎬ i0 ()5912 cmft 64 fSc 34223 σρπ decreases progressively. This is due ⎩⎪ ⎣⎢ ⎦⎥⎭⎪ to the intensity increase of the If the conductor is a supermalloy rod magnetic field upon the body while it ( × × 40011 mm) then μ = ,000100 penetrates the field. In order to r 3 6 understand this phenomenon we (initial); ρ = 8770 m/kg ;σ ×= 1061 m/S. might, based on (43), think of the and S ×= 101 − m26 . Substitution of inertial mass as being formed by two these values into the equation above parts: one positive and another yields the following expression for the negative. Thus, when the body 28 gravitational mass of the supermalloy Δ Pp 2 rod = 32 i0 cm 2ρ cv ⎧ ⎡ −12 434 ⎤⎫ m()smg ⎨ ()×+−= 0 π −12sin1071.5121 ⎬mftfi ()smi Substitution of this expression into ⎩ ⎣⎢ ⎦⎥⎭ (41) gives Some oscillators like the HP3325A 2 ⎧ ⎡ 2 ⎤⎫ (Op.002 High Voltage Output) can ⎪ ⎢ ⎛ P ⎞ ⎥⎪ m ⎨ 121 +−= ⎜ ⎟ − ⎬m ()601 generate sinusoidal voltages with g ⎢ ⎜ 32 ⎟ ⎥ i0 ⎪ ⎢ ⎝ 2ρ cv ⎠ ⎥⎪ extremely-low frequencies down to ⎩ ⎣ ⎦⎭ f ×= 101 −6 Hz and amplitude up to This expression shows that in the case of sound waves the decreasing 20V (into 50Ω load). The maximum of gravitational mass is relevant for output current is 080 A. . pp very strong pressures only.

Thus, for i0 = 0 ( 08004 A.A. pp ) It is known that in the nucleus and ×< 10252 −6 Hz.f the equation of the Earth the pressure can reach values greater than 13 2 above shows that the gravitational 10 m/N . The mass of the rod becomes negative at equation above tells us that sound waves produced by pressure πft = π 22 ; for ×≅ 1071 −6 Hz.f at variations of this magnitude can 5 ≅×== 8401047141 h.s.ft it shows cause strong decreasing of the that ()smg −≅ mm ()smi . gravitational mass at the This leads to the idea of the surroundings of the point where the Gravitational Motor. See in Fig. IV a sound waves were generated. This type of gravitational motor (Rotational obviously must cause an abrupt Gravitational Motor) based on the decreasing of the pressure at this possibility of gravity control on a place since pressure = weight /area = ferromagnetic wire. mgg/area). Consequently a local It is important to realize that instability will be produced due to the this is not the unique way of opposite internal pressure. The decreasing the gravitational mass of conclusion is that this effect may a body. It was noted earlier that the cause Earthquakes. expression (53) is general for all Consider a sphere of radius r types of waves including non- around the point where the sound electromagnetic waves like sound waves were generated (at ≈ 0001 km, waves for example. In this case, the depth; the Earth's radius is 3786 km, ). velocity v in (53) will be the speed of If the maximum pressure, at the sound in the body and D the intensity explosion place ( sphere of radius r0 ), of the sound radiation. Thus from (53) 13 2 is Pmax ≈ 10 m/N and the pressure we can write that at the distance = 10kmr is Δp VD D 2 == = ( ) ≈10 m/NPrrP 29 then we can cm cm ρcv 2 min 0 max ii consider that in the sphere It can easily be shown that 11 2 222 2 ≈= 10 m/NPPP .Thus assuming = 2 ρπ vAfD where = 2πρλ vPA ; minmax A and P are respectively the ≈103 s/mv and ρ ≈10 m/kg 33 we can amplitude and maximum pressure calculate the variation of gravitational variation of the sound wave. mass in the sphere by means of the

Therefore we readily obtain equation of mg , i.e.,

⎧ ⎡ 4 ⎤⎫ ⎪ i μ ⎪ α ⎨ ⎢ 121 +−= −1⎥⎬ ⎢ 64 fSc 34223 σρπ ⎥ ⎩⎪ ⎣ ⎦⎭⎪ Axis of the Rotor Ferromagnetic wire F = mg g = α mi g

P = mi g

~ One plate of ferromagnetic wire of the Rotor ELF current source

Several plates of ferromagnetic wire

Rotor of the Motor

Fig. IV - Rotational Gravitational Motor

r force, Fi , is given by Eq.(6), and from mmm =−=Δ ()initialgg g Eq.(13) we can obtain the r r r ⎧ ⎡ 2 ⎤⎫ gravitational force, F . Thus, ≡ FF ⎪ ⎛ P2 ⎞ ⎪ g gi m ⎢ 121 +−−= ⎜ ⎟ −1⎥ m = i0 ⎨ ⎜ 32 ⎟ ⎬ i0 leads to ⎪ ⎢ ⎝ 2ρ cv ⎠ ⎥⎪ ⎣ ⎦ mg m′g mg ⎩ ⎭ r ≡Ga ≡ 3 2 22 ⎡ 2 ⎤ 22 2 ⎛ 22 ⎞ 1− cV ⎛ P2 ⎞ ()1− cV ⎜ ′ 1− cVr ⎟ ⎢ 12 += ⎜ ⎟ − ⎥ρ ≈101 11kgV ⎝ ⎠ ⎜ 32 ⎟ ⎢ ⎝ 2ρ cv ⎠ ⎥ ⎛ m′ ⎞ m m ⎣ ⎦ ≡⎜G g ⎟ g ≡ gr g 61 ⎜ 2 ⎟ 3 3 () The transitory loss of this great ⎝ r′ ⎠()1− cV 22 2 ()1− cV 22 2 amount of gravitational mass may whence results evidently produce a strong pressure r ≡ ga r (62) variation and consequently a strong Consequently, the equivalence is Earthquake. evident, and therefore Einstein's Finally, we can evaluate the equations from the General Relativity energy necessary to generate those continue obviously valid. sound waves. From (48) we can write 16 2 The new expression for Fi max max ≈= 10 m/WvPD . Thus, the 212 (Eqs. (5) and (6)) shows that the released power is 0 max( π 0 ) ≈= 104 WrDP inertial forces are proportional to the and the energy ΔE released at the gravitational mass, mg . This means time interval Δt must be Δ = ΔtPE . 0 that these forces result from the Assuming Δ ≈10−3 st we readily obtain gravitational interaction between the 18 4 0ΔΔ tPE ≈= 10 joules ≈ 10 Megatons particle and the other gravitational This is the amount of energy released masses of the Universe, just as by an earthquake of magnitude 9 Mach’s principle predicts. Therefore ()+ 4415 M. 18 the new expression for the inertial (M=9), i.e., .E 10741 s ≅×= 10 joules. s forces incorporates the Mach’s The maximum magnitude in the principle into Gravitation Theory, and Richter scale is 12. Note that the sole furthermore reveals that the inertial releasing of this energy at 1000km effects upon a particle can be depth (without the effect of reduced because, as we have seen, gravitational mass decreasing) the gravitational mass may be cannot produce an Earthquake, since reduced. When = mm the the sound waves reach 1km depth ig 0 with pressures less than 10N/cm2. nonrelativistic equation for inertial r r Let us now return to the Theory. forces, = gi amF , reduces to r The equivalence between = amF r . This is the well-known frames of non-inertial reference and ii 0 gravitational fields assumed ≡ mm Newton's second law for motion. ig In Einstein's Special Relativity because the inertial forces were given Theory the motion of a free-particle is r r by = ii amF , while the equivalent described by means of δS = 0 [29]. r gravitational forces, by = gmF r . Now based on Eq. (1), δS = 0 will be gg given by the following expression Thus, to satisfy the equivalence rr r δ −= g δ ∫ dscmS = .0 (63) ( ≡ ga and ≡FF gi ) it was necessary which also describes the motion of that ≡ mm . Now, the inertial ig the particle inside the gravitational 31 field. Thus, Einstein's equations from number n is very large (Classical the General Relativity can be derived limit). Therefore, just under these starting fromδ ( SS gm ) =+ 0 , where circumstances the Einstein's S and S refer to the action of the equations from the General Relativity g m can be used in order to “classicalize” gravitational field and the action of the quantum theory by means of the matter, respectively [30]. approximated description of the The variations δS g and δSm can spacetime. be written as follows [31]: Later on we will show that the 3 c length dmin of Eq. (29) is given by δS = − 1 δ ik dggRgR Ω− ()64 g ∫()2 ikik ~~ 1 16πG == ( cGklkd 3 )2 ≈10−34 m ()70 1 min planck h δS −= δ ik dggT Ω− ()65 m 2c ∫ ik (See Eq. (100)). On the other hand, where R is the Ricci's tensor; g the we will find in the Eq. (129) the length ik ik scale of the initial Universe, i.e., metric tensor and Tik the matter's −14 initial ≈10 md . Thus, from the Eq. (29) energy-momentum tensor: −− 203414 we get: = initial ddn min = ≈101010 ik (PT += ε )μ μ + Pg ikkig (66) 2 this is the quantum number of the where P is the pressure and =ρε gg c spacetime at initial instant. That is now, the density of gravitational quantum number is sufficiently large energy, E g , of the particle; ρ g is then for the spacetime to be considered the density of gravitational mass of approximately “continuous” starting the particle, i.e., M at the volume from the beginning of the Universe. g Therefore Einstein's equations can be unit. used even at the Initial Universe. Substitution of (64) and (65) into Now, it is easy to conclude why δ δSS gm =+ 0 yields the attempt to quantize gravity 3 starting from the General Relativity c 1 8πG ik ()ik ik −− 4 ik dggTRgR Ωδ =− 0 was a bad theoretical strategy. 16πG ∫ 2 c Since the gravitational whence, interaction can be repulsive, besides ⎛ 1 8πG ⎞ ⎜ ikik −− TRgR ik ⎟ = ()670 attractive, such as the ⎝ 2 c4 ⎠ electromagnetic interaction, then the because the δg are arbitrary. ik graviton must have spin 1 (called Equations (67) in the following form graviphoton) and not 2. 1 8πG ikik =− TRgR ik (68) Consequently, the gravitational forces 2 c4 are also gauge forces because they or are yielded by the exchange of the k 1 k 8πG k i δi =− TRgR i . (69) 2 c4 so-called "virtual" quanta of spin 1, are the Einstein's equations from the such as the electromagnetic forces General Relativity. and the weak and strong nuclear It is known that these equations forces. are only valid if the spacetime is Let us now deduce the Entropy continuous. We have shown at the Differential Equation starting from Eq. beginning of this work that the (55). Comparison of Eqs. (55) and spacetime is not continuous it is (41) shows that r = ΔpcUn . For small quantized. However, the spacetime velocities, i.e., ( << cV ), we have can be considered approximately 2 << cmUn . Under these “continuous” when the quantum r i0 32

circumstances, the development of (∂ i ∂ ) = 2ETET Ki (77) 2 Eq. (55) in power of ( ir 0cmUn ) gives However, Eq.(18) shows that 2 2 = −EEE .Therefore Eq.(77) becomes ⎛ Un ⎞ giKi ⎜ r ⎟ mm ig 0 −= 2 mi0 ()71 ⎜ ⎟ E = ig − ()i ∂∂ TETE (78) ⎝ i0cm ⎠ In the particular case of thermal Here, we can identify the energy Ei radiation, it is usual to relate the with the free-energy of the system-F energy of the photons to the and E g with the internal energy of temperature, through the relationship the system-U. Thus we can write the −23 ν ≈ kTh where ×= 10381 KJ.k is Eq.(78) in the following form: the Boltzmann's constant. Thus, in = −TFU ()∂ ∂TF (79) that case, the energy absorbed by the This is the well-known equation of particle will be ≈= ηνη kThU , Thermodynamics. On the other hand, st where η is a particle-dependent remembering that ∂Q τ ∂+∂= U (1 absorption/emission coefficient. principle of Thermodynamics) and Therefore, Eq.(71) may be rewritten = −TSUF (80) in the following form: (Helmholtz's function), we can easily 2 2 obtain from (79), the following ⎡⎛ ηkn ⎞ T ⎤ mm −= ⎢⎜ r ⎟ ⎥m ()72 equation ig 0 c2 2 i0 ⎢⎣⎝ ⎠ mi0 ⎥⎦ ∂ = ∂τ ∂+ .STQ (81) For electrons at T=300K, we have For isolated systems,∂τ =0, we have 2 ⎛ ηkn ⎞ T 2 ∂ ∂= STQ (82) ⎜ r ⎟ ≈ 10−17 2 2 which is the well-known Entropy c ⎠⎝ me Comparing (72) with (18), we obtain Differential Equation. 2 2 Let us now consider the 1 ⎛ rηkn ⎞ T Eq.(55) in the ultra-relativistic case EKi = ⎜ ⎟ . ()73 2 ⎝ c ⎠ mi0 where the inertial energy of the 2 The derivative of E Ki with respect to particle = ii cME is much larger than 2 temperature T is its inertial energy at rest i0cm . ∂E Ki = η 2 mTckn 74 Comparison of (4) and (10) leads to ()r ()i0 () 2 ∂T Δ = i cVEp which, in the ultra- Thus, relativistic case, gives 2 2 ∂EKi ( ηkTn ) T = r 75 Δ i i ≅≅= icMcEcVEp . On the other 2 () ∂T i0cm hand, comparison of (55) and (41)

Substitution of −= EEE iiKi 0 into (75) shows that Unr = Δpc. Thus 2 2 gives r i >>≅Δ= i0cmcMpcUn . Consequently, ⎛ ∂E ∂E ⎞ ()ηkTn 2 Eq.(55) reduces to T⎜ + ii 0 ⎟ = r ()76 2 ⎝ ∂T ∂T ⎠ i0cm 2 By comparing the Eqs.(76) and (73) ig 0 −= 2 r cUnmm (83) and considering that ∂ i0 ∂TE =0 because E does not depend on T , Therefore, the action for such i 0 particle, in agreement with the Eq.(2), the Eq.(76) reduces to is 33

t2 222 complete description of the ∫ g 1 dtcVcmS =−−= t1 electromagnetic field. This means t2 2222 that from the present theory for ∫ ()+−= ri 12 dtcVccUnm =− t1 gravity we can also derive the t2 222 22 equations of the electromagnetic ∫ []i r 121 −+−−= .dtcVUncVcm ()84 t1 field. The integrant function is the Due to Δ ≅= cMpcUn 2 the Lagrangean, i.e., r i 222 22 second term on the right hand side of i0 r 121 −+−−= cVUncVcmL (85) Eq.(86) can be written as follows Starting from the Lagrangean we can ⎡( cV 22 − 24 )⎤ find the Hamiltonian of the particle, by Δpc⎢ ⎥ = 22 means of the well-known general ⎣⎢ 1− cV ⎦⎥ formula: ⎡ 22 ⎤ ()−∂∂= .LVLVH ()cV − 24 2 = ⎢ ⎥ i cM = The result is 22 ⎣⎢ 1− cV ⎦⎥ 2 ⎡ 22 ⎤ i0cm ()cV −24 QQ ′ QQ ′ H = +Unr ⎢ ⎥. ()86 Qϕ == = cV 22 ⎢ cV 22 ⎥ 4πε R 22 1− ⎣ 1− ⎦ 0 πε0 14 − cVr The second term on the right hand whence side of Eq.(86) results from the QQ ′ 22 24 cMcV 2 =− particle's interaction with the ( ) i 4πε 0 r electromagnetic field. Note the 22 similarity between the obtained The factor ( cV − 24 ) becomes Hamiltonian and the well-known equal to 2 in the ultra-relativistic case, Hamiltonian for the particle in an then it follows that electromagnetic field [32]: 2 QQ ′ 2 i cM = ()88 4πε 0 r 2 22 From (44), we know that there is a i0 1 +−= QcVcmH ϕ. (87) minimum value for M given by i in which Q is the electric charge and ()mini = mM ()mini . Eq.(43) shows that ϕ , the field's scalar potential. The and Eq.(23) gives g()min = mm i0 (min) quantity Q ϕ expresses, as we ()ming ±= cLhm max ±= 838 cdh max . know, the particle's interaction with Thus we can write the electromagnetic field in the same way as the second term on the right i()min i0 ()min ±== 83 cdhmM max (89) hand side of the Eq. (86). 2 According to (88) the value 2 ()mini cM It is therefore evident that it is is correlated to ′ 4πε = 2 4πε rQrQQ , the same quantity, expressed by ( 0 )min min 0 max different variables. i.e., Thus, we can conclude that, in Q 2 min = 2 cM 2 ()90 ultra-high energy conditions 4πε r ()mini 2 2 0 max ( ir >≅ i0cmcMUn ), the gravitational where Qmin is the minimum electric and electromagnetic fields can charge in the Universe ( therefore be described by the same equal to minimum electric charge of Hamiltonian, i.e., in these the quarks, i.e., 1 e ); r is the circumstances they are unified ! 3 max It is known that starting from maximum distance between Q and that Hamiltonian we may obtain a Q′, which should be equal to the so- 34

matter in the Universe is called "diameter",dc , of the visible 46 3 we can conclude Universe ( = 2ld where l is obtained Vn UU ≅ 10 / mparticles cc c that these particles fill all space in the from the Hubble's law for = cV , i.e., Universe, by forming a Continuous4 ~ −1 c = cl H ). Thus, from (90) we readily Universal Medium or Continuous obtain Universal Fluid (CUF), the density of which is Q = πε 24()ddhc = min 0 maxc mn iU ()min0 −27 3 ρCUF = ≅ /10 mkg 2 ~ −1 VU = ()πε0hc 96 dH max = Note that this density is much smaller 1 = 3 e ()91 than the density of the Intergalactic −26 3 whence we find Medium (ρ IGM ≅ /10 mkg ). 30 max ×= 1043 m.d The extremely-low density of the This will be the maximum "diameter" that Continuous Universal Fluid shows that its the Universe will reach. Consequently, local gravitational mass can be strongly Eq.(89) tells us that the elementary affected by electromagnetic fields quantum of matter is (including gravitoelectromagnetic fields), pressure, etc. (See Eqs. 57, 58, 59a, ±= 83 cdhm ×±= 1093 −73 kg 59b, 55a, 55c and 60). The density of i0()min max . this fluid is clearly not uniform along the Universe, since it can be strongly This is, therefore, the smallest indivisible compressed in several regions (galaxies, particle of matter. stars, blackholes, planets, etc). At the Considering that, the inertial mass normal state (free space), the mentioned of the Observable Universe is fluid is invisible. However, at super 3 53 U 0GHcM ≅= 102 kg and that its volume compressed state it can become visible by giving origin to the known matter is 4 RV 3 == 4 ππ Hc 3 ≅ 10 m 379 , U 3 U 3 ()0 since matter, as we have seen, is −− 118 quantized and consequently, formed by where H 0 ×= 1075.1 s is the Hubble an integer number of elementary constant, we can conclude that the number of these particles in the quantum of matter with mass mi ()min0 . Observable Universe is Inside the proton, for example, there are M 45 U 125 = mmn ipp min0 ≅10 elementary quanta nU ≅= 10 particles () mi ()min0 of matter at supercompressed state, with

By dividing this number by VU , we get volume nV pproton and “radius”

nU 46 3 3 −30 ≅ 10 / mparticles nR pp ≅ 10 m . V U Therefore, the solidification of the Obviously, the dimensions of the matter is just a transitory state of this smallest indivisible particle of matter Universal Fluid, which can back to the depend on its state of compression. In primitive state when the cohesion free space, for example, its volume is conditions disappear. . Consequently, its “radius” is nV UU Let us now study another aspect 3 −15 nR UU ≅10 m . of the present theory. By combination of If N particles with diameter φ fill gravity and the uncertainty principle we will derive the expression for the Casimir all space of 1m3 then Nφ 3 =1 . Thus, if force. −15 φ ≅ 10 m then the number of particles, An uncertainty Δmi in mi with this diameter, necessary to fill produces an uncertainty Δp in p and all1m3 is N ≅ 10 45 particles . Since the number of smallest indivisible particles of 4 At very small scale. 35

therefore an uncertainty Δmg in mg , ⎛ 2 ⎞ hc 2 ΔF −= ⎜ ⎟ l planck = which according to Eq.(41) , is given ⎝ π ⎠ ()Δr 4 by ⎛ π ⎞ hc ⎡⎛ 960 ⎞ 2 ⎤ 2 ⎡ ⎤ −= ⎜ ⎟ 4 ⎢⎜ 2 ⎟l planck ⎥ = ⎛ Δp ⎞ 480 Δr π ΔΔ mm ⎢ 12 +−= ⎜ ⎟ − ⎥Δm ()921 ⎝ ⎠ ()⎣⎝ ⎠ ⎦ g i ⎢ ⎜ ⎟ ⎥ i ⎝ Δ icm ⎠ π hcA ⎣ ⎦ ⎛ 0 ⎞ −= ⎜ ⎟ 4 ()97 From the uncertainty principle for ⎝ 480 ⎠ ()Δr position and momentum, we know or that the product of the uncertainties of ⎛ π 0 ⎞ hcA F0 −= ⎜ ⎟ ()98 the simultaneously measurable ⎝ 480 ⎠ r 4 values of the corresponding position which is the expression of the Casimir and momentum components is at force for AA == (960 π )l 22 . least of the magnitude order of h , 0 planck i.e., This suggests that A0 is an Δ Δ ~rp h elementary area related to the Substitution of Δ Δr~p into (92) yields existence of a minimum length h ~ d = lk what is in accordance ⎡ Δ cm 2 ⎤ min planck ⎢ ⎛ h i ⎞ ⎥ ΔΔ mm ig 12 +−= ⎜ ⎟ − Δmi ()931 with the quantization of space (29) ⎢ Δr ⎥ ⎣ ⎝ ⎠ ⎦ and which points out to the existence

Therefore if of d min . Δr << h ()94 It can be easily shown that the Δ icm minimum area related to d min is the then the expression (93) reduces to: area of an equilateral triangle of side

2h length d min ,i.e., Δmg −≅ ()95 Δrc 3 2 3 ~ 22 min ( )dA min == ( ) lk planck Note that Δm does not depend on 4 4 g On the other hand, the maximum mg . area related to d min is the area of a Consequently, the uncertainty sphere of radius d min ,i.e., ΔF in the gravitational 2 ~ 22 2 max dA min == ππ lk planck force −= ′gg rmGmF , will be given by Thus, the elementary area 2 ~ 22 0 dA == δδ AminA lk planck (99) Δ Δmm ′gg Δ −= GF 2 = must have a value between A and ()Δr min A , i.e., ⎡ 2 ⎤ hc ⎛ G ⎞ max h 3 −= ⎢ ⎥ 22 ⎜ 3 ⎟ ()96 << πδ ⎣ ()r ⎦ ()ΔΔπ r ⎝ c ⎠ 4 A The previous assumption that 22 1 A = (960 π )l shows that 3 2 −35 0 planck The amount ()h ×= 10611 m.cG ~ 2 2 δ Ak = 960 π what means that is called the Planck length, l planck ,( the ~ . << .k 91465 length scale on which quantum fluctuations of the metric of the space Therefore we conclude that ~ −34 time are expected to be of order d min lk planck ≈= 10 .m (100) unity). Thus, we can write the The − esimaln area after A is expression of ΔF as follows 0

2 2 30 =δ ( minA ) = AnndA 0 (101) we conclude that Lmax ≅ 10 m . From It can also be easily shown that the Hubble's law and (22) we have that the minimum volume related to d min is the volume of a regular tetrahedron ~ ~ ~ Vmax lH max == (dH max ) = ( 232 ) LH max of edge length d min , i.e., 2 3 2 ~ 33 Ω min (12 )d min == (12 ) lk planck ~ where ×= 1071 s.H −− 118 . Therefore we The maximum volume is the volume obtain of a sphere of radius d min , i.e., 12 ~ Vmax ≅ 10 / sm . Ω 4π d 3 == 4π lk 33 max ()3 min ()3 planck Thus, the elementary volume This is the speed upper limit imposed 3 ~ 33 0 d minV == δδΩ V lk planck must have a by the quantization of velocity (Eq. 36). It is known that the speed upper value between Ω and Ω , i.e., min max limit for real particles is equal to c . 2 4π (12 ) δ V << 3 However, also it is known that On the other hand, the − esimaln imaginary particles can have volume after Ω is velocities greater than c 0 (Tachyons). Thus, we conclude that 3 3 =δΩ ()minV = Ω0 = 321 max.n,...,,,nnnd is the speed upper limit for Vmax The existence of nmax given by imaginary particles in our ordinary (26), i.e., space-time. Later on, we will see that

max = max min = max ddLLn min = also exists a speed upper limit to the ~ imaginary particles in the imaginary . ×= 1043 30 lk ≈1064 ()planck space-time. shows that the Universe must have a Now, multiplying Eq. (98) (the finite volume whose value at the expression of F ) by n 2 we obtain present stage is 0 2 3 3 3 3 ⎛π ⎞ hcAn ⎛ π ⎞ hcA ΩΩ == ()= δδ ddddn 2FnF −== ⎜ 0 ⎟ −= 102 Up Up 0 pVminVminp 0 ⎜ ⎟ 4 ⎜ ⎟ 4 () 480 r ⎝ 480⎠ r where d is the present length scale ⎝ ⎠ p This is the general expression of the of the Universe. In addition as Casimir force. 2 4π ( 12 ) δ V << 3 we conclude that the Thus, we conclude that the Universe must have a polyhedral Casimir effect is just a gravitational space topology with volume between effect related to the uncertainty the volume of a regular tetrahedron principle. Note that Eq. (102) arises only of edge length d p and the volume of when Δmi and Δmi′ satisfy Eq.(94). If the sphere of diameter d p . only Δ m satisfies Eq.(94), i.e., A recent analysis of i astronomical data suggests not only Δ i <> h Δrc then that the Universe is finite, but also Δmg and Δm′g will be respectively that it has a dodecahedral space given by topology [33,34], what is in strong accordance with the previous Δm ≅ − 2 Δ Δ ′ ≅ Δmmandrc theoretical predictions. g h g i From (22) and (26) we have that 3 == 3.dndL Since Consequently, the expression (96) max max minmax becomes −34 64 (100) gives min ≅10 md and nmax ≅ 10 37

2 hc ⎛ ΔmG ′ ⎞ hc ⎛ Δ ′cmG ⎞ Δmi >> h Δrc ΔF = i = ⎜ i ⎟ = 3 ⎜ 2 ⎟ 3 ⎜ 4 ⎟ ()Δr ⎝ πc ⎠ ()Δr ⎝ πc ⎠ mi′ >> h ΔΔ rc hc ⎛ ΔEG ′ ⎞ In this case, Δ ≅ Δmm ig and = 3 ⎜ 4 ⎟ ()103 ()Δr ⎝ πc ⎠ Δ ′g ≅ Δmm i′ . Thus, However, from the uncertainty ΔΔ mm ′ ()2 ()ΔΔ ′ cEcE 2 Δ −= GF ii −= G = principle for energy and time we know ()Δr 2 ()Δr 2 that G ()Δt 2 G hc 1 Δ Δt~E 104 ⎛ ⎞ h ⎛ h ⎞ ⎛ ⎞ h ( ) −= ⎜ 4 ⎟ 2 −= ⎜ 3 ⎟ 2 ⎜ 22 ⎟ = c c Δtc Therefore, we can write the ⎝ ⎠ ()Δr ⎝ ⎠ ()Δr ⎝ ⎠ expression (103) in the following ⎛ 1 ⎞ hc 2 −= ⎜ ⎟ l planck = form: ⎝ 2π ⎠ ()Δr 4

hc ⎛ Gh ⎞⎛ 1 ⎞ ⎛ π ⎞ hc ⎛ 960 2 ⎞ ⎛ π 0 ⎞ hcA ΔF = ⎜ 33 ⎟⎜ ⎟ = −= ⎜ ⎟ ⎜ l planck ⎟ −= ⎜ ⎟ ()Δr ⎝ c ⎠⎝ Δπ ′ct ⎠ ⎝1920 ⎠ ()Δr 4 ⎝ π 2 ⎠ ⎝1920 ⎠ ()Δr 4 hc ⎛ 1 ⎞ whence = l 2 ⎜ ⎟ ()105 3 planck Δπ ′ct ⎛ πA ⎞ hc ()Δr ⎝ ⎠ F −= ⎜ ⎟ ()107 ⎜ ⎟ 4 From the General Relativity Theory ⎝1920⎠ r we know that −= gcdtdr 00 . If the The force will be attractive and its 2 intensity will be the fourth part of the field is weak then g 00 −−= 21 φ c and 2 22 intensity given by the first expression (1 φ ) (1−=+= crGmcdtccdtdr ). (102) for the Casimir force. For crGm 22 <<1 we obtain ≅ cdtdr . We can also use this theory to Thus, if = rddr ′ then = tddt ′ . This explain some relevant cosmological means that we can change (Δ ′ct ) by phenomena. For example, the recent discovery that the cosmic expansion (Δr) into (105). The result is of the Universe may be accelerating, hc ⎛ 1 2 ⎞ and not decelerating as many Δ F = ⎜ l planck ⎟ = ()Δ r 4 ⎝ π ⎠ cosmologists had anticipated [35]. We start from Eq. (6) which ⎛ π ⎞ hc ⎛ 480 2 ⎞ r = ⎜ ⎟ ⎜ 24 l planck ⎟ = shows that the inertial forces, F , ⎝ 480 ⎠ ()Δ r ⎝ π ⎠ i 1 4243 whose action on a particle, in the 1 A 2 0 case of force and speed with same ⎛ π A ⎞ hc = ⎜ 0 ⎟ direction, is given by 960 4 m ⎝ ⎠ ()Δ r r g r Fi = 3 a or ()1− cV 22 2 ⎛ πA0 ⎞ hc F0 = ⎜ ⎟ Substitution of mg given by (43) into ⎝ 960 ⎠ r 4 the expression above gives whence ⎛ ⎞ ⎛πA ⎞ hc r ⎜ 3 2 ⎟ r Fi = 3 − 2 i0 am F = ⎜ ⎟ ()106 ⎜ 22 2 22 ⎟ ⎜ ⎟ 4 ()1 − cV ()1 − cV ⎝960⎠ r ⎝ ⎠ Now, the Casimir force is repulsive, whence we conclude that a particle and its intensity is half of the intensity with rest inertial mass, mi0 , subjected r previously obtained (102). to a force, Fi , acquires an Consider the case when both Δmi acceleration ar given by and Δmi′ do not satisfy Eq.(94), and 38

r r Fi a = ⎛ ⎞ ⎜ 3 2 ⎟ 3 − 2 m i 0 ⎜ 22 2 22 ⎟ ⎝ ()1 − cV ()1 − cV ⎠ δ By substituting the well-known β expression of Hubble’s law for ~ ~ S C β velocity, = lHV , ( H . ×= 1071 s −− 118 is the Hubble constant) into the Photons expression of ar , we get the acceleration for any particle in the expanding Universe, i.e., Fig. V – Gravitational deflection of light about r the Sun. r Fi a = Since δ and β are very small we can ⎛ 3 2 ⎞ ⎜ ⎟ write that 3 − 2 mi0 ⎜ ~ 222 2 ~ 222 ⎟ ⎝ ()1− clH ()1− clH ⎠ Vy δ = 2β and β = Obviously, the distance l increases c with the expansion of the Universe. Then Under these circumstances, it is easy 2Vy to see that the term δ = c ⎛ ⎞ ⎜ 3 2 ⎟ Consider the motion of the 3 − 2 ~ 222 2 ~ 222 photons at some time t after it has ⎜ ()1− clH ()1− clH ⎟ ⎝ ⎠ passed the point of closest approach. decreases, increasing the acceleration of We impose Cartesian Co-ordinates the expanding Universe. with the origin at the point of closest Let us now consider the approach, the x axis pointing along its phenomenon of gravitational path and the y axis towards the Sun. deflection of light. The gravitational pull of the Sun is A distant star’s light ray, MM gpgS −= GP under the Sun’s gravitational force 2 field describes the usual central force r hyperbolic orbit. The deflection of the where M gp is the relativistic light ray is illustrated in Fig. V, with gravitational mass of the photon and the bending greatly exaggerated for a M the relativistic gravitational mass better view of the angle of deflection. gS The distance CS is the of the Sun. Thus, the component in a distance d of closest approach. The perpendicular direction is angle of deflection of the light ray,δ , MM gpgS y −= GF sin β = is shown in the Figure V and is r 2 δ = π − 2β . MM gpgS d −= G where β is the angle of the 222 + tcd tcd 222 asymptote to the hyperbole. Then, it + follows that According to Eq. (6) the expression of the force is δ tantan (π 2β ) −=−= tan 2β Fy From the Figure V we obtain ⎛ ⎞ ⎜ mgp ⎟ dVy V F = y y ⎜ 3 ⎟ β = .tan ⎜ 1− cV 22 2 ⎟ dt c ⎝ ()y ⎠ By substituting Eq. (43) into this expression, we get 39

⎛ ⎞ 4 ⎛ hf ⎞ dV m += ⎜ ⎟i ⎜ 3 2 ⎟ y gp 2 Fy = 22 − 3 M ip 3 ⎝ c ⎠ ⎜ 22 ⎟ ⎜ ()1− y cV 2 ⎟ dt ⎝ ()1− y cV ⎠ This means that the gravitational and inertial masses of the photon are For y << cV , we can write this imaginaries, and invariants with expression in the following form respect to speed of photon, i.e. = ( yipy dtdVMF ). This force acts on = mM ipip and = mM gpgp .On the other the photons for a time t causing an hand, we can write that increase in the transverse velocity 2 ⎛ hf ⎞ Fy ()realipip += mmm ip (imaginary )= ⎜ ⎟i dV = dt 3 ⎝ c 2 ⎠ y M ip and Thus the component of transverse 4 ⎛ hf ⎞ velocity acquired after passing the ()realgpgp += mmm gp (imaginary )= ⎜ ⎟i 3 ⎝ c2 ⎠ point of closest approach is This means that we must have M gp (− GMd gS ) V = dt = ()realip = mm ()realgp = 0 y M ∫ 3 ip ()+ tcd 222 2 The phenomenon of gravitational deflection of light about the Sun − GM ⎛ M gpgS ⎞ − GM ⎛ mgpgS ⎞ = ⎜ ⎟ = ⎜ ⎟ shows that the gravitational dc ⎜ M ⎟ dc ⎜ m ⎟ ⎝ ip ⎠ ⎝ ip ⎠ interaction between the Sun and the Since the angle of deflection δ photons is attractive. Thus, due to the is given by gravitational force between the Sun

2Vy and a photon can be expressed by 2βδ == 2 c −= ()mMGF gpSung (imaginary )r , where we readily obtain mgp(imaginary) is a quantity positive and V y − 22 GM ⎛ mgpgS ⎞ δ == ⎜ ⎟ imaginary, we conclude that the 2 ⎜ ⎟ c dc ⎝ mip ⎠ force F will only be attractive if the matter ()M has negative If mm ipgp = 2 , the expression above (Sung ) gives imaginary gravitational mass. The Eq. (41) shows that if the 4GM gS δ −= inertial mass of a particle is null then its 2 dc gravitational mass is given by As we know, this is the correct g = ± 2Δ cpm formula indicated by the experimental where Δp is the momentum variation due results. to the energy absorbed by the particle. If Equation (4) says that the energy of the particle is invariant, ⎧ ⎡ 2 ⎤⎫ then Δp = 0 and, consequently, its ⎪ ⎢ ⎛ Δp ⎞ ⎥⎪ m 121 +−= ⎜ ⎟ −1 m gp ⎨ ⎢ ⎜ ⎟ ⎥⎬ ip gravitational mass will also be null. This ⎪ ⎝ ip cm ⎠ ⎪ ⎩ ⎣⎢ ⎦⎥⎭ is the case of the photons, i.e., they have an invariant energy hf and a momentum Since mm ipgp = 2 then, by making =Δ hp λ into the equation above we h λ . As they cannot absorb additional get energy, the variation in the momentum will be null Δp = 0 and, therefore, their 2 ⎛ hf ⎞ ( ) m += ⎜ ⎟i ip 2 gravitational masses will also be null. 3 ⎝ c ⎠ However, if the energy of the Due to mm ipgp = 2 we get particle is not invariant (it is able to absorb energy) then the absorbed energy will transfer the amount of motion 40

(momentum) to the particle, and making the vortex (particle) gain or lose consequently its gravitational mass will mass. If real motion is what makes real be increased. This means that the mass then, by analogy, we can say that motion generates gravitational mass. imaginary mass is made by imaginary On the other hand, if the motion. This is not only a simple gravitational mass of a particle is null generalization of the process based on then its inertial mass, according to Eq. the theory of the imaginary functions, but (41), will be given by also a fundamental conclusion related to 2 Δp the concept of imaginary mass that, as it m ±= i c will be shown, provides a coherent 5 explanation for the materialization of the From Eqs. (4) and (7) we get fundamental particles, in the beginning of ⎛ Eg ⎞ ⎛ p ⎞ the Universe. p =Δ ⎜ ⎟ V =Δ 0 ΔV ⎜ 2 ⎟ ⎜ ⎟ c ⎝ c ⎠ It is known that the simultaneous ⎝ ⎠ disappearance of a pair Thus we have (electron/positron) liberates an amount of ⎛ 2p0 ⎞ 2 ⎛ p0 ⎞ 2 m ±= ⎜ ⎟ΔV and m ±= ⎜ ⎟ΔV energy, 2 0 ()realei cm , under the form of g 2 i ⎜ 2 ⎟ ⎝ c ⎠ 5 ⎝ c ⎠ two photons with frequency f , in such a Note that, like the gravitational mass, the way that inertial mass is also directly related to the motion, i.e., it is also generated by the 2 motion. 2 0 ()realei = 2hfcm Thus, we can conclude that is the Since the photon has imaginary masses motion, or rather, the velocity is what associated to it, the phenomenon of makes the two types of mass. transformation of the energy 2 cm 2 In this picture, the fundamental 0 ()realei particles can be considered as into 2hf suggests that the imaginary immaterial vortex of velocity; it is the 2 energy of the photon, mip()imaginary c , velocity of these vortexes that causes the fundamental particles to have masses. comes from the transformation of That is, there exists not matter in the imaginary energy of the electron, 2 usual sense; but just motion. Thus, the m 0 (imaginaryei )c , just as the real energy of difference between matter and energy the photon, hf , results from the just consists of the diversity of the motion transformation of real energy of the direction; rotating, closed in itself, in the electron, i.e., matter; ondulatory, with open cycle, in the energy (See Fig. VI). 2 2 Under this context, the Higgs 2m 0 ()imaginaryei + 2 0 ()realei cmc = mechanism† appears as a process, by = 2m 2 + 2hfc which the velocity of an immaterial vortex 0 ()imaginarypi can be increased or decreased by

Then, it follows that † The Standard Model is the name given to the current theory of fundamental particles and m 0 ()()imaginaryei = −mip imaginary how they interact. This theory includes: Strong interaction and a combined theory of weak and electromagnetic interaction, known as The sign (-) in the equation above, is due electroweak theory. One part of the Standard to the imaginary mass of the photon to Model is not yet well established. What causes be positive, on the contrary of the the fundamental particles to have masses? The imaginary gravitational mass of the simplest idea is called the Higgs mechanism. This matter, which is negative, as we have mechanism involves one additional particle, already seen. called the Higgs boson, and one additional force type, mediated by exchanges of this boson.

Real Particles Imaginary Particles (Tardyons) (Tachyons)

Real Inertial Mass Imaginary Inertial Mass

Non-null Null Non-null Null

* V < c c < V ≤ Vmax * * v = c v = ∞

V 0≤v

Vortex Anti-vortex Vortex Anti-vortex (Particle) (Anti-Particle) Real Photons (Particle) (Anti-particle) Imaginary Photons ( “virtual” photons )

(Real Bodies) (Real Radiation) (Imaginary Bodies) (Imaginary Radiation)

* Vmax is the speed upper limit for Tachyons with non-null imaginary inertial mass. It has been previously obtained starting from the Hubble's law and Eq.(22). The result is: ~ 12 −1 . Vmax = ( 23 ) LH max ≅ 10 .sm ** In order to communicate instantaneously the interactions at infinite distance the velocity of the quanta (“virtual” photons) must be infinity and consequently their imaginary masses must be null .

Fig. VI - Real and Imaginary Particles.

Thus, we then conclude that Thus, the electron, the neutron and the proton have respectively, m 0 ()()imaginaryei −= mip imaginary = the following masses: Electron 2 −= 2 ichf = m ×= 1011.9 −31kg 3 ()e 0 ()realei −= 2 ()λ −= 2 imich 3 e 3 0 ()realei 2 m 0 ()imei −= 0 ()realei im 3 where λ = 0 ()realeie cmh is the Broglies’ ⎧ ⎡ 2 ⎤⎫ wavelength for the electron. ⎛ ⎞ ⎪ ⎢ U ()reale ⎥⎪ By analogy, we can write for m 121 +−= ⎜ ⎟ −1 m = ()realge ⎨ ⎢ ⎜ 2 ⎟ ⎥⎬ 0 ()realei the neutron and the proton the ⎪ ⎢ ⎝ 0 ()realei cm ⎠ ⎥⎪ ⎣ ⎦ following masses: ⎩ ⎭ = χ m m −= 2 m i 0 ()realeie 0neutroni ()imaginary 3 0neutroni ()real 2 m 0 protoni ()imaginary += m 0 ()realprotoni i 3 ⎧ ⎡ 2 ⎤⎫ ⎛ ⎞ The sign (+) in the expression of ⎪ ⎢ ⎜ U ()ime ⎟ ⎥⎪ m ⎨ 121 +−= −1 ⎬m = m is due to the fact ()imge ⎢ ⎜ 2 ⎟ ⎥ 0 ()imei 0 protoni (imaginary) ⎪ ⎢ ⎝ 0 ()imei cm ⎠ ⎥⎪ ⎩ ⎣ ⎦⎭ that m 0neutroni (imaginary) and = χ m 0 ()imeie m 0 protoni (imaginary) must have contrary signs, as will be shown later on. 42

where η n , η pr and η e are the Neutron absorption factors respectively, for the −27 m 0 ()realni ×= 106747.1 kg neutrons, protons and electrons; k ×= −23 /º1038.1 KJ is the Boltzmann

2 constant; Tn , Tpr and Te are the m 0 ()imni −= 0 ()realni im 3 temperatures of the Universe, respectively when neutrons, protons ⎧ ⎡ 2 ⎤⎫ and electrons were created. ⎛ ⎞ ⎪ ⎢ ⎜ U ()realn ⎟ ⎥⎪ In the case of the electrons, it m realgn ⎨ 121 +−= −1 ⎬m 0 realni = () ⎢ ⎜ 2 ⎟ ⎥ () was previously shown that ηe ≅ 1.0 . ⎪ ⎢ ⎝ 0 ()realni cm ⎠ ⎥⎪ ⎩ ⎣ ⎦⎭ Thus, by considering that T ×≅ 102.6 31K , we get = χ m 0 ()realnin e 7 U () η eeime ikT ×== 105.8 i ⎧ ⎡ 2 ⎤⎫ It is known that the protons were ⎛ ⎞ ⎪ ⎢ ⎜ U ()imn ⎟ ⎥⎪ created at the same epoch. Thus, we m ⎨ 121 +−= −1 ⎬m = ()imgn ⎢ ⎜ 2 ⎟ ⎥ 0 ()imni will assume that ⎪ ⎢ ⎝ 0 ()imni cm ⎠ ⎥⎪ ⎩ ⎣ ⎦⎭ 7 U ()=η prprimpr ikT ×= 105.8 i = χ m 0 ()imnin Then, it follows that Proton 21 χ e ×−= 108.1 −27 m0 ()realpri ×= 106723.1 kg 17 χ pr ×−= 107.9 m += 2 im 0 ()impri 3 0 ()realpri Now, consider the gravitational forces, due to the imaginary masses of two ⎧ ⎡ 2 ⎤⎫ ⎛ ⎞ electrons, F , two protons, F , and ⎪ ⎢ ⎜ U ()realpr ⎟ ⎥⎪ ee prpr m ()realgpr ⎨ 121 +−= −1 ⎬m0 ()realpri = ⎢ ⎜ 2 ⎟ ⎥ one electron and one proton, Fepr , all at ⎪ ⎢ ⎝ 0 ()realpri cm ⎠ ⎥⎪ ⎩ ⎣ ⎦⎭ rest. =χ m 2 2 2 0 ()realpripr m − 0 realei im ()imge 2 ( 3 ()) ee −= GF −= Gχ e = ⎧ ⎡ 2 ⎤⎫ 2 2 ⎛ ⎞ r r ⎪ ⎢ U ()impr ⎥⎪ m 121 +−= ⎜ ⎟ −1 m = m2 −28 ()imgpr ⎨ ⎢ 2 ⎥⎬ 0 ()impri 4 2 0 ()realei ×+ 103.2 ⎜ cm ⎟ += Gχ = repulsion)( ⎪ ⎢ ⎝ 0 ()impri ⎠ ⎥⎪ e 2 2 ⎩ ⎣ ⎦⎭ 3 r r

=χ m 0 ()impripr 2 ⎛ 2 ⎞ 2 ⎜+ 0 ()realpri im ⎟ m ()imgpr 2 ⎝ 3 ⎠ prpr −= GF −= Gχpr = where U (real) and U ()im are r2 r2 respectively, the real and imaginary 2 −28 4 2 m 0 ()realpri ×+ 103.2 energies absorbed by the particles. += Gχpr = repulsion)( 3 r2 r2 When neutrons, protons and electrons were created after the Big- ()mm ()imgprimge bang, they absorbed quantities of −= GF = epr 2 electromagnetic energy, respectively r given by ⎛ 2 ⎞ ⎛ 2 ⎞ ⎜− 0 ()realei ⎟ ⎜+ 0 ()realpri imim ⎟ ⎝ 3 ⎠ ⎝ 3 ⎠ U ()=η UkT ()imaginarynnnrealn =η nn ikT −= G χχ pre = r2 U ()realpr =η UkT prprpr ()imaginary =η prpr ikT −28 4 ()mm 00 ()realprirealei ×− 103.2 −= G χχ = pre 2 2 U ()=η UkT ()imaginaryeeereale =η ee ikT 3 r r atraction)(

Note that charge of the neutron is null. Thus, it is e2 ×103.2 −28 necessary to assume that Felectric == 4πε 2 rr 2 0 qqq −+ =+= 4πε mG + i + Therefore, we can conclude that nnn 0 gn()imaginary e2 − FFF +=≡= repulsion)( + 4πε0 mG gn()imaginary i = ee prpr electric 2 4πε0r ⎛ + ⎞ = 4 ⎜χπε mG i⎟+ and 0 ⎝ n 0 ()imaginaryni ⎠ e2 FF −=≡ atraction)( − ep electric 2 + 4 0 (χπε mG 0 ()imaginarynin i)= 4πε0r 2 2 2 2 These correlations permit to define the = 4 0 []χπε (+ 0 imG ) χ (−+ 0ninnin im ) = 0 electric charge by means of the 3 3 following relation: We then conclude that in the neutron,

half of the total amount of elementary q = 4πε 0 mG g imaginary i () quanta of electric charge,,qmin is negative, while the other half is positive. For example, in the case of the In order to obtain the value of electron, we have the elementary quantum of electric charge, qmin , we start with the qe = 4πε0 mG ge()imaginary i = expression obtained here for the electric charge, where we = 4 χπε mG i = 0 ()0 ()imaginaryeie change m (imaginaryg )by its quantized 2 2 4 −= χπε imG = expression 2 0 ( 3 0 ()realeie ) m ()imaginaryg = mn i0 ()(imaginary min), 2 −19 derived from Eq. (44a). Thus, we get = 4 0 χπε mG 0 ()realeie ×−= 106.1 C ( 3 ) In the case of the proton, we get q = 4πε 0 mG g ()imaginary i =

2 q pr = 4πε0 mG gpr()imaginary i = = 4πε 0 mnG i 0 ()()imaginary min i = = 4πε 2 ± 2 iimnG = = 4 0 ()χπε mG 0 pripr ()imaginary i = 0 []( 3 i ()min0 ) 2 2 2 4 0G(+= χπε 0 ()realpripr im )= = 2 4πε mnG 3 m 3 0 i ()min0 2 −19 4 0G(−= χπε m 0 ()realpripr ) ×+= 106.1 C 3 This is the quantized expression of the electric charge. For the neutron, it follows that For n =1 we obtain the value of qn = 4πε 0 mG gn()imaginary i = the elementary quantum of electric charge, q , i.e., = 4 χπε mG i = min 0 (0 ()imaginarynin ) 2 2 2 −83 = 4 0G (− χπε 0 ()realnin im )= q = 4πε mG ×= 108.3 C 3 min m 3 i ()min00 m 2 = 4 0 (χπε mG 0 ()realnin ) 3 where m is the elementary However, based on the quantization of i (min0 ) the mass (Eq. 44), we can write that quantum of matter, whose value previously calculated, is χ 2 = 2mnm n ≠ 0 3 0 ()realnin i ()min0 −73 mi ()min0 ×±= 109.3 kg. Since n can have only discrete values The existence of imaginary mass different of zero (See Appendix B), we associated to a real particle suggests conclude that χ n cannot be null. the possible existence of imaginary However, it is known that the electric 44 particles with imaginary masses in the whole space must be finite − Nature. inasmuch as the particle is somewhere. In this case, the concept of wave On the other hand, if associated to a particle (De Broglie’s +∞ 2d =Ψ 0 waves) would also be applied to the ∫ ∞− V imaginary particles. Then, by analogy, the interpretation is that the particle will the imaginary wave associated to an not exist. However, if imaginary particle with imaginary +∞ 2 dV ∞=Ψ ()108 masses miψ and mgψ would be ∫ ∞− described by the following expressions The particle will be everywhere r r simultaneously. = hkp ψψ In Quantum Mechanics, the wave function Ψ corresponds, as we know, Eψ = hωψ to the displacement y of the Henceforth, for the sake of simplicity, we will use the Greek letterψ to stand undulatory motion of a rope. However, Ψ , as opposed to y , is not a for the word imaginary; pr is the ψ measurable quantity and can, hence, momentum carried by the ψ wave and be a complex quantity. For this reason, r it is assumed that Ψ is described in the Eψ its energy; kψ = 2 λπ ψ is the − directionx by propagation number and λψ the −(2π )()− pxEthi Ψ=Ψ 0e wavelength of the ψ wave; ωψ = 2πfψ This is the expression of the wave is the cyclical frequency. function for a free particle, with total According to Eq. (4), the energy E and momentum pr , moving in r momentum pψ is the direction + x . r r = VMp As to the imaginary particle, the ψ gψ imaginary particle wave function will be where V is the velocity of the ψ denoted by Ψψ and, by analogy the particle. expression of Ψ , will be expressed by: By comparing the expressions of r pψ we get −(2π )( − ψψ xptEhi ) Ψ=Ψ 0ψψ e h λψ = gψ VM Therefore, the general expression of It is known that the variable the wave function for a free particle can quantity which characterizes the De be written in the following form Broglie’s waves is called wave function, −(2π )( − xptEhi ) usually indicated by symbol Ψ . The ()real ()real Ψ=Ψ 0()real e + wave function associated with a −()2π ()− ψψ xptEhi material particle describes the dynamic Ψ+ 0ψ e state of the particle: its value at a particular point x, y, z, t is related to the It is known that the uncertainty probability of finding the particle in that principle can also be written as a place and instant. Although Ψ does function of ΔE (uncertainty in the not have a physical interpretation, its energy) and Δt (uncertainty in the 2 * square ΨΨ (or Ψ ) calculated for a time), i.e., particular point x, y, z, t is proportional to the probability of finding the particle Δ ΔtE ≥ in that place and instant. . h 2 Since Ψ is proportional to the This expression shows that a probability P of finding the particle variation of energy ΔE , during a described by Ψ , the integral of Ψ 2 on 45 time interval Δt , can only be ⎧ ⎡ 2 ⎤⎫ χ ⎨ 121 ()Δ+−= i0cmp − 1 ⎬ detected if Δ ≥ h ΔEt . Consequently, ⎩ ⎣⎢ ⎦⎥⎭ a variation of energy ΔE , during a time interval Δ < h ΔEt , cannot be Since the condition to make experimentally detected. This is a the particle imaginary is limitation imposed by Nature and not by our equipments. λg λ < Thus, a quantum of energy i 2π =Δ hfE that varies during a time and interval =Δ 1 = λ h Δ< Ecft (wave period) cannot be experimentally λg λ hh === i detected. This is an imaginary 2π g χ i cMcM 2πχ photon or a “virtual” photon. Now, consider a particle with 2 Then we get energy g cM . The DeBroglie’s gravitational and inertial wavelengths 1 χ =< 159.0 are respectively λg = g cMh and 2π λ = cMh . In Quantum Mechanics, i i However, χ can be positive or particles of matter and quanta of negative ( or ).This radiation are described by means of χ < + 159.0 χ −> .0 159 wave packet (DeBroglie’s waves) means that when with average wavelength λ . i − 1590 < χ +< 1590 Therefore, we can say that during a . . time interval =Δ λ ct , a quantum of i the particle becomes imaginary. 2 energy =Δ gcME varies. According Under these circumstances, we can to the uncertainty principle, the say that the particle made a particle will be detected if ≥Δ h ΔEt , transition to the imaginary space- time. i.e., if λ ≥ cMc 2 or λ ≥ λ 2π . i h g gi Note that, when a particle This condition is usually satisfied becomes imaginary, its gravitational when = MM ig . In this case, λ = λig and inertial masses also become imaginary. However, the factor and obviously, λ > λii 2π . However, χ = M g ()(imaginary M imaginaryi ) remains when M g decreases λg increases real because and λ 2π can become bigger than g

λi , making the particle non- M iM M χ = g ()imaginary g g === real detectable or imaginary. M ()imaginaryi i iM M i According to Eqs. (7) and (41) we can write M in the following g form:

m χ m M = g = i = χM g i 1− 22 1− cVcV 22 where

Body χ Ordinary Space-time 0 ≤ < cV +0.159 0 ≤ V ≤ ∞

Imaginary Body

0 Imaginary Space-time

“Virtual” Photons (V = ∞ ) −0.159 Ordinary Space-time

Real Photons ( = cV )

Fig. VII – Travel in the imaginary space-time. Similarly to the “virtual” photons, imaginary bodies can have infinite speed in the imaginary space-time.

Real particle Real particle Δt1 ΔE

Δt2 c (speed upper limit)

Δt3

(a)

Imaginary Space-time Ordinary Space-time

Δt3

Δt2 Vmax (speed upper limit)

ΔE Δt1 Imaginary particle Imaginary particle (b) Fig. VIII – “Virtual” Transitions – (a) “Virtual” Transitions of a real particle to the imaginary space-time. The speed upper limit for real particle in the imaginary space-time is c. (b) - “Virtual” Transitions of an imaginary particle to the ordinary space-time. The speed upper limit for imaginary particle in the ordinary space-time is 10 smV −112 max ≈ .

Note that to occur a “virtual” transition it is necessary thatΔt=Δt1+ Δt2+ Δt3 <ℏ/ΔE Thus, even at principle, it will be impossible to determine any variation of energy in the particle (uncertainty principle).

Thus, if the gravitational mass of the mg()imaginary particle is reduced by means of the M g()imaginary = = 1− cV 22 absorption of an amount of im im electromagnetic energy U , for = g = g example, we have cVi 22 −1 cV 22 −1 This expression shows that M g ⎧ 2 ⎫ χ ⎡ 121 +−== cmU 2 −1⎤ imaginary particles can have ⎨ ⎢ ()i0 ⎥⎬ Mi ⎩ ⎣ ⎦⎭ velocities V greater than c in our ordinary space-time (Tachyons). This shows that the energyU of the The quantization of velocity (Eq. 36) electromagnetic field remains acting shows that there is a speed upper limit . As we have already on the imaginary particle. In max > cV practice, this means that calculated previously, 10 smV −112 , max ≈ . electromagnetic fields act on (Eq.102). imaginary particles. Note that this is the speed The gravity acceleration on a upper limit for imaginary particles in imaginary particle (due to the rest of our ordinary space-time not in the the imaginary Universe) are given imaginary space-time (Fig.7) by because the infinite speed of the “virtual” quanta of the interactions ′j = χ j jgg = 321 n.,...,,, shows that imaginary particles can have infinite speed in the imaginary space-time. where χ = M g ()(imaginary M imaginaryi ) While the speed upper limit and −= Gmg r 2 . Thus, j gj()imaginary j for imaginary particles in the the gravitational forces acting on the ordinary space-time is −112 particle are given by V ≈10 .sm , the speed upper limit max for real particles in the imaginary = gMF ′ = gj g()imaginary j space-time is c , because the 2 = M g()imaginary (− χGmgj ()imaginary rj )= relativistic expression of the mass shows that the velocity of real −= χ riGmiM 2 += χ rmGM 2 g ()jgj jgjg . particles cannot be larger than c in any space-time. The uncertainty Note that these forces are real. principle permits that particles make Remind that, the Mach’s principle “virtual” transitions, during a time says that the inertial effects upon a interval Δt , if Δ h Δ< Et . The particle are consequence of the “virtual” transition of mesons emitted gravitational interaction of the from nucleons that do not change of particle with the rest of the Universe. mass, during a time interval Then we can conclude that the <Δ cmt 2 , is a well-known inertial forces upon an imaginary h π particle are also real. example of “virtual” transition of Equation (7) shows that , in particles. During a “virtual” the case of imaginary particles, the transition of a real particle, the relativistic mass is speed upper limit in the imaginary space-time is c , while the speed upper limit for an imaginary particle 49 in the our ordinary space-time is of the bodies ; ≈10 smV −112 . (See Fig. 8). 22 max . ii 22 1−= 2 cVmM and mi2 is the There is a crucial total inertial mass of the bodies of cosmological problem to be solved: region 2. the problem of the hidden mass. Now consider that from Most theories predict that the Eq.(7), we can write amount of known matter, detectable 2 g g cME 2 and available in the universe, is only ξ === ρ g c about 1/100 to 1/10 of the amount VV needed to close the universe. That where ξ is the energy density of is, to achieve the density sufficient matter. to close-up the universe by Note that the expression of ξ maintaining the gravitational only reduces to the well-known curvature (escape velocity equal to expression ρc 2 , where ρ is the the speed of light) at the outer sum of the inertial masses per boundary. volume unit, when = mm . Eq. (43) may solve this ig problem. We will start by substituting Therefore, in the derivation of the the expression of Hubble's law for well-known difference ~ 8πGρ ~ velocity, V = lH , into Eq.(43). The U − H 2 expression obtained shows that 3 particles which are at distances which gives the sign of the curvature ~ 26 of the Universe [36], we must use 0 == ( )()cll ×= 103135 m.H have = ρξ c 2 instead of = ρξ c 2 .The quasi null gravitational mass gU U result obviously is = mm (mingg ); beyond this distance, 8πGρgU ~ the particles have negative − H 2 ()109 gravitational mass. Therefore, there 3 are two well-defined regions in the where Universe; the region of the bodies gU g1 + MMM g 2 ρ == 110 with positive gravitational masses gU () VU VU and the region of the bodies with M and are respectively the negative gravitational mass. The gU VU total gravitational mass of the first total gravitational mass and the region, in accordance with Eq.(45), volume of the Universe. will be given by Substitution of M g1 and M g 2

mi1 into expression (110) gives MM ig 11 =≅ ≅ mi1 1− cV 22 ⎡ ⎤ 1 ⎢ 3 2 ⎥ miU + − −mm ii 22 where m is the total inertial mass ⎢ 22 1− cV 22 ⎥ i1 ⎣ 1− 2 cV 2 ⎦ of the bodies of the mentioned ρgU = VU region; << cV is the average 1 velocity of the bodies at region 1. where = + mmm is the total The total gravitational mass of the iiiU 21 inertial mass of the Universe. second region is The volume V1 of the region 1 ⎛ 1 ⎞ ⎜ ⎟ and the volume 2 of the region 2, M g 2 −= 21 − 1 M i2 V ⎜ 22 ⎟ are respectively given by ⎝ 1 − 2 cV ⎠ where V2 is the average velocity 50

2 3 32 the anomalies in the spectral red- V 1 = 2π 0 andl V 2 2π lc −= V 1 shift of certain galaxies and stars. ~ Several observers have where cl ×== 1081 26 m.H is the c noticed red-shift values that cannot so-called "radius" of the visible be explained by the Doppler- Universe. Moreover, ρ = mii V111 and Fizeau effect or by the Einstein effect (the gravitational spectrum ρ = mii V222 . Due to the hypothesis shift, supplied by Einstein's theory). of the uniform distribution of matter in the space, it follows that This is the case of the so- called Stefan's quintet (a set of five ρ =ρ .Thus, we can write ii 21 galaxies which were discovered in 3 m ⎛ l ⎞ 1877), whose galaxies are located i1 V1 == ⎜ 0 ⎟ = .380 m ⎜ l ⎟ at approximately the same distance i2 V 2 ⎝ c ⎠ from the Earth, according to very Similarly, reliable and precise measuring mmm iU 2 == ii 1 methods. But, when the velocities of U 2 VVV 1 the galaxies are measured by its Therefore, red-shifts, the velocity of one of 3 them is much larger than the ⎡ ⎛ l ⎞ ⎤ V 2 mm ⎢1 −== ⎜ 0 ⎟ ⎥ = 620 m.m velocity of the others. i2 iU ⎜ l ⎟ iU iU VU ⎣⎢ ⎝ c ⎠ ⎦⎥ Similar observations have and = 380 m.m . been made on the Virgo i1 iU constellation and spiral galaxies. Substitution of mi2 into the Also the Sun presents a red-shift expression of ρ gU yields greater than the predicted value by the Einstein effect. It seems that some of these .861 .241 anomalies can be explained if we miU + − − .620 miU 22 1− cV 22 consider the Eq.(45) in the 1− 2 cV 2 ρgU = calculation of the gravitational mass VU of the point of emission. The expression of the

Due to 2 ≅ cV , we conclude that the gravitational spectrum shift was term between bracket is much larger previously obtained in this work. It is the same supplied by Einstein's than 10miU . The amount miU is the mass of matter in the universe (1/10 theory [37], and is given by to 1/100 of the amount needed to φ −φ12 ωωω 21 =−=Δ ω0 = close the Universe). c2 Consequently, the total g +− g rGmrGm 1122 mass = ω0 ()111 c2

.861 .241 where ω1 is the frequency of the miU + − − .620 miU 22 1− cV 22 light at the point of emission ; ω is 1− 2 cV 2 2 the frequency at the point of must be sufficient to close observation; φ1 and φ2 are the Universe. respectively, the Newtonian There is another cosmological gravitational potentials at the point problem to be solved: the problem of of emission and at the point of observation. 51

In Einstein theory, this been fully defined as yet, but it is expression has been deduced from known that it is located between

−= gtT 00 [38] which correlates 1.8M~ and 2.4M~. Thus, if the mass of the star exceeds 2.4M , the own time (real time),t , with the ~ temporal coordinate x0 of the space- contraction will continue. 0 According to Hawking [40] time ( = cxt ). collapsed objects cannot have mass When the gravitational field is less than ×= 10114 −8 kg.Gc . This weak, the temporal component g h 00 means that, with the progressing of of the metric tensor is given by the compression, the neutrons −−= 21 φ c/g 2[39].Thus, we readily oo cluster must become a cluster of obtain superparticles where the minimal 2 inertial mass of the superparticle is −= 21 g rcGmtT ()112

This is the same equation that we ×= 1011 −8 .kg.m 113 have obtained previously in this ()spi ( ) work. Curiously, this equation tell us Symmetry is a fundamental attribute of the Universe that that we can have < tT when mg > 0 enables an investigator to study ; and > tT for m < 0 . In addition, if g particular aspects of physical 2 2 g = 2Grcm , i.e., if = 2 g cGmr systems by themselves. For (Schwarzschild radius) we obtain T=0. example, the assumption that space Let us now consider the well- is homogeneous and isotropic is known process of stars' gravitational based on Symmetry Principle. Also contraction. It is known that the here, by symmetry, we can assume destination of the star is directly that there are only superparticles −8 correlated to its mass. If the star's with mass ()spi ×= 1011 kg.m in the mass is less than 1.4M~ cluster of superparticles. (Schemberg-Chandrasekhar's limit), Based on the mass-energy of it becomes a white dwarf. If its mass the superparticles ( ~1018 GeV ) we exceeds that limit, the pressure can say that they belong to a produced by the degenerate state of putative class of particles with mass- the matter no longer energy beyond the supermassive counterbalances the gravitational Higgs bosons ( the so-called X pressure, and the star's contraction bosons). It is known that the GUT's continues. Afterwards there occurs theories predict an entirely new the reactions between protons and force mediated by a new type of electrons (capture of electrons), boson, called simply X (or X boson where neutrons and anti-neutrinos ). The X bosons carry both are produced. electromagnetic and color charge, in The contraction continues order to ensure proper conservation until the system regains stability of those charges in any interactions. (when the pressure produced by the The X bosons must be extremely neutrons is sufficient to stop the massive, with mass-energy in the gravitational collapse). Such systems unification range of about 1016 GeV. are called neutron stars. If we assume the There is also a critical mass superparticles are not hypermassive for the stable configuration of Higgs bosons then the possibility of neutron stars. This limit has not the neutrons cluster become a 52

Higgs bosons cluster before the superparticles' relativistic inertial becoming a superparticles cluster mass M ()spi is must be considered. On the other hand, the fact that superparticles Un η kTn −8 M rr ≈=≅ 10 kg ()115 must be so massive also means that ()spi c2 c2 it is not possible to create them in Comparing with the superparticles' any conceivable particle accelerator inertial mass at rest (113), we that could be built. They can exist as conclude that free particles only at a very early stage of the Big Bang from which the universe emerged. −8 () ()spispi ×=≈ 1011 kg.mM (116) Let us now imagine the

Universe coming back to the past. There will be an instant in which it From Eqs.(83) and (115), we obtain will be similar to a neutrons cluster, the superparticle's gravitational such as the stars at the final state of mass at rest: gravitational contraction. Thus, with = − 2Mmm ≅ the progressing of the compression, () ()spispg ()spi η kTn the neutrons cluster becomes a M −≅−≅ r ()117 superparticles cluster. Obviously, ()spi c 2 this only can occur before 10-23s Consequently, the superparticle's (after the Big-Bang). relativistic gravitational mass, is The temperature T of the m Universe at the 10-43s< t < 10-23s ()spg M ()spg = = period can be calculated by means 1− cV 22 of the well-known expression[41]: η kTn = r ()118 − 1 2 22 ≈ 22 ()tT 1010 −23 2 ()114 1− cVc Thus at ≅ 10 −43 st (at the first Thus, the gravitational forces spontaneous breaking of symmetry) between two superparticles , 32 according to (13), is given by: the temperature was ≈ 10 KT 19 ' (∼10 GeV).Therefore, we can rr ()MM ()spgspg −=−= GFF μˆ = assume that the absorbed 12 21 r 2 21 electromagnetic energy by each 2 −43 ⎡⎛ M ⎞ ⎤ superparticle, before ≅ 10 st , was ⎢⎜ ()spi ⎟ ⎛ G ⎞ 2 ⎥ hc = ⎜ ⎟()r Tn μκη ˆ 21 ()119 9 ⎢⎜ m ⎟ ⎝c5 ⎠ ⎥ r 2 η ×>= 101 JkTU (see Eqs.(71) and ⎣⎝ ()spi ⎠ h ⎦ (72)). By comparing with 2 ×≅ 109 8 Jcm , we conclude that ()spi Due to the unification of the > ()spi cmU . Therefore, the gravitational and electromagnetic 22 interactions at that period, we have unification condition ( >≅ iir cmcMUn ) is satisfied. This means that, before ≅ 10 −43 st ,the gravitational and electromagnetic interactions were unified. From the unification condition 2 ( ≅ ir cMUn ), we may conclude that 53

' component of its position at that rr ()MM ()spgspg 12 21 =−= GFF μˆ 21 = time ,i.e., a particle cannot be r 2 precisely located in a particular ⎡ 2 ⎤ ⎛ M ()spi ⎞ ⎛ G ⎞ 2 c direction without loss of all = ⎢⎜ ⎟ ⎜ ⎟()T ⎥ h μηκ ˆ = ⎜ ⎟ 5 2 21 knowledge of its momentum ⎢ m ()spi ⎝ c h⎠ ⎥ r ⎣⎝ ⎠ ⎦ component in that direction . This e2 means that in intermediate cases = ()120 2 the product of the uncertainties of 4πε0r From the equation above we can the simultaneously measurable write values of corresponding position and momentum components is at 2 ⎡ ⎤ 2 least of the magnitude order of h , ⎛ M spi ⎞ G e ⎢⎜ ()⎟ ⎛ ⎞ 2 ⎥ ⎜ 5 ⎟()ηκ hcT = ()121 ⎢⎜ m ⎟ ⎝c ⎠ ⎥ 4πε ⎣⎝ ()spi ⎠ h ⎦ 0 Δ Δr.p ≥ h (127) Now assuming that 2 This relation, directly obtained here ⎛ M ()spi ⎞ ⎛ G ⎞ 2 ⎜ ⎟ ⎜ ⎟()ηκT =ψ ()122 from the Unified Theory, is the well- ⎜ ⎟ 5 ⎝ m ()spi ⎠ ⎝ c h ⎠ known relation of the Uncertainty the Eq. (121) can be rewritten in the Principle for position and following form: momentum. According to Eq.(83), the e2 1 ψ == ()123 gravitational mass of the 4πε0hc 137 superparticles at the center of the which is the well-known reciprocal cluster becomes negative when fine structure constant. 2 2η r > mckTn ()spi , i.e., when For = 1032 KT the Eq.(122) gives 2 ()spi cm 32 TT critical ≈=> K.10 2 2η rkn M ()spi ⎞⎛ G ⎞⎛ 2 1 ψ = ⎜ ⎟ ⎜ ⎟()κη Tn ≈ ()124 ⎜ m ⎟ c 5 r 100 ⎝ ()spi ⎠ ⎝ h ⎠ According to Eq. (114) this temperature corresponds tot ≈10−43s. This value has the same order of c magnitude as the exact value(1/137) With the progressing of the of the reciprocal fine structure compression, more superparticles constant. into the center will have negative From equation (120) we can gravitational mass. Consequently, write: there will be a critical point in which the repulsive gravitational forces ⎛ MM ' ⎞ ⎜G () ()spgspg ⎟r = 125 between the superparticles with ⎜ r ⎟ h () ⎝ ψ rc ⎠ negative gravitational masses and The term between parentheses has the superparticles with positive the same dimensions as the linear gravitational masses will be so momentum pr . Thus, (125) tells us strong that an explosion will occur. that This is the event that we call the Big r r Bang. ⋅ =h.rp (126) A component of the momentum of a Now, starting from the Big particle cannot be precisely Bang to the present time. specified without loss of all Immediately after the Big Bang, the knowledge of the corresponding superparticles' decompression 54 begins. The gravitational mass of −43 during a period of time c ≈10 st the most central superparticle will .Thus, only be positive when the GM temperature becomes smaller than 1 2 ()Ui 2 cr initial 2 ()tgdd c ==− ( χ ) ()tc (130 ) 32 dd initialcr the critical temperature,Tcritical ≈10 K . At the maximum state of The Eq.(83), gives compression (exactly at the Big m ()spg 2Un 2η kTn Bang) the volumes of the χ 1 r 1−=−== r m 2 2 superparticles were equal to ()spi m ()spi c ()spi cm 3 the elementary volume Ω 0 = δV d min and the volume of the Universe was Calculations by Carr, B.J =δΩ ()nd 3 =δ d 3 where d [41], indicate that it would seem minV initialV initial reasonable to suppose that the was the initial length scale of the fraction of initial primordial black Universe. At this very moment the hole mass ultimately converted into average density of the Universe was photons is about 11.0 . This means equal to the average density of the that we can take superparticles, thus we can write η = 11.0 3 2 ⎛d ⎞ M Thus, the amount η iU cM , ⎜ initial⎟ = ()Ui ()128 ⎜ ⎟ where M is the total inertial mass ⎝ dmin ⎠ m()spi iU of the Universe, expresses the total where M ≈ 10 53 kg is the inertial ()Ui amount of inertial energy converted mass of the Universe. It has already into photons at the initial instant of been shown that the Universe(Primordial Photons). ~ −34 dmin kl planck ≈= 10 .m Then, from It was previously shown that Eq.(128), we obtain: photons and also the matter have imaginary gravitational masses −14 dinitial ≈ 10 m (129) associated to them. The matter has After the Big Bang the negative imaginary gravitational mass, while the photons have Universe expands itself from d initial positive imaginary gravitational up to d (when the temperature cr mass, given by decrease reaches the critical 32 temperatureTcritical ≈ 10 K , and the 4 ⎛ hf ⎞ M gp()imaginary = 2M ip ()imaginary += ⎜ ⎟i gravity becomes attractive). Thus, it 3 ⎝ c 2 ⎠ expands by cr − dd initial , under effect of the repulsive gravity where M ip imaginary)( is the imaginary inertial mass of the photons. ggg minmax == Then, from the above we can conclude that, at the initial instant of 1 1 2 1 1 2 = []()2 ()Ug ()2 initial []2 () ()2 dMGdMG crUi = the Universe, an amount of imaginary gravitational mass, total 2 ()MMG ()UiUg 2 ()MmG ()Uispg M = = ∑ = gm imaginary)( , which was associated dd initialcr dd initialcr to the fraction of the matter 2 χ MmG transformed into photons, has been ∑ () ()Uispi 2GM()Ui χ = = converted into imaginary dd dd initialcr initialcr gravitational mass of the primordial

total where m (the imaginary photons, M gp imaginary)( , while an gp(imaginary) amount of real inertial mass of the gravitational mass of the photon) is total 2 a quantity positive and imaginary, matter, M realim )( =η iUcM , has been and M gSun (imaginary ) (the imaginary converted into real energy of the N gravitational mass associated to the matter of the Sun) is a quantity primordial photons, p = ∑ hfE j , i.e., j=1 negative and imaginary. The fact of the gravitational interaction between the imaginary M total M total =+ gravitational masses of the gm imaginary realim )()( primordial photons and the total total = M gp imaginary + M realip )()( imaginary gravitational mass of the 43421 Ep matter be attractive is highly c2 relevant, because it shows that it is necessary to consider the effect of where M total = M total and this gravitational interaction, which is gm imaginary gp imaginary)()( equivalent to the gravitational effect produced by the amount of real 2 total total p realip MMcE realim )()( ηMiU ≅==≡ 11.0 MiU total gravitational mass, M ()realgp ≅ 22.0 MiU , sprayed by all the Universe. It was previously shown that, for the This means that this amount, photons equation: gp = 2MM ip , is which corresponds to 22% of the valid. This means that total inertial mass of the Universe, must be added to the overall computation of the total mass of the M gp imagimary + M realgp )()( = 142444 43444 matter (stars, galaxies, etc., gas and Mgp dust of interstellar and intergalactic media). Therefore, this additional = 2(Mip imagimary + M realip )()( ) 142444 4444 3portion corresponds to what has Mip been called Dark Matter (See Fig. By substituting Mgp()imaginary = 2Mip ()imaginary IX). into the equation above, we get On the other hand, the total amount of gravitational mass at the total initial instant, M g , according to M ()realgp = 2M ()realip Eq.(41), can be expressed by total Therefore we can write that M g = χ MiU total total M ()realgp = 2M ()realip = 22.0 M iU This mass includes the total The phenomenon of negative gravitational mass of the total gravitational deflection of light about matter, M gm −)( , plus the total the Sun shows that the gravitational total gravitational mass, M , interaction between the Sun and the realgp )( photons is attractive. This is due to converted into primordial photons. the gravitational force between the This tells us that we can put Sun and a photon, which is given by total total total g gm − += MMM realgp )()( = χ M iU 2 −= MGF gSun()()imaginary mgp imaginary r , whence 56

At the Initial Instant

M iU

Imaginary spacetime

Real spacetime total total total total Mgp()imaginary+Mip()real Mgm()imaginary+Mim()real

Primordial Photons total total M gp()imaginary + M ip()real 43421 Ep c2

∼ 10-14 m

Fig. IX – Conversion of part of the Real Gravitational Mass of the Primordial Universe into Primordial Photons. The gravitational effect caused by the gravitational interaction of imaginary gravitational masses of the primordial photons with the imaginary gravitational mass associated to the matter is equivalent to the effect produced by the amount of real gravitational mass, total M ()realgp ≅ 22.0 M iU , sprayed by all Universe. This additional portion of mass corresponds to what has been called Dark Matter.

total − 1 M gm −)( χ M iU −= 22.0 M iU M ⎛ ⎡ 121 −−= 22 2 ⎤⎞McV spg )( ⎜ ⎢( ) ⎥⎟ spi )( ⎝ ⎣ ⎦⎠ In order to calculate the value of χ we can start from the By comparing this expression with expression previously obtained for the equation above, we obtain χ , i.e., 1 m spg )( 2η kTn T ≅ 5.1 χ 1−== r 1−= 22 2 1 − cV m spi )( cm Tcritical spi )( where Substitution of this value into the total 2 expressions of χ and M gm −)( results spi )( cm 32 Tcritical = ×= 103.3 K in 2η r kn χ = − 5.0 and and ⎛ m ⎞ ⎜ spi )( ⎟c2 2 ⎜ 22 ⎟ spi )( cM ⎝ 1− cV ⎠ Tcritical total T == = M gm −)( −≅ 72.0 M iU 2η kn 2η kn 22 r r 1− cV This means that 72% of the total We thus obtain energy of the Universe ( cM 2 ) is iU due to negative gravitational mass 1 χ 1 −= of the matter created at the initial 1 − cV 22 instant. Since the gravitational mass By substitution of this expression is correlated to the inertial mass (Eq. (41)), the energy related to the into the equation of M total , we get gm −)( negative gravitational mass is where there is inertial energy (inertial ⎛ ⎞ mass). In this way, this negative total ⎜ 1 ⎟ M gm −)( 78.0 −= M iU gravitational energy permeates all ⎜ 22 ⎟ ⎝ 1 − cV ⎠ space and tends to increase the rate of expansion of the Universe due to On the other hand, the Unification produce a strong gravitational 2 repulsion between the material condition ( r =Δ≅ iUcMpcUn ) previously shown and Eq. (41) show that at the particles. Thus, this energy initial instant of the Universe, corresponds to what has been M has the following value: called Dark Energy (See Fig. X). spg )( The value of χ −= 5.0 at the

initial instant of the Universe shows ⎧ ⎡ ⎤⎫ ⎪ ⎛ Un ⎞ ⎪ that the gravitational interaction was M ⎢ 121 +−= ⎜ r ⎟ − ⎥ ≅ 1.01 MM spg )( ⎨ ⎜ ⎟ ⎬ spi spi )()( repulsive at the Big-Bang. It remains ⎢ M spi )( ⎥ ⎩⎪ ⎣ ⎝ ⎠ ⎦⎭⎪ repulsive until the temperature of the Universe is reduced down to the Similarly, Eq.(45) tells us that critical limit, Tcritical . Below this temperature limit, 58

Negative Gravitational Mass of Matter

72% 22%

6% Positive Gravitational Mass carried by the primordial photons Positive Gravitational Mass of M atter

Fig. X - Distribution of Gravitational Masses in the Universe. The total energy related to negative gravitational mass of all the matter in the Universe corresponds to what has been called Dark Energy. While the Dark Matter corresponds to the total gravitational mass carried by the primordial photons, which is manifested in the interaction of the imaginary gravitational masses of the primordial photons with the imaginary mass of matter.

the attractive component of the imaginary bodies. Just as the real gravitational interaction became particles constitute the real bodies. greater than the repulsive The idea that we make about component, making attractive the a consciousness is basically that of resultant gravitational interaction. an imaginary body containing Therefore, at the beginning of the psychic energy and intrinsic Universe – before the temperature knowledge. We can relate psychic decreased down to Tcritical , there energy with psychic mass (psychic 2 occurred an expansion of the mass= psychic energy/c ). Thus, by Universe that was exponential in analogy with the real bodies the time rather than a normal power-law psychic bodies would be constituted expansion. Thus, there was an by psychic particles with psychic evident Inflation Period during the mass. Consequently, the psychic beginning of the expansion of the particles that constitute a Universe (See Fig. XI). consciousness would be equivalent With the progressing of the to imaginary particles, and the decompression the superparticles psychic mass ,mΨ ,of the psychic cluster becomes a neutrons cluster. particles would be equivalent to the This means that the neutrons are imaginary mass, i.e., created without its antiparticle, the antineutron. Thus, this solves the Ψ = mm ()imaginaryi (131) matter/antimatter dilemma that is unresolved in many cosmologies. Now a question: How did the Thus, the imaginary masses primordial superparticles appear at associated to the photons and the beginning of the Universe? electrons would be elementary It is a proven quantum fact psyche actually, i.e., that a wave function may collapse, and that, at this moment, all the m Ψphoton = m ()imaginaryi photon = possibilities that it describes are 2 ⎛ hf ⎞ suddenly expressed in reality. This = ⎜ ⎟ i 132 2 () means that, through this process, 3 ⎝ c ⎠ particles can be suddenly materialized. mΨelectron = m ()imaginaryi electron = The materialization of the primordial superparticles into a 2 ⎛ hfelectron ⎞ −= ⎜ ⎟ i = critical volume denotes knowledge 3 ⎝ c2 ⎠ of what would happen with the 2 −= m 0()reali electron i ()133 Universe starting from that initial 3 condition, a fact that points towards the existence of a Creator. The idea that electrons have It was shown previously the elementary psyche associated to possible existence of imaginary themselves is not new. It comes particles with imaginary masses in from the pre-Socratic period. Nature. These particles can be By proposing the existence of associated with real particles, such as psyche associated with matter, we in case of the photons and electrons, as are adopting what is called we have shown, or they can be panpsychic posture. Panpsychism associated with others imaginary particles by constituting the dates back to the pre-Socratic period; 60

χ ≅ − 5.0 t ≈15 billion years t = 0 ≈10−43 st

32 TT ×== 103.3 K = 7.2 KT Inflation Period critical

Fig. XI – Inflation Period. The value of χ ≅ − 5.0 at the Initial Instant of the Universe shows that the gravitational interaction was repulsive at the Big-Bang. It remains repulsive until the temperature of the Universe is reduced down to the critical limit, Tcritical . Below this temperature limit, the attractive component of the gravitational interaction became greater than the repulsive component, making attractive the resultant gravitational interaction. Therefore, at beginning of the Universe − before the temperature to be decreased down toTcritical , there occurred an expansion of the Universe that was exponential in time rather than a normal power-law expansion. Thus, there was an evident inflation period during the beginning of the expansion of the Universe.

remnants of organized panpsychism function must have been generated may be found in the Uno of in a consciousness with a psychic Parmenides or in Heracleitus’s mass much greater than that Divine Flow. The scholars of needed to materialize the Universe Miletus’s school were called (material and psychic). hylozoists, that is, “those who This giant consciousness, in believe that matter is alive”. More its turn, would not only be the recently, we will find the greatest of all consciences in the panpsychistic thought in Spinoza, Universe but also the substratum of Whitehead and Teilhard de Chardin, everything that exists and, among others. The latter one obviously, everything that exists admitted the existence of proto- would be entirely contained within it, conscious properties at the including all the spacetime. elementary particles’ level. Thus, if the consciousness we We can find experimental refer to contains all the space, its evidences of the existence of volume is necessarily infinite, psyche associated to electron in an consequently having an infinite experiment similar to that commonly psychic mass. used to show the wave duality of This means that it contains all light. (Fig. XII). One merely the existing psychic mass and, substitutes an electron ray (fine therefore, any other consciousness electron beam) for the light ray. Just that exists will be contained in it. as in the experiment mentioned Hence, we may conclude that It is above, the ray which goes through the Supreme Consciousness and the holes is detected as a wave if a that there is no other equal to It: It is wave detector is used (it is then unique. observed that the interference Since the Supreme pattern left on the detector screen is Consciousness also contains all time; analogous with that produced by the past, present and future, then, for It light ray), and as a particle if a the time does not flow as it flows for particle detector is used. us. Since the electrons are Within this framework, when detected on the other side of the we talk about the Creation of the metal sheet, it becomes obvious Universe, the use of the verb “to then that they passed through the create” means that “something that holes. On the other hand, it is also was not” came into being, thus evident that when they approached presupposing the concept of time the holes, they had to decide which flow. For the Supreme one of them to go through. Consciousness, however, the How can an electron “decide” instant of Creation is mixed up with which hole to go through? Where all other times, consequently there there is “choice”, isn’t there also being no “before” or “after” the psyche, by definition? Creation and, thus, the following If the primordial superparticles question is not justifiable: “What did that have been materialized at the the Supreme Consciousness do before beginning of the Universe came Creation?” from the collapse of a primordial On the other hand, we may wave function, then the also infer, from the above that the psychic form described by this wave 62

Light

(a)

Light

(b)

Electrons ? (c)

Fig. XII – A light ray, after going through the holes in the metal sheet, will be detected as a wave(a) by a wave detector Dw or as a particle if the wave detector is substituted for the wave detector Dp. Electron ray (c) has similar behavior as that of a light ray. However, before going through the holes, the electrons must “decide” which one to go through.

existence of the Supreme such as the electromagnetic forces and Consciousness has no defined limit the weak and strong nuclear forces. (beginning and end), what confers Thus, the gravitational forces are upon It the unique characteristic of produced by the exchange of “virtual” uncreated and eternal. photons. Consequently, this is If the Supreme Consciousness is precisely the origin of the gravity. eternal, Its wave function ΨSC shall Newton’s theory of gravity does never collapse (will never be null). not explain why objects attract one Thus, for having an infinite psychic another; it simply models this observation. Also Einstein’s theory mass, the value of Ψ 2 will be always SC does not explain the origin of gravity. infinite and, hence, we may write that Einstein’s theory of gravity only describes gravity with more precision +∞ 2 dV ∞=Ψ than Newton’s theory does. ∫ ∞− SC Besides, there is nothing in both theories explaining the origin of the By comparing this equation with Eq. energy that produces the gravitational (108) derived from Quantum forces. Earth’s gravity attracts all Mechanics, we conclude that the objects on the surface of our planet. Supreme Consciousness is This has been going on for well over simultaneously everywhere, i.e., It is 4.5 billions years, yet no known energy omnipresent. source is being converted to support Since the Supreme Consciousness this tremendous ongoing energy contains all consciences, it is expected expenditure. Also is the enormous that It also contain all the knowledge. continuous energy expended by Therefore, It is also omniscient. Earth’s gravitational field for Consequently, It knows how to maintaining the Moon in its orbit - formulate well-defined mental images millennium after millennium. In spite of with psychic masses sufficient for its the ongoing energy expended by contents to materialize. In this way, It Earth’s gravitational field to hold can materialize everything It wishes objects down on surface and the Moon (omnipotence). in orbit, why the energy of the field All these characteristics of the never diminishes in strength or drains Supreme Consciousness (infinite, its energy source? Is this energy unique, uncreated, eternal, expenditure balanced by a conversion omnipresent, omniscient and of energy from an unknown energy omnipotent) coincide with those source? traditionally ascribed to God by most The energy W necessary to religions. support the effort expended by the It was shown in this work that gravitational forces F is well-known the “virtual” quanta of the gravitational and given by interaction must have spin 1 and not 2, and that they are “virtual” photons r mM (graviphotons) with zero mass outside −== GFdrW gg ∫∞ the coherent matter. Inside the r coherent matter the graviphoton mass is non-zero. Therefore, the According to the principle of energy gravitational forces are also gauge conservation, this energy expenditure forces, because they are yielded by the must be balanced by a conversion of exchange of "virtual" quanta of spin 1, energy from another energy type. 64 The Uncertainty Principle tells us never diminishes in strength or that, due to the occurrence of drains its energy source. exchange of graviphotons in a time If an experiment involves a large interval h Δ<Δ Et (where ΔE is the number of identical particles, all energy of the graviphoton), the energy described by the same wave function variation ΔE cannot be detected in the Ψ , real density of mass ρ of these system − mM gg . Since the total energy particles in x, y, z, t is proportional to 2 2 W is the sum of the energy of the n the corresponding value Ψ (Ψ is known as density of probability. If Ψ is graviphotons, i.e., Δ+Δ= EEW 21 +...+ΔEn, 2 * then the energy W cannot be detected complex then ΨΨ=Ψ . Thus, 2 * as well. However, as we know it can be ρ .ΨΨ=Ψ∝ ). Similarly, in the case converted into another type of energy, of psychic particles, the density of for example, in rotational kinetic psychic mass, ρΨ , in x, y, z, will be energy, as in the hydroelectric plants, 2 expressed by ρ ΨΨ=Ψ∝ * . It is or in the Gravitational Motor, as shown Ψ Ψ ΨΨ 2 in this work. known that ΨΨ is always real and

It is known that a quantum of positive while ρΨ = Ψ Vm is an energy =Δ hfE which varies during a imaginary quantity. Thus, as the time interval 1 ==Δ λ < h ΔEcft modulus of an imaginary number is (wave period) cannot be always real and positive, we can 2 experimentally detected. This is an transform the proportion ρΨ Ψ∝ Ψ , in imaginary photon or a “virtual” photon. equality in the following form: Thus, the graviphotons are imaginary photons, i.e., the energies ΔE of the 2 i Ψ =Ψ k ρ Ψ (134) graviphotons are imaginaries energies and therefore the energy where k is a proportionality constant 1 EEW 2 ... Δ++Δ+Δ= En is also an (real and positive) to be determined. imaginary energy. Consequently, it In Quantum Mechanics we have belongs to the imaginary space-time. studied the Superpositon Principle, According to Eq. (131), which affirms that, if a particle (or imaginary energy is equal to psychic system of particles) is in a dynamic energy. Consequently, the imaginary state represented by a wave function space-time is, in fact, the psychic Ψ1 and may also be in another space-time, which contains the dynamic state described by Ψ then, Supreme Consciousness. Since the 2 Supreme Consciousness has infinite the general dynamic state of the psychic mass, then the psychic space- particle may be described by Ψ , time has infinite psychic energy. This is where Ψ is a linear highly relevant, because it confers to combination(superposition)of Ψ1 and Ψ2 , the psychic space-time the i.e., characteristic of unlimited source of Ψ = Ψ cc Ψ+ 2211 (135) energy. Complex constants c and c This can be easily confirmed by 1 2 respectively indicates the percentage the fact that, in spite of the enormous of dynamic state, represented amount of energy expended by Earth’s by Ψ and Ψ in the formation of the gravitational field to hold objects down 1 2 on the surface of the planet and general dynamic state described by Ψ . maintain the Moon in its orbit, the In the case of psychic particles energy of Earth’s gravitational field (psychic bodies, consciousness, etc.), 65 2 by analogy, if ΨΨ1 , ΨΨ2 ,..., ΨΨn refer 2 pΨ Ψ 2 Ψ =Ψ+Ψ∇ 0 ()137 to the different dynamic states the h psychic particle assume, then its general dynamic state may be Because the wave functions are described by the wave function ΨΨ , capable of intertwining themselves, the given by: quantum systems may “penetrate” each other, thus establishing an internal relationship where all of them Ψ Ψ cc Ψ2211 ... c Ψ++Ψ+Ψ=Ψ Ψnn (136) are affected by the relationship, no

longer being isolated systems but The state of superposition of wave becoming an integrated part of a larger functions is, therefore, common for system. This type of internal both psychic and material particles. In relationship, which exists only in the case of material particles, it can be quantum systems, was called verified, for instance, when an electron Relational Holism [44]. changes from one orbit to another. The equation of quantization of Before effecting the transition to mass (33), in the generalized form, another energy level, the electron leads us to the following expression: carries out “virtual transitions” [42]. A 2 kind of relationship with other electrons m ()imaginaryi = mn i0 (imagynary )()min before performing the real transition. Thus, we can also conclude that the During this relationship period, its wave psychic mass is also quantized, due to function remains “scattered” by a wide Ψ = mm (imaginaryi ) (Eq. 131), i.e., region of the space [43] thus superposing the wave functions of the = 2 mnm (138) other electrons. In this relationship the Ψ Ψ ()min electrons mutually influence one where 2 2 another, with the possibility of mΨ()min −= ( min ) ichf = intertwining their wave functions‡‡. 3 −= 2 m i ()139 When this happens, there occurs the 3 ()reali min0 so-called Phase Relationship according to quantum-mechanics It was shown that the minimum concept. quantum of real inertial mass in the In the electrons “virtual” Universe, m ()reali min0 , is given by: transition mentioned before, the “listing” of all the possibilities of the m 0()reali min ±= 83 cdh max = electrons is described, as we know, by ×±= 109.3 − 73 kg ()140 Schrödinger’s wave equation. By analogy to Eqs. (132) and (133), Otherwise, it is general for material the expressions of the psychic masses particles. By analogy, in the case of associated to the proton and the psychic particles, we may say that the neutron are respectively given by: “listing” of all the possibilities of the psyches involved in the relationship will mΨproton = m ()imaginaryi proton = be described by Schrödinger’s equation – for psychic case, i.e., += 2 2 ichf = 3 ()proton += 2 m i ()141 3 0()protonreali ‡‡ Since the electrons are simultaneously waves and particles, their wave aspects will interfere with each other; besides superposition, there is also the possibility of occurrence of intertwining of their wave functions. 66 mΨneutron = m ()imaginaryi neutron = 2 2 2 = MGF ()()imaginaryi m imaginaryi r = −= ()neutron ichf = 3 ⎛ ΔE ⎞ 2 ⎜ ()mmm pne +−± ⎟ −= m 0()reali neutron i ()142 2 () ()realireali imiM 3 ≅ ⎜ c ⎟G The imaginary gravitational ⎜ ΔE ⎟ r 2 ⎜ mmm npe +++ ⎟ masses of the atoms must be much ⎝ c2 ⎠ smaller than their real gravitational Therefore, the total gravity is masses. On the contrary, the weight of M ()reali real Δ+ gg ()imaginary −= G − the bodies would be very different of r 2 the observed values. This fact shows ⎛ ΔE ⎞ that m ()imaginaryi proton and m ()imaginaryi neutron ⎜ ()mmm pne +−± ⎟ ⎜ c 2 ⎟ M ()reali must have contrary signs. In this way, − G ⎜ ΔE ⎟ r 2 the imaginary gravitational mass of an ⎜ mmm npe +++ ⎟ c 2 atom can be expressed by means of ⎝ ⎠ the following expression Thus, the imaginary gravitational mass of a body produces an excess of ⎛ ΔE ⎞ m ()imaginaryi atom ⎜ ()mmmN pne +−±= ⎟i gravity acceleration, Δg , given by ⎝ c 2 ⎠ where, ΔE , is the interaction energy. ⎛ ΔE ⎞ By comparing this expression with the ⎜ ()mmm pne +−± ⎟ 2 M ()reali following expression g ≅Δ ⎜ c ⎟G ⎜ ΔE ⎟ r 2 ⎛ ΔE ⎞ ⎜ mmm npe +++ ⎟ m ()atomreali ⎜ mmmN npe +++= ⎟ ⎝ c 2 ⎠ ⎝ c 2 ⎠

Thus, In the case of soft atoms we can m << m ()imaginaryi atom ()atomreali consider E ×≅Δ 102 −13 joules . Thus, in Now consider a monatomic body with this case we obtain real mass M (reali ) and imaginary mass M g ×≅Δ 106 −4 G i ()143 M (imaginaryi ). Then we have r 2 In the case of the Sun, for example, ⎛ Δ iE ⎞ there is an excess of gravity Σ⎜m + a ⎟ ()imaginaryi atom 2 acceleration, due to its imaginary M ()imaginaryi ⎝ c ⎠ = = gravitational mass, given by M ()reali ⎛ ΔEa ⎞ Σ⎜m ()atomreali + ⎟ −4 M iS ⎝ c 2 ⎠ g ( ×≅Δ 106 )G r 2 ⎛ ΔE ΔEa ⎞ ⎜ ()mmmn pne +−± + ⎟i At a distance from the Sun of c c 22 13 = ⎝ ⎠ ≅ r ×= 100.1 m the value of Δg is ⎛ ΔE ΔE ⎞ ⎜ mmmn +++ + a ⎟ npe 22 ⎝ c c ⎠ g ×≅Δ .108 sm −− 210 ⎛ ΔE ⎞ mmm +−± ⎜ ()pne 2 ⎟ ≅ ⎜ c ⎟i Experiments in the pioneer 10 ⎜ ΔE ⎟ spacecraft, at a distance from the Sun ⎜ mmm npe +++ ⎟ ⎝ c 2 ⎠ of about 67 AU or r ×= 100.1 13 m [45], measured an excess acceleration Since a Δ<<Δ EE . The intensity of the gravitational towards the Sun of forces between M and an g (imaginary ) g ×±=Δ .1033.174.8 sm −− 210 imaginary particle with mass mg (imaginary ), both at rest, is given by 67 Note that the general expression for electromagnetic interaction which, the gravity acceleration of the Sun is as we know, is conveyed by the exchange of “virtual” photons. M If electrons, protons and neutrons g ()×≈+= 1061 −4 G iS r 2 have psychic mass, then we can infer Therefore, in the case of the that the psychic mass of the atoms are *** gravitational deflection of light about Phase Condensates . In the case of the Sun, the new expression for the the molecules the situation is similar. deflection of the light is More molecular mass means more 4GM atoms and consequently, more psychic δ ()×≈+= 1061 −4 iS ()144 2dc mass. In this case the phase Thus, the increase inδ due to the condensate also becomes more excess acceleration towards the Sun structured because the great amount can be considered negligible. of elementary psyches inside the Similarly to the collapse of the condensate requires, by stability real wave function, the collapse of the reasons, a better distribution of them. psychic wave function must suddenly Thus, in the case of molecules with also express in reality all the very large molecular masses possibilities described by it. This is, (macromolecules) it is possible that therefore, a point of decision in which their psychic masses already constitute there occurs the compelling need of the most organized shape of a Phase realization of the psychic form. Thus, Condensate, called Bose-Einstein this is moment in which the content of Condensate†††. the psychic form realizes itself in the The fundamental characteristic of space-time. For an observer in space- a Bose-Einstein condensate is, as we time, something is real when it is in the know, that the various parts making up form of matter or radiation. Therefore, the condensed system not only behave the content of the psychic form may as a whole but also become a whole, realize itself in space-time exclusively i.e., in the psychic case, the various under the form of radiation, that is, it consciousnesses of the system does not materialize. This must occur become a single consciousness with when the Materialization Condition is psychic mass equal to the sum of the not satisfied, i.e., when the content of psychic masses of all the the psychic form is undefined consciousness of the condensate. This (impossible to be defined by its own obviously, increases the available psychic) or it does not contain enough knowledge in the system since it is psychic mass to materialize§§ the proportional to the psychic mass of the respective psychic contents. Nevertheless, in both cases, *** Ice and NaCl crystals are common examples of there must always be a production of imprecisely-structured phase condensates. Lasers, “virtual” photons to convey the psychic super fluids, superconductors and magnets are interaction to the other psychic examples of phase condensates more structured. particles, according to the quantum ††† Several authors have suggested the possibility of field theory, only through this type of the Bose-Einstein condensate occurring in the brain, and that it might be the physical base of quanta will interaction be conveyed, memory, although they have not been able to find a since it has an infinite reach and may suitable mechanism to underpin such a hypothesis. be either attractive or repulsive, just as Evidences of the existence of Bose-Einstein condensates in living tissues abound (Popp, F.A Experientia, Vol. 44, p.576-585; Inaba, H., New §§ By this we mean not only materialization proper Scientist, May89, p.41; Rattermeyer, M and Popp, but also the movement of matter to realize its F. A. Naturwissenschaften, Vol.68, Nº5, p.577.) psychic content (including radiation). 68 consciousness. This unity confers an 2 2 ‡‡‡ correlated to ΨΨ1 and ΨΨ 2 . Only individual character to this type of a simple algebraic form fills the consciousness. For this reason, from requirements of interchange of the now on they will be called Individual indices, the product Material Consciousness. 2 . 2 2 . 2 =ΨΨ=ΨΨ We can derive from the above 1 2 2 ΨΨΨΨ 1 that most bodies do not possess 1,22,1 === AAA ()145 individual material consciousness. In In the above expression, A is due to an iron rod, for instance, the cluster of 2 2 elementary psyches in the iron the product 1.ΨΨ ΨΨ 2 will be always molecules does not constitute Bose- positive. From equations (143) and Einstein condensate; therefore, the (134) we get iron rod does not have an individual 2 22 A 1. ΨΨ 2 =ΨΨ= k ρρ ΨΨ 21 = consciousness. Its consciousness is m m consequently, much more simple and = k 2 1 ΨΨ 2 ()146 constitutes just a phase condensate V1 V2 imprecisely structured made by the The psychic interaction can be consciousness of the iron atoms. described starting from the psychic The existence of consciousnesses mass because the psychic mass is the in the atoms is revealed in the source of the psychic field. Basically, molecular formation, where atoms with the psychic mass is gravitational mass, strong mutual affinity (their Ψ ≡ mm g(imaginary). In this way, the consciousnesses) combine to form equations of the gravitational molecules. It is the case, for instance interaction are also applied to the of the water molecules, in which two Psychic Interaction. However, due to Hydrogen atoms join an Oxygen atom. the psychic mass, m , to be an Well, how come the combination Ψ imaginary quantity, it is necessary to between these atoms is always the put m into the mentioned equations same: the same grouping and the Ψ same invariable proportion? In the in order to homogenize them, because case of molecular combinations the as we know, the module of an phenomenon repeats itself. Thus, the imaginary number is always real and chemical substances either mutually positive. attract or repel themselves, carrying Thus, based on gravity theory, out specific motions for this reason. It we can write the equation of the is the so-called Chemical Affinity. This psychic field in nonrelativistic phenomenon certainly results from a Mechanics. specific interaction between the =ΔΦ 4 G ρπ Ψ (147) consciousnesses. From now on, it will be called Psychic Interaction. Mutual Affinity is a dimensionless psychic quantity with ‡‡‡ Quantum Mechanics tells us that Ψ do not have which we are familiar and of which we a physical interpretation or a simple meaning and have perfect understanding as to its also it cannot be experimentally observed. However meaning. The degree of Mutual such restriction does not apply to Ψ 2 , which is Affinity, A , in the case of two known as density of probability and represents the consciousnesses, respectively probability of finding the body, described by the wave function Ψ , in the point x, y, z at the described by ΨΨ1 and ΨΨ2 , must be moment t. A large value of Ψ 2 means a strong possibility to find the body, while a small value of Ψ 2 means a weak possibility to find the body. 69 It is similar to the equation of the attractive, if A is positive gravitational field, with the difference (expressing positive mutual affinity that now instead of the density of between the two psychic bodies), and gravitational mass we have the density repulsive if A is negative (expressing of psychic mass. Then, we can write negative mutual affinity between the the general solution of Eq. (147), in the two psychic bodies). Contrary to the following form: interaction of the matter, where the ρ dV opposites attract themselves here, the −=Φ G Ψ ()148 ∫ r 2 opposites repel themselves. This equation expresses, with A method and device to obtain nonrelativistic approximation, the images of psychic bodies have been potential of the psychic field of any previously proposed [46]. By means of distribution of psychic mass. this device, whose operation is based Particularly, for the potential of on the gravitational interaction and the the field of only one particle with piezoelectric effect, it will be possible psychic mass m , we get: to observe psychic bodies. Ψ1 Expression (146) can be mG −=Φ Ψ1 ()149 rewritten in the following form: r m m A = k 2 1 ΨΨ 2 ()153 Then the force produced by this field V V upon another particle with psychic 1 2 The psychic masses m and m are mass m is Ψ1 Ψ2 Ψ2 imaginary quantities. However, the rr Φ∂ Ψ12 Ψ21 −=−= mFF Ψ2 = product .mm ΨΨ 21 is a real quantity. One ∂r can then conclude from the previous mm −= G ΨΨ 21 ()150 expression that the degree of mutual r 2 affinity between two consciousnesses By comparing equations (150) and depends basically on the densities of (146) we obtain their psychic masses, and that:

1) If mΨ1 > 0 and mΨ2 > 0 then rr VV A > 0 (positive mutual affinity −=−= AGFF 21 ()151 Ψ12 Ψ 21 k r 22 between them)

In the vectorial form the above 2) If mΨ1 < 0 and mΨ2 < 0 then equation is written as follows A > 0 (positive mutual affinity

rr VV 21 between them) Ψ12 FF Ψ 21 −=−= GA 22 ˆμ ()152 k r 3) If mΨ1 > 0 and mΨ2 < 0 then Versor ˆμ has the direction of the line A < 0 (negative mutual affinity connecting the mass centers (psychic between them) mass) of both particles and oriented 4) If mΨ1 < 0 and mΨ2 > 0 then from mΨ1 to mΨ2 . A < 0 (negative mutual affinity In general, we may distinguish between them) and quantify two types of mutual In this relationship, as occurs in the affinity: positive and negative case of material particles (“virtual” (aversion). The occurrence of the first transition of the electrons previously type is synonym of psychic attraction, mentioned), the consciousnesses (as in the case of the atoms in the interact mutually, intertwining or not water molecule) while the aversion is their wave functions. When this synonym of repulsion. In fact, Eq. (152) happens, there occurs the so-called r r Phase Relationship according to shows that the forces FΨ12 and FΨ21 are quantum-mechanics concept. 70 Otherwise a Trivial Relationship takes place. The psychic forces such as the gravitational forces, must be very weak when we consider the interaction between two particles. However, in spite of the subtleties, those forces stimulate the relationship of the consciousnesses with themselves and with the Universe (Eq.152). From all the preceding, we perceive that Psychic Interaction – unified with matter interactions, constitutes a single Law which links things and beings together and, in a network of continuous relations and exchanges, governs the Universe both in its material and psychic aspects. We can also observe that in the interactions the same principle reappears always identical. This unity of principle is the most evident expression of monism in the Universe.

APPENDIX A: Allais effect explained ×≅ −18 .103.1 mkg −3 that can work as a A Foucault-type pendulum slightly gravitational shielding and explain the increases its period of oscillation at sites Allais and pendulum effects. Below this experiencing a solar eclipse, as compared layer, the density of the lunar atmosphere with any other time. This effect was first increases, making the effect of observed by Allais [47] over 40 years ago. gravitational shielding negligible. Also Saxl and Allen [48], using a torsion During the solar eclipses, when the pendulum, have observed the Moon is between the Sun and the Earth, phenomenon. Recently, an anomalous two gravitational shieldings Sh1 and Sh2, eclipse effect on gravimeters has become are established in the top layer of the lunar well-established [49], while some of the atmosphere (See Fig. 1A). In order to pendulum experiments have not. Here, understand how these gravitational we will show that the Allais gravity and shieldings work (the gravitational shielding pendulum effects during solar eclipses effect) see Fig. II. Thus, right after result from a shielding effect of the Sun’s Sh1 gravity when the Moon is between the Sun (inside the system Moon-Lunar atmosphere), the Sun’s gravity and the Earth. r r The interplanetary medium includes acceleration, g S , becomes χ g S where, interplanetary dust, cosmic rays and hot according to Eq. (57) χ is given by plasma from the solar wind. Its density is inversely proportional to the squared ⎧ ⎡ 2 ⎤⎫ distance from the Sun, decreasing as this ⎪ ⎛ 2 Dn ⎞ ⎪ χ ⎢ 121 +−= ⎜ r ⎟ − ⎥ 11 A distance increases. Near the Earth-Moon ⎨ ⎜ 3 ⎟ ⎬ () ⎪ ⎢ ⎝ ρc ⎠ ⎥⎪ system, this density is very low, with ⎩ ⎣ ⎦⎭ 3 − 321 values about /5 cmprotons ( ×103.8 mkg ). The total density of solar radiation D However, this density is highly variable. It arriving at the top layer of the lunar can be increased up to atmosphere is given by 100~ / cmprotons 3 ( ×107.1 −19 mgk 3 ) [50]. σTD 4 ×== 7 /1032.6 mW 2 The atmosphere of the Moon is Since the temperature of the surface of the very tenuous and insignificant in Sun is T ×= 10778.5 3 K and comparison with that of the Earth. The σ ×= −8 ..1067.5 KmW −− 42 . The density of average daytime abundances of the −18 −3 elements known to be present in the lunar the top layer is ρ ×≅ .103.1 mkg then atmosphere, in atoms per cubic Eq. (1A) gives§§§ centimeter, are as follows: H <17, He 2- 3 4 40x10 , Na 70,K 17, Air 4x10 , yielding χ −= 1.1 ~8x104 total atoms per cubic −16 −3 centimeter (≅ .10 mkg )[51]. According to The negative sign of χ shows that Öpik [52], near the Moon surface, the χgr , has opposite direction to gr . As density of the lunar atmosphere can reach S S values up to −12 .10 mkg −3 .The minimum previously showed (see Fig. II), after the second gravitational shielding possible density of the lunar atmosphere is in the top of the atmosphere and is §§§ essentially very close to the value of the The text in red in wrong. But the value of interplanetary medium. χ −= 1.1 is correct. It is not the solar radiation Since the density of the that produces the phenomenon. The exact interplanetary medium is very small it description of the phenomenon starting from the cannot work as gravitational shielding. same equation (1A) is presented in the end of my However, there is a top layer in the lunar paper: “Scattering of Sunlight in Lunar Exosphere atmosphere with density Caused by Gravitational Microclusters of Lunar Dust” (2013). 72 Sh2 the gravity acceleration χgr 2 () S ′ gg ⊕ ( 4.0 )()g S −−−−= 4.0 g Moon = becomes χ 2 gr . This means that χ 2 gr −− 24 S S g ⊕ ×−≅ .106.9 sm = r −5 has the same direction of g S . In ()×−= 107.91 g ⊕ ()4 A addition, right after (Sh2) the lunar This decrease in g increases the r gravity becomes χg moon . Therefore, the pendulum’s period by about total gravity acceleration in the Earth g ′ = TT ⊕ = 000048.1 T will be given by ()×− 104.91 −5 g ⊕ 2 This corresponds to %0048.0 increase ′ −−= χχ gggg rrrr (2A) ⊕ S moon in the pendulum’s period. Jun’s abstract [53] tells us of a relative − 23 Since g S ×≅ /109.5 sm and change less than 0.005% in the − 25 pendulum’s period associated with the g moon ×≅ /103.3 sm Eq. (2A), gives 1990 solar eclipse. For example, if the density of the ′ gg 1.1 2 g −−−−= 1.1 g = ⊕ ()S ()Moon top layer of the lunar atmosphere −− 23 −18 3 g ⊕ ×−≅ .101.7 sm = increase up to ×100917.2 mgk , the −4 value for χ becomes ()×−= 103.71 g ⊕ ()3A

This decrease in g increases the χ ×−= 105.1 −3 period = 2π glT of a paraconical Thus, we obtain pendulum (Allais effect) in about −3 2 −3 ′ gg ⊕ ( 105.1 )(gS ×−−×−−= 105.1 )gMoon = g ⊕ ′ −− 28 = TT −4 = 00037.1 T g⊕ ×−≅ .103.6 sm = ()×− 103.71 g ⊕ −9 This corresponds to %037.0 increase ()×−= 104.61 g⊕ ()5A in the period, and is roughly the value So, the total gravity acceleration in the (0.0372%) obtained by Saxl and Allen Earth will decrease during the solar during the total solar eclipse in March eclipses by about 1970 [48]. −9 As we have seen, the density of ×104.6 g ⊕ the interplanetary medium near the Moon is highly variable and can reach The size of the effect, as measured 3 values up to 100~ protons / cm with a gravimeter, during the 1997 −19 3 −9 ( ×107.1 mgk ). eclipse, was roughly ()×− 1075 g ⊕ When the density of the [54, 55]. interplanetary medium increases, the The decrease will be even top layer of the lunar atmosphere can -18 -3 smaller forρ ≳ 0917.2 ×10 kg.m . The also increase its density, by absorbing lower limit now is set by Lageos particles from the interplanetary satellites, which suffer an anomalous medium due to the lunar gravitational acceleration of only about ×103 −13 g , attraction. In the case of a density ⊕ increase of roughly 30% during “seasons” where the satellite ( ×107.1 −18 mgk 3 ), the value for χ experiences eclipses of the Sun by the Earth [56]. becomes

χ −= 4.0 Consequently, we get

Top layer of the Moon’s atmosphere (Gravitational Shielding) Interplanetary medium ≈10 −19 mkg 3

≈10−18 kg.m-3 Solar radiation ≈10−17 kg.m-3 ≈10−12 kg.m-3 χgr Sh1 Moon Sh2 Moon Earth r r r r 2 r 2 r r g S χg S gMoon χg Moon χ g S χ g S g ⊕

Fig. 1A – Schematic diagram of the Gravitational Shielding around the Moon – The top layer of the Moon’s atmosphere with density of the order of 10-18 kg.m-3 , produces a gravitational shielding r r when subjected to the radiation from the Sun. Thus, the solar gravity g S becomes χ g S after the 2 r r first shielding Sh1 and χ g S after the second shielding Sh2 . The Moon gravity becomes χ g Moon after Sh2 . Therefore the total gravity acceleration in the Earth will be given by 2 rrrr ′ ⊕ S −−= χχ gggg moon .

APPENDIX B hn 22 hn 22 En 2 == 22 ()3B In this appendix we will show g Lm 88 η i Lm why, in the quantized gravity equation (Eq.34), n = 0 is excluded from the From this equation we can easily sequence of possible values of n . conclude that η cannot be 0 Obviously, the exclusion of n = 0, zero ( n → ∞ EorE n → 0 ). On the means that the gravity can have only other hand, the Eq. (2B) shows that the discrete values different of zero. exclusion of η = 0 means the exclusion Equation (33) shows that the of m = 0 as a possible value for the gravitational mass is quantized and g given by gravitational mass. Obviously, this also means the exclusion of M g = 0 2 g = mnM g()min (Relativistic mass). Equation (33) tells 2 us that g = mnM g(min), thus we can Since Eq. (43) leads to conclude that the exclusion of Mg =0

implies in the exclusion of n = 0 since g()min = mm io ()min g()min = mm i ()min0 = finite value (elementary

quantum of mass). Therefore Eq. (3B) where is only valid for values of n and η

−73 different of zero. Finally, from the ±= cdhm ×±= 109.383 kg i ()min0 max quantized gravity equation (Eq. 34),

GM ⎛ Gm ⎞ is the elementary quantum of inertial g g n 2 ⎜ −=−= g ()min ⎟ = 2 ⎜ 2 ⎟ mass. Then the equation r ⎝ ()max nr ⎠ = 4 gn for M g becomes min we conclude that the exclusion of n = 0 2 == 2 mnmnM means that the gravity can have only g g()min i ()min discrete values different of zero. On the other hand, Eq. (44) shows that 2 = mnM iii ()min0 Thus, we can write that

2 M g ⎛ n ⎞ = ⎜ ⎟ =η 2 MMor ()1B M ⎜ n ⎟ g i ⎝ ii ⎠

where η = nn i is a quantum number different of n . By multiplying both members of

Eq. (1B) by 1− cV 22 we get

2 g =η mm i (2B) By substituting (2B) into Eq. (21) we get

REFERENCES

[1] Isham, C. J. (1975) Quantum gravity, an [19] Verheijen, A.A., van Ruitenbeek, J.M., Oxford Symposium, OUP. de Bruyn Ouboter, R., and de Jongh, [2] Isham, C.J., (1997) "Structural Problems L.J., (1990) Measurement of the London Facing Quantum Gravity Theory'', in M, Moment for Two High Temperature Francaviglia, G, Longhi, L, Lusanna, and Superconductors. Nature 345, 418-419. E, Sorace, eds., Proceedings of the 14th [20] Sanzari, M.A., Cui, H.L., and Karwacki, International Conference on General F., (1996) London Moment for Heavy- Relativity and Gravitation, 167-209, Fermion Superconductors. Appl. Phys. (World Scientific, Singapore, 1997). Lett., 68(26), 3802-3804. [3] Landau, L. and Lifchitz, E. (1969) Theorie [21] Liu, M., (1998) Rotating Superconductors du Champ, Ed.MIR, Moscow, Portuguese and the Frame-Independent London version (1974) Ed. Hemus, S.Paulo, p.35. Equation. Phys. Rev. Lett. 81(15), 3223-3226. [4] Landau, L. and Lifchitz, E. (1969) [22] Jiang, Y., and Liu, M. Rotating (2001) Mecanique, Ed.MIR, Moscow, 15. Superconductors and the London Moment: Thermodynamics versus [5] Landau, L. and Lifchitz, E.[3], p.35. Microscopics. Phys. Rev. B 63, 184506. [6] Landau, L. and Lifchitz, E.[3], p.36. [23] Capellmann, H., (2002) Rotating [7] Beiser, A. (1967) Concepts of Modern Superconductors: Ginzburg-Landau Physics, McGraw-Hill, Portuguese version Equations.Europ. Phys. J. B 25, 25-30. (1969) Ed. Poligno, S.Paulo, p.151. [24] Berger, J., (2004) Nonlinearity of the [8] Schiff, L.I. (1981) Quantum Mechanics, Field Induced by a Rotating McGraw-Hill, p.54. Superconducting Shell. Phys. Rev. B 70, [9] V.B. Braginsky, C.M. Caves, and 212502. K.S.Thorne (1977) Phys. Rev. D15, 2047. [25] Podkletnov, E. and Nieminen, R. (1992) [10] L.D. Landau and E.M. Lifshitz,(1951) Physica C, 203, 441. The Classical Theory of Fields, [26] Ference Jr, M., Lemon, H.B., Stephenson, 1st. Edition (Addison-Wesley), p.328. R.J., Analytical Experimental Physics, [11] R.L. Forward,(1961) Proc. IRE 49, 892. Chicago Press, Chicago, Illinois,. [12] Tajmar, M. et at., (2006) Experimental Portuguese version (Eletromagnetismo), Detection of the Gravitomagnetic Ed. Blücher, S.Paulo, Brazil, p.73. London Moment. gr-qc/0603033. [27] Modanese, G. (1996), gr-qc/9612022, p.4 [13] Tate, J., Cabrera, B., Felch, S.B., [28] Quevedo, C.P.(1978) Eletromagnetismo, Anderson, J.T., (1989) Precise McGraw-Hill, p. 102. Determination of the Cooper-Pair Mass. Phys. Rev. Lett. 62(8), 845-848. [29] Landau, L. and Lifchitz, E.[3], p.324. [14] Tate, J., Cabrera, B., Felch, S.B., [30] Landau, L. and Lifchitz, E.[3], p.357. Anderson, J.T., (1990) Determination of [31] Landau, L.and Lifchitz,E.[3],p.p.358-359. the Cooper-Pair Mass in Niobium. Phys. [32] Landau, L. and Lifchitz, E.[3], p.64. Rev. B 42(13), 7885-7893. [33] Luminet, J. at al.,(2003)Nature 425,593–595. [15] Tajmar, M. and De Matos, C.J (2006) [34] Ellis, G.F.R (2003) Nature 425, 566–567. Local Photon and Graviton Mass and its [35] Hogan, C. et al. (1999) Surveying Space- Consequences, gr-qc / 0603032. time with Supernovae, Scientific America, [16] Landau, L. and Lifchitz, E.[3], p.426-427. vol.280, January, pp. 46-51. [17] Hildebrandt, A.F.,(1964) Magnetic [36] Landau, L. and Lifchitz, E.[3], p.440-441. Field of a Rotating Superconductor. [37] Landau, L. and Lifchitz, E.[3], p.333. Phys. Rev.Lett. 12(8), 190-191. [38] Landau, L. and Lifchitz, E.[3], p.331. [39] Landau, L. and Lifchitz, E.[3], p.327. [18] Verheijen, A.A., van Ruitenbeek, J.M., [40] Hawking, S.W (1971) MNRAS., 152, 75. de Bruyn Ouboter, R., and de Jongh, [41] Carr, B.J.(1976) Astrophys.J., 206,10. L.J., (1990) The London Moment for [42] Bohm, D. (1951) Quantum Theory, High Temperature Superconductors. Prentice-Hall, N.Y, p.415. Physica B 165-166, 1181-1182.

[43] D’Espagnat, B. The Question of Quantum Reality, Scientific American, 241,128. [44] Teller, P. Relational Holism and Quantum Mechanics, British Journal for the Philosophy of Science, 37, 71-81. [45] Anderson, J.D, et al., (2002) Study of the anomalous acceleration of Pioneer 10 and 11, Phys. Rev.D65, 082004. [46] De Aquino, F. (2007) “Gravity Control by means of Electromagnetic Field through Gas at Ultra-Low Pressure”, physics/0701091. [47] Allais, M. (1959) Acad. Sci. URSS 244,2469;245,1875(1959);245,2001 (1959);245,2170(1959);245,2467 (1959). [48] Saxl, E. and Allen, M. (1971) Phys. Rev.D3,823-825. [49] http://science.nasa.gov/newhome/headli nes/ast06aug991.htm [50] Martin, M. and Turyshev, S.G. (2004) Int. J. Mod. Phys.D13, 899-906. [51] Stern, S.A. (1999) Rev. Geophys. 37, 453 [54] Öpik, E,J (1957) The Density of the Lunar Atmosphere, Irish Astr. J., Vol4 (6), p.186. [52] Öpik, E,J (1957) The Density of the Lunar Atmosphere, Irish Astr. J., Vol4(6), p.186. [53] Jun, L., et al. (1991) Phys. Rev. D 44,2611-2613. [54] http://www.spacedaily.com/news/ china-01zi.httml [55] Wang. Q et al., (2000) Phys. Rev. D 62, 041101(R). [56] Rubincam, D. P. (1990) J. Geophys. Res., [Atmos.] 95, 4881.