The Unification of Gravitation and Electromagnetism

The Unification of Gravitation and Electromagnetism Shlomo Barak To cite this version: Shlomo Barak. The Unification of Gravitation and Electromagnetism. 2020. ฀hal-02616267฀ HAL Id: hal-02616267 https://hal.science/hal-02616267 Preprint submitted on 24 May 2020 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. The Unification of Gravitation and Electromagnetism Shlomo Barak Taga Innovations 28 Beit Hillel St. Tel Aviv 67017 Israel Corresponding author: [email protected] Abstract All past efforts to unify gravitation and electromagnetism failed because they considered energy/momentum to be the common denominator in this unification. The problem was that nobody knew how mass curves space and that charge, as we have shown in one of our papers, also curves space. It was naively believed that if mass, namely energy, curves space then so will the electromagnetic energy and in the same way. We took an entirely different approach. Our common denominator, as we show, is the deformation (distortion, curving) of space by both mass and charge. This approach yields the expected result. Key Words: Gravitation; General Relativity; Riemannian Geometry; Electromagnetism Introduction We have achieved the unification of gravitation and electromagnetism by reassessing almost all we know about gravitation, electromagnetism and the geometry of space. Our Perception of Space and General Relativity (GR) Space in GR is considered a continuous 3D manifold, bent (curved) in a 4D hyperspace - bent by energy/momentum. We, however, consider space to be a 3D deformed lattice rather than a bent continuous manifold. We present in another paper a new geometry for this kind of space [1]. This new geometry uses the same terms as Riemannian Geometry, the geometry of bent 1 manifolds. In General Relativity (GR), therefore, instead of considering space distortion by mass to be bending, we consider it a contraction, with no need to change the mathematical formalism, based on Riemannian Geometry. Note that in this paper we sometimes use the term curving although we mean deforming or specifically contraction or dilation. The text will clarify what we mean. The Problem with the Right-hand Side Term of the GR Equation Einstein complained that the left-hand side of the GR equation expresses curving whereas the right-hand side merely expresses energy/momentum and not curving. He expected that future physics will show us how energy/momentum really curves space and will also express it in terms of curvature. We show that the right-hand side term of the GR equation expresses contraction (curving) as does the left. And we also show that this contraction is due to the angular momentum of the particles that compose a mass. Gravitation – Space Contraction - by Angular Momentum of Particles We show that the rotation (spin) of an elementary particle is the basic mechanism by which it gravitates - curves space - in addition to the curving by its charge. This idea came to us following Project “Gravity Probe B” [2] recent validation of the GR predicted “Frame Dragging” [3], [4], phenomenon. This phenomenon is considered to be a result of an additional mechanism by which a rotation of a macroscopic mass contracts (curves) space around it and contributes to gravitation, as the Kerr Metric shows [5], [6]. We asked ourselves: if Frame Dragging is a macroscopic phenomenon could it be that it is also a microscopic phenomenon related to the spin of elementary particles. Our affirmative answer paved the way to unification. 2 Electric Charge and its Field as Space Contraction or Dilation Our paper [7] presents a model of the positive elementary electric charge and its field as a contracted zone of space, and the negative elementary electric charge and its field as a dilated zone of space. Relating to space as a lattice (cellular structure), we define Space Density  as the number of space cells per unit volume (denoted 0 for space with no deformations). Based on this definition, we define electric charge density. With this definition alone, we derive electrostatics, without any phenomenology, and together with the Lorentz Transformation - the entire Maxwell theory. Based on [7] we show how charge curves space [8]. The Common Denominator Needed for Unification By attributing space contraction or dilation to charge, and space contraction to mass, we have created a common denominator for both electromagnetism and gravitation. This common denominator enables us to extend Einstein’s GR field equation to accommodate electromagnetism. Our Work-plan Prove that the right-hand side term of mass/energy, in the GR equation, expresses contraction (curving). Einstein’s dream was to show that this is indeed the case. Prove that it is the angular momentum of the elementary particles that contract space around them. Thus, revealing how mass creates gravitation. Present our result that positive charge contracts space whereas negative charge dilates it. Calculate the ratio of curvature (deformation) - by charge to curvature (deformation) - by mass/energy. 3 Use the ratio of curvature by angular momentum to the curvature by charge to get the factor needed to incorporate the contribution of charge in the GR equation. Incorporate the charge/current tensor in the GR equation to extend it. The Right-Hand Side of GR Equation Expresses Space Contraction The right-hand side of Einstein’s field equation (1) of GR (below) should express curvature exactly as the left-hand side does. This paper shows, for the first time, that this is indeed the case. The need to express curvature by Riemannian geometry and obtain a covariant formulation of physical laws, by using tensors, led Einstein to the equation of General Relativity (GR): 8G 1 R ij − Rg ij + g ij = 4  Tij 2 c (1) Rij is the Ricci contracted Riemannian tensor and R is the Ricci scalar, Gaussian Curvature [9] and Ʌ is the cosmological constant. The 1 R is the familiar 2 1 Rg ij term is added to give a 2 covariant divergence, which is identically zero, and T 00 = j(0 )0 = ϵ is the energy density of space (see Appendix C). In this paper we ignore the Cosmological Constant term Ʌ g ij . It was Einstein’s vision that “future physics” will show, as we do below, that the right handside of (1): 8G ∙ Tij is also curvature (and not simply have the same dimensionality). c4 In equation (2), see [10] and [11]: Rc is the Gaussian radius of curvature, and: K = 1/Rc2 is the Gaussian curvature at the radial distance r from the center of a mass with Schwarzschild Radius rS (our notations are different from those in [10] and [11]): 4 K=1/Rc2 = rs/r3 (2) The right-hand side of the GR equation is 8πG/c4∙Tij. Relation (2) enables us, for the first time to show that 8πG/c4∙Tij expresses curvature. According to [11] p.290 “Ignoring curvature effects, the Schwarzschild radius of a body of uniform density m and radius r is rS = 2GM/c2 ~ 8πG/3c2∙mr3 ” , where m is the mass density. But m = ϵ/c2 Hence: K=1/Rc2 = rS/r3 = (1/3) 8πGm /c2 = (1/3) 8πG ϵ /c4 where ϵ = T00 is the energy density. We thus conclude that 8πG/c4∙Tij is indeed curvature (ignoring the factor 1/3, which is related to the suggested uniform density). An elementary particle, besides its direct contraction or dilation of space which is related to its charge, has an additional mechanism by which it contracts space around itself (creates positive curvature). This contraction is. This mechanism of contraction, related to its energy content, is (see Section 3) due to the torsion in space created by the spin of the elementary particle, and hence related to its inertial mass. All this proves that gravitational mass is inertial mass. Note that energy is purely electromagnetic [8] and that elementary charges are kinds of black and white holes that curve space drastically [7], [8]. The Positive Curving of Space by the Angular Momentum of an Elementary Particle In [8] we consider the electrons/positrons as circulating longitudinal wavepackets of dilation/ contraction, which are their elementary electric charges. Thus, hinted by the validation of “Frame Dragging”, it is logical to explore the possibility that gravitation is the result of their angular momentum due to their circulation. 5 Our conjecture is that a longitudinal wavepacket applies pressure on the space lattice and bends its sides perpendicular to the wavepacket propagation direction. This bending is the “Frame Dragging” and it causes the contraction of space around the circulation track. As a result, particles carrying the same or different elementary charges, have an identical minute positive additional curvature that depends on their Angular Momentum. The sum of two additional curvatures created by a pair of particles, carrying different types of charge, is the gravitational curvature of neutral matter. Fig. (1a) shows how an anti-clockwise rotating macroscopic mass (the inner circle) drags a radial frame; this is the kind of figure that usually appears in the literature. But this dragging causes space torsion around the mass and hence contraction, which is discussed in the literature on elasticity [12]. This contraction is presented in Fig.(1b); it shows that the concentric frame circles in Fig. (1a) should be corrected to show the contraction due to the torsion that takes place mainly in the gray zone around the mass. These figures represent the dragging and contraction created by a longitudinal wavepacket of radius re represented by the thickness of the gray annular ring of the circulating wavepacket with radius Re . Note that this type of space contraction, due to lattice bending, is not dependent on the direction of rotation of the wavepacket. Note also that the contraction is spherically symmetric only on average. This is also relevant to trios or duos of quarks, which are twisted electrons or positrons according to our theory of matter [13]. Our model of quarks [13] leads to the conclusion that in neutral matter the number of electrons equals the number of positrons. Thus, the long-standing issue of “where is anti-matter?” is resolved. 6 a b Fig. (1) Frame Dragging The question that we have to ask now becomes “why almost all of the nucleuses in the universe are positive?” It seems that since positive charge curves space positively, atoms with a positive nucleus are much more stable, but this issue is out of the scope of this paper. Rotation induces Frame Dragging [3], [4], [5], [6], which is a torsional contraction (positive curving). This phenomenon is expressed by the Kerr metric [6] equation (14.22) p.303. We, however, repeat the procedure used in Section 2, in which we rewrote the Schwarzschild radius, rS = 2GM/c2, by considering mass to be purely electromagnetic [8]. Here we rewrite the Schwarzschild radius by expressing the mass of an elementary particle by its angular momentum J=1/2 ℏ. Since J= MRv we get for the elementary mass: Me=1/2 ℏ/Rec where Re is defined in [8]. This gives: rS =Gℏ/Rec3 (4) Using equation (2) we get for the curvature created by spin: KS=1/Rc2 = rS /r3 = Gℏ/Rec3 r3 7 (5) Note that M in this discussion is the inertial mass. Thus, it is the inertial mass that creates gravitation, and we are allowed to eliminate the concept of “gravitational mass”. Note that instead of proving the equivalence of inertial mass and gravitational mass we are proving their identity. The Kerr contribution to the square of the line-element that appears in equation (14.22 p303) in [6] is: 4GJ/c3r2 Sin2 θ(r dφ) cdt By taking J in (6) as J = 1/2∙ℏ and the average of Sin2θ as 1/2, we are getting an expression compatible with (4). Note that, since both charge and gravitation are quantized, the total space curvature is quantized. Thus, “singularity” turns into merely a mathematical issue. Space Contraction and Dilation by Charge In a recent paper [6] titled “On the Essence of Electric Charge” Part 1 we consider positive electric charge to be a contracted zone of space and negative electric charge to be a dilated zone of space. This yields the Maxwell theory of electromagnetism, with no phenomenology. We show that deformed spaces are represented by our geometry [1] in the same way as are bent manifolds by Riemannian geometry. Hence, we can attribute positive curvature to a contracted zone of space, namely to a positive charge, and negative curvature to a negative charge. This attribution enables us, in Part 2 [8], to show that the bivalent charges are kinds of electrical black and white holes, and derive and calculate their radii. We further derive and calculate the masses of the leptons and quarks. All this serves as the basis for our discussion on gravitation in this paper. 8 According to [7], [8] the energy of an elementary particle is purely electromagnetic. The absolute values of the bivalent charges are the same, and their absolute values of curving space are precisely equal. The curvature, contributed by pairs of these bivalent charges, to a neutral macroscopic body is therefore zero. What is left over is the additional curving due to their angular momentum, which is very small in comparison, as is the gravitation Note that we consider quarks to be sub-tracks of topologically-twisted electrons or positrons and not elementary particles in their own right. This is how we understand confinement and the assignment of 1/3 Qe or 2/3 Qe to quarks. This enables us to derive and calculate the quarks’ masses [13]. The equation: 1/Rc2= rsρ/r3 is a corrected (2), that takes into account the space density ρ, see [7] and [8], as explained below. The absolute charge values of the bivalent elementary charges are equal |Q+| = |Q-|. Hence for a spherical elementary charge with radius r and charge density q this implies, according to [7], r r the following: Since r+<r- (dilation versus contraction) and |Q+|=|Q-| = |∫0 + q + dr |=|∫0 − q − dr | , necessarily on the average |q+| < |q-| . We define, [7], Electric Charge in a given zone of space τ as: Q =  qdτ Electric Charge in the GDM has the dimensions of volume [Q] = L3 τ Note that by omitting the factor 1/4π in equation (1) in [7] - the definition of charge density we get for the spherical symmetric case dτ =4πr2dr and hence: r Q = ∫0 q4πr 2 dr = 4π/3∙ r3(1−0/ρ) =V – V’. Thus  > 0 gives V >V’ and Q > 0 whereas 9  < 0 gives V < V’ and Q < 0. Thus, Q becomes a pure volumetric change, which is positive for contraction and negative for dilation. Note also that the equality |Q+| = |Q-|, of the absolute values of the bivalent elementary charges, means (1– 0/ρ+) = – (1– 0/ρ-) and hence 2/0 = 1/ρ+ + 1/ρ- . It also means that: |Q+| = |Q-| = |V – V’|. For example, if ρ+ = 20 then ρ- = 2/3∙0 and |V – V’| = 1/2∙V. This means that V+/V- = 1/3 and hence r+/r- = (1/3)1/3 = 0.69 whereas according to [8]: rp / re = 0.64 The exact equality |Q+| = |Q-| is related to the essence of the photon and Pair Production, and is a theoretical result of our photon model, whereas here it is considered a phenomenological result. According to [8]: r- = 3/2 r+. Hence |Q+| = |Q-| implies q-= 2/3 q+ , as discussed at the beginning of this section. This means that the |curvature| created by Q+ is the same as that created by Q-. As a result, the |curvature| for both of the pure bivalent charges, according to (2), and denoting rS(Charge) as rQ , is: KQ=1/Rc2 = rQ / r3 = √2/2s2 ∙ √G Q / r3 (6) where s = 1 and [s] = LT-1, see [7] and [8]. The bending of a light beam by charge, due to its induced space curvature, is discussed in Appendix B. In Appendix C we suggest an experiment designed to verify or falsify our theory. The Ratio of Curvature-by Charge to Curvature-by Mass/Energy Dividing (6) by (5) gives the ratio of the curving by charge KQ to that by spin KS. This ratio for a given distance r from the center of the elementary particle is the ratio: KQ / KS = rQ / rS = (√2/2s2)√G Qe /(Gℏ/Rec3) ~ c2 / s2 10 (7) The ratio of the electric force to the gravitational force between two electrons, at a distance r from each other is: FQ/FG = Qe2 /GMe2 ~ 4c4 / s4 (8) The comparison of equations (7) and (8) reveals the connection between the curving of space and the applied force: FQ/FG ~ (KQ/KS)2 which means that Force is the multiplication of Curvatures. For masses M1 and M2 the multiplication of their curvatures is: KS1KS2 = rS1/r12 3 ∙ rS2/r12 3 = [4G/(c4 r12 4)]∙G M1M2/r12 2 and thus gravitational force is: GM1M2/r12 2 = [(c4 r12 4)/4G]∙KS1KS2 For charges Q1 and Q2 the multiplication of their curvatures is: KQ1KQ2 = rQ1/r12 3 ∙ rQ2/r12 3 = [(½)G/(s4 r12 4)]∙Q1Q2/ r12 2 and thus the electric force is: Q1Q2/ r12 2 = [2(s4 r12 4)/G]∙KQ1KQ2 The Curving by Both Charge and Angular Momentum The total curvature, K= 1/Rc2 , created by an elementary particle is the sum of its curvature KQ, due to charge and its curvature, KS, due to angular momentum, K = KQ + KS , or according to (5) and (6) for the same r: K = rQ/r3 + rS/r3 = √G Q/s2r3 + (2GM/c2)/r3 (9) Thus, for two very close elementary particles, carrying the bivalent elementary charges, the charge contributions to curvature at a distance r cancel out, whereas the energy/angular momentum contributions add up. Note that in annihilation of a pair the residual curvature does not cancel out. This is an open issue in the current paradigm that asks about the gravitational energy disappearance. We, 11 however, wonder if this very small energy is not carried by a graviton created in the annihilation. Section 2 has shown that (2GM/c2)/ r3 in equation (9) appears in the GR equation (1) as: 8G ∙ c4 ϵ where ϵ = T00 and in general becomes: 8πG/c4∙Tij . Similarly, (√2/2)√G Q/ s2r3 (√2/2)√G Q/ s2r3 = (2√2π/3) √G Q/ s2(4π/3) r3 = (1/3) 2√2π√G/s2 ∙ q in (9) where q is the charge density. Thus: 2√2π√G /s2 ∙ q becomes analogous to (10) 8G ∙ c4 ϵ and is used in Section 7 in the extension of the GR equation. Note that we ignore the factor (1/3), as we did in Section 2. The Extension of Einstein’s Field Equation of GR to Incorporate the Curving by Charge / Current This extension turns the GR equation from an equation that deals with spacetime curving by energy/momentum into a universal equation that also deals with the curving by charge/current. As a result, the equation becomes relevant not only to the macroscopic world but also to the microscopic world of elementary particles. Note that in this microscopic world the quantization of spacetime curvature also becomes relevant, since both angular momentum and charge are quantized. To construct the extended equation, we adopt the idea expressed by (9) and add a new term to the GR equation, which is the multiplication of (10) by the tensor: charge density/current density. 12 The 4 –Vector of Electric Current Density Electric Current: J  = (cQ, J ) , J = Qv hence: J  = Q(c, v ) Electric Current Density: j = (cq, j) , j = qv hence: j = q (c, v ) q = q 0 j = q 0 (c, v ) = q 0 v  Q is not a relativistic quantity, whereas q is a relativistic quantity (note the volume). The Tensor Tq of Charge Density / Current Density The 4-current densities j(0 ) , j(1) , j(2 ) , j(3 ) transform into each other under Lorentz ( ) Transformation (LT). Thus, they themselves form a 4-vector, p = p0 , pi . Tq is the tensor:  j(0 )0   j(0 )μ   j(1)0 μν T =  (i )μ  =  (2 )0  j  j  j(3)0  T 00 = j(0 )0 . (11) j(0 )1 j(1)1 j(2 )1 j(3)1 j(0 )2 j(1)2 j(2 )2 j(3)2 T0i = j(0 )i = the i th component of the current . q0k 0  T = c v1 = q cv 1 = qcv 1 01 13 j(0 )3   j(1)3   j(2 )3  j(3)3  T10 = cq 0 v1 = qcv 1 The Extended Field Equation of General Relativity for Energy/Momentum and Charge/Current Using (9), (10) and (11) the extended Einstein GR field equation becomes: R-1/2Rg = 8πG/c4 ∙ Tm + 4π√G /s2 ∙ Tq (12) Summary We have shown how Charge and Angular Momentum curve space (curve: contract or dilate in the case of charge and contract in the case of angular momentum). Based on this we extended Einstein’s field equation to become the equation of both energy/momentum and charge/current. This way the equation becomes an equation of the macroscopic/microscopic reality. Acknowledgements We would like to thank Mr. Roger M. Kaye for his linguistic contribution and technical assistance References [1] Shlomo Barak: On Bent Manifolds and Deformed Spaces Journal of Mathematics Research; Vol. 11, No. 4; 2019 S. Barak: On Bent Manifolds and Deformed Spaces hal-01498435 (2017) https://hal.archives-ouvertes.fr/hal-01498435 14 [2] Everitt; et al: Gravity Probe B: Final Results of a Space Experiment to Test General Relativity. Physical Review Letters. 106 (22) (2011). arXiv:1105.3456. [3] H. Thirring: Über die Wirkung rotierender ferner Massen in der Einsteinschen Gravitationstheorie. Physikalische Zeitschrift. 19: 33. (1918). [On the Effect of Rotating Distant Masses in Einstein’s Theory of Gravitation] [4] J. Lense, and H. Thirring: Über den Einfluss der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie. Physikalische Zeitschrift. 19: 156–163 (1918). [On the Influence of the Proper Rotation of Central Bodies on the Motions of Planets and Moons According to Einstein's Theory of Gravitation] [5] R. P. Kerr: Gravitational field of a spinning mass as an example of algebraically special metrics. Physical Review Letters. 11 (5): 237–238. (1963). [6] J. B. Hartle: Gravity, Addison Wesley (2003) [7] Shlomo Barak: Electric Charge and Its Field as Deformed Space Applied Physics Research; Vol. 11, No. 4; 2019 or S. Barak: On the Essence of Electric Charge, Part 1: Charge as Deformed Space. hal-01401332 (2016) https://hal.archives-ouvertes.fr/hal-01401332 [8] Shlomo Barak: The Essence of the Elementary Charge and the Derivation and Calculation of the Electron Inertial Mass Applied Physics Research; Vol. 11, No. 4; 2019 or S. Barak: On the Essence of Electric Charge Part 2: How Charge Curves Space. hal-01402667 (2016) https://hal.archives-ouvertes.fr/hal-01402667 [9] 15 D. Landau and E. M. Lifshitz : The Classical Theory of Fields, p304 Pergamon (1962) [10] P. G. Bergmann: The Riddle of Gravitation, p195, Dover (1992) [11] A.M. Steane: Relativity Made Relatively Easy, p276, Oxford (2012) [12] R. Feynman: The Feynman Lectures on Physics Vol. II Ch. 38: Elasticity [13] S. Barak: Where is Anti-Matter? hal-01423547 (Dec 2016) https://hal.archives-ouvertes.fr/hal-01423547 [14] Ta-Pei Cheng: Relativity Gravitation and Cosmology, p208, Oxford (2005) Appendix - A The Energy Momentum Tensor According to [14]: 4-Vectors in Gravitomagnetism  Position: x = (ct, x, y, z ) Interval: x  = η x  = (− ct, x, y, z ) S2 = x  x  = −c 2 t 2 + x 2 + y 2 + z 2 = c 2 2 Proper time: τ (a Lorentz scalar) Velocity: v =   x  but t =  hence v =   x  v  =  (c, v x , v y , v z ) =  (c, v ) v 2 = v  v  =  2 (− c 2 + v 2 ) Quantities which transform a LT, like x  , are 4-vectors. Momentum: 16 P  Mv  =  (Mc, Px , Py , Pz )  v2  P0 = Mc = Mc1 − 2   c  − 1 2  1 v2   1  = Mc1 + + ..... =  Mc 2 + Mv 2 + ........ 2 2   2c   cP 0 = Mc 2 = U Rest energy: Relativistic energy: cP 0 = Mc 2 = U U  P =  , P  c  ( ) P 2 = P P = M 2 v v = M 2 − c2 ( ) 2 2 Hence: U = Mc + (cP) 2 2 Covariant force: F = d P = Md  v or:   U U  F = d t P = d t  , P  =   , F  c  c  Rate of energy change – power: 1 W = F v = Md  v  v = Md  v v = 0 2 since v2 is a constant. F v =  2 (− U + F  v ) = 0 hence:  Fv  F =  ,F  c  Mass current: J  = (cM, J ) , J = Mv hence: J  = M (c, v ) Mass current density: j = (cm, j) , j = mv hence: j = m(c, v ) m = m 0 Energy current–momentum: P 0 = 17 U c j = m 0 (c, v ) = m 0 v   U J (0 ) =  U,   c Energy current - momentum density:   j(0 ) = ,   c The 4-current density for the i th momentum component (pi): j(i ) = (cpi , pi ) Note that in this section M and m stand for inertial mass and mass density. With a proper modification they also stand for gravitational mass and mass density. The Energy Momentum Tensor  j(0 )0   j(0 )μ   j(1)0 μν  T in (1) is: T =  (i )μ  =  (2 )0 j  j  j(3)0  j(0 )1 j(1)1 j(2 )1 j(3)1 j(0 )2 j(1)2 j(2 )2 j(3)2 j(0 )3   j(1)3  j(2 )3   j(3)3  The 4-current density j(0 ) , j(1) , j(2 ) , j(3 ) transform into each other under Lorentz Transformation. Thus, they ( ) themselves form a 4-vector, p = p0 , pi . T 00 = j(0 )0 = energy density T 0i = j(0 )i = the i th component of the m 0c 2 0 v1 = m 0cv1 = mcv1 T =  v1 or  c c 01  current c T10 = cm 0 v1 = mcv1 We see that the momentum density T i0 is equal to the energy density current T 0i Tii = momentum current = momentum force = = pressure ΔS  Δt ΔS The off-diagonal terms are shear forces. Tij = T ji is a symmetric tensor.   T  = 0 for an isolated system is the conservation of energy – momentum 18 Appendix - B The Bending of a Light Beam by an Electric Charge The bending of a light beam by mass M is given by the known equation [15]: αM = 2GM/bc2 (B1) where αM is the bending angle in radians and b is the impact parameter (the distance of the passing beam from the center of the mass). Note that 1 milliradian (10-3 radian) equals 206” arc second. We now derive a similar equation to (B1) that expresses the bending angle αQ , caused by the presence of charge Q rather than a mass M. Equation (7), which is the ratio of the curvatures KQ / KS = rQ / rS ~ c2/ s2 for the electron/positron, enables us to modify (B1) and convert it to become an equation for αQ. Since the curving, and hence the bending, by the elementary charge Qe is c2/ s2 times larger than that by the lowest elementary rest mass Me of the electron/positron we can set a lower limit to αQ. This limit is achieved by replacing M in (B1) by Q∙ c2/ s2 , which gives: αQ = 2GQ/bs2 (B3) For a grazing light beam to a sphere of radius b = 10cm the bending by αQ=1miliradian =1∙10-3 is achieved by charging the sphere by Q = αQ bs2/ 2G = 7.5∙104esu. The potential of the surface is then: φ = Q/b = 7.5∙103esu or v = 300∙7.5∙103 = 2.25∙106 volt. See also Appendix C. Appendix C A Practical Experiment to Test Our Theory We suggest creating an “accumulated flowing charge” at the focal zone of an electron beam. It is a simple task to take a slow beam of several milliamps current and create a focal zone of, say, 19 several microns. Thus, we create at the focal zone a charge, similar to the charge in our example above and observe a similar bending angle. The details are as follows: 1 ampere = (1 coulomb / 1 cm2 )∙(1/1sec ) = (1 coulomb / 1 cm3 )∙(1 cm/1 sec) I = Q/(At) = (Q/V) ∙ v I - current, A - cross section area, Q - charge, V - volume, v - speed. Q = I (V/v ) The current: I = 0.1 amps = 10-1 ∙ 3∙109 = 3∙108esu cm-2sec-1. The “radius” of the focal zone volume is r ~3∙10-1cm. The velocity of the electrons is v = 104cm sec-1. The “accumulated flowing charge” at the focal zone is thus: Q ~ I (r/1cm) /v ~ 1∙104esu A laser beam, with a radius r ~3∙10-1cm, crosses the electron beam perpendicularly, but close to and above the focal zone. Using (B3) we get the exact (and not the lower limit, since we are dealing with electrons) bending angle αQ. From the center of the focal zone to the center of the laser beam: b = 2r. αQ = 2GQ/bs2 = 2∙6.67∙10-8∙104/(2∙3∙10-1) ~ 2∙10-3 = 2 milliradians. This experiment is doable. The result can verify or falsify our theory. 20