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- 1. Correlation Between Gravitational and Inertial Mass: Theory and Experimental Test. Fran De Aquino Maranhao State University, Physics Department, 65058-970 S.Luis/MA, Brazil.The physical property of mass has two distinct aspects, gravitational mass and inertial mass.The weight of a particle depends on its gravitational mass. According to the weak form of theequivalence principle, the gravitational and inertial masses are equivalent. But, we show herethat they are correlated by a dimensionless factor, which can be different of one . The factordepends on the electromagnetic energy absorbed or emitted by the particle and of the index ofrefraction of the medium around the particle. This theoretical correlation has beenexperimentally proven by means of a very simple apparatus, presented here.The gravitational mass, mg , produces δmβ represents also the inertial massand responds to gravitational fields. It shift . It can be seen by repeating thesupplies the mass factors in Newtons renormalization of the externalfamous inverse-square law of electromagnetic vertex at finiteGravitation (F12 = Gmg 1mg 2 r12 ). Inertial 2 temperature. On the other hand, it is easy to see that the inertial mass shiftmass mi is the mass factor in Newtons is related to inertial Hamiltonian shift2nd Law of Motion (F = mi a ). δΗ . Thus we can obtain the general Several experiments1-6, have expression of δmβ by means of thebeen carried out since Newton to try to inertial Hamiltonian shift δΗ ,i.e.,establish a correlation betweengravitational mass and inertial mass. δΗ c δp 2 + mi2 c 2 − mi c 2 δ mβ = = = Recently J.F.Donoghue and c2 c2B.R. Holstein7 have shown that the 2 δp renormalized mass for temperature = mi 1 + m c − mi [1]T = 0 is expressed by mr = m + δm0 i where δm0 is the temperature- where δp is the correspondentindependent mass shift. In addition, for particles momentum shift.T > 0 , mass renormalization leads to Consequently, the generalthe following expressions for inertial and expression of correlation betweengravitational masses, respectively: gravitational and inertial mass can be mi = m + δm0 + δmβ ; mg = m +δm0 −δmβ , write in the following form 2 where δm β is the temperature- mg = mi − 2δmβ = 1 − 2 1 + δp − 1m [2] mc idependent mass shift given by i δmβ = παT 2 3mi . We can look on this change in This means that a particle’s momentum as due to thegravitational mass decreases with the electromagnetic energy absorbed orincreasing temperature and that only emitted by the particle ( absorbed orin absolute zero (T = 0 K ) are emitted radiation by the particle and/orgravitational mass and inertial mass Lorentzs force upon charged particleequivalent. due to electromagnetic field). The expression of δm β In the case of radiation,obtained by Donoghue and Holstein according to Quantum Mechanics, werefers solely to thermal radiation. But, can write
- 2. 2δp = N k = N ω /( ω / k ) = U /( dz / dt ) = speed of light is reduced to <0.1m/s, =U / v [3] the Eq.(7) tell us that the gravitational masses of the atoms of the Bose-Where U is the electromagnetic Einstein condensate become negative.energy absorbed or emitted by the If the absorbed (or emitted)particle and v is the velocity of the radiation is monochromatic and haselectromagnetic waves, which can frequency f , we can put U = nhf inbe write as follows Equation(7), where n is the number of v= c [4] incident (or radiated) photons on the ε r µr 1 + (σ ωε ) + 1 particle of mass mi . Thus we obtain 2 2 2 ε ,µ and σ, are the electromagnetic nhf characteristics of the medium in which mg = mi − 2 1 + nr − 1m [8] mi c 2 the incident (or emitted) radiation is propagating ( ε = ε r ε 0 where ε r is the In that case, according to the Statisticalrelative electric permittivity and Mechanics, the calculation of n can −12ε 0 = 8.854 × 10 F / m ; µ = µ r µ0 where be made based on the well-known µ r is the relative magnetic permeability method of Distribution Probability . If all the particles inside the body have theand µ 0 = 4π × 10 −7 H / m ). For an atom same mass mi , the result isinside a body , the incident(or emitted)radiation on this atom will be n= N a [9]propagating inside the body , and Sconsequently , σ = σbody , ε = εbody, where S is the average density ofµ =µbody. absorbed (or emitted) photons on the From the Eq.(3) follows that body; a is the area of the surface of a U U c U particle of mass mi from the body.δp = = = nr [5] v c v c Obviously the power P of thewhere nr is the index of refraction, absorbed radiation must begiven by P = Nhf / ∆t = Nhf , thus we can write 2 εµ N = P / hf 2 . Substitution of N into nr = = r r 1 + (σ ωε ) + 1 c 2 [6] Eq.(9) gives v 2 a P ac is the speed in a vacuum and v is n= = D [10]the speed in the medium. hf 2 S hf 2 By the substitution of Eq.(5) where D is the power density of theinto Eq.(2), we obtain incident( or emitted) radiation. Thus 2 Eq.(8) can be rewritten in the following mg = 1 − 2 1 + U n − 1m 2 r [7] form: mc i i 2 aD Recently, L.V. Hau et al.,8 mg = mi − 2 1 + − 1m [11]succeeded in reducing the speed of mi cvf light to 17 m/s by optically inducing a quantum interference in a Bose- For σ >> ωε Eq.(4) reduces toEinstein condensate. This means an 4πfenormous index of refraction ( nr ≈ 107 ) v= [12]at ~1014Hz. µσ Light can be substantially slowed By substitution of Eq.(12) into Eq.(11)down or frozen completely 9. If t he we obtain
- 3. 3 π Rr = (∆z ) [17 ] 2 aD µσ 2 σ i µ i f 3 3mg = mi − 2 1 + − 1mi [13] 9 mi c 4πf 3 The ohmic resistance of the dipole is 11 ∆z This equation shows clearly that, Rohmic ≅ RS [18]atoms (or molecules) can have their 2πr0gravitational masses strongly where r0 is the radius of the crossreduced by means of Extra-Low section of the dipole, and RS is theFrequency (ELF) radiation. We have built an apparatus to surface resistance , ωµ dipoleproduce ELF radiation ( transmitter and RS = [19]antenna) and to check the effects of 2σ dipolethis radiation upon the gravitationalmass of a material surrounding the Thus, ∆z µ dipole fantenna ( see Fig. 1). Rohmic ≅ [20] The antenna is a half-wave r0 4πσ dipoledipole, encapsulated by a iron sphere(purified iron, 99.95% Fe; µi = 5,000µ 0 ; Where µ dipole = µ copper ≅ µ 0 andσ i = 1.03 × 107 S / m ). σ dipole = σ copper = 5.8 ×107 S / m . The radiation resistance of the Let us now consider theantenna for a frequency ω = 2πf , can apparatus ( System H ) presented inbe written as follows 10 Fig.1. The radiated power for an ωµ i β i 2Rr = ∆z [14] effective (rms) current I is then 6π P = Rr I and consequently 2where ∆z is the length of the dipole and P (∆zI ) π 2 εµ [21] β i = ω i i 1 + (σ i ωε i ) + 1 = D= = σ i µi f 3 3 2 2 S S 9 ω ε ri µ ri where S is the effective area. It can be = 1 + (σ i ωε i ) + 1 = 2 easily shown that S is the outer c 2 area of the iron sphere, i.e., ω ωc ω S = 4πrouter = 0.19m 2 . = (nr ) = = [15] 2 c c vi vi The iron surrounding the dipolewhere vi is the velocity of the radiation increases its inductance L . However,through the iron. for series RLC circuit the resonance Substituting (15) into (14) gives frequency is f r = 1 2π LC , then when 2π µ i f = fr , Rr = (∆zf )2 [16] 3 vi 1 L L X L − X C = 2πf r L − = − = 0. Note that when the medium 2πf r C C Csurrounding the dipole is air and Consequently, the impedance of theω >> σ ε , β ≅ ω ε 0 µ0 , v ≅ c and Rr antenna, Z ant , becomes purelyreduces to the well-know expression resistive, i.e., Rr ≅ (∆zω ) 6πε 0 c 3 . Z ant = Rant + (X L − X C ) = Rant = Rr + Rohmic . 2 2 2 Here, due to σ i >> ωε i , vi is For f = fr = 9.9mHz the lengthgiven by the Eq.(12). Then Eq.(16) can of the dipole isbe rewritten in the following form ∆z = λ 2 = v 2 f = π µiσ i f = 0.070m = 70mm.
- 4. 4Consequently, the radiation resistance Above this critical current, mgi Rr , according to Eq.(17), is becomes negative. Rr = 4.56µΩ and the ohmic The Table 1 presents theresistance, for r0 = 13mm , according to experimental results obtained from the System H for the gravitational mass ofEq.(20), is Rohmic ≅ 0.02µΩ . Thus, the iron sphere, mg (iron sphere ) , as a Z ant = Rr + Rohmic = 4.58µΩ and the function of the current I , forefficiency of the antenna is miron sphere = 60.50kg ( inertial mass e = Rr Rr + Rohmic = 99.56 (99.56% ). The radiation of frequency of the iron sphere ). The values for f = 9.9mHz is totally absorbed by mg (iron sphere ) , calculated by means ofthe iron along a critical thickness Eq.(24), are on that Table to beδ = 5 z = 5 π fµ iσ i ≅ 0 .11m = 110 mm . compared with those supplied by the experiment.Therefore, from the Fig.1 we concludethat the iron sphere will absorb REFERENCESpractically all radiation emitted from thedipole. Indeed, the sphere has been 1. Eötvos, R. v. (1890), Math. Natur. Ber.designed with this purpose, and in Ungarn, 8,65.such a manner that all their atoms 2. Zeeman, P. (1917), Proc. Ned. Akad. Wet.,should be reached by the radiation. In 20,542.this way, the radiation outside of the 3. Eötvos, R. v., Pékar, D., Fekete, E. (1922)sphere is practically negligible. Ann. Phys., 68,11. When the ELF radiation strikes 4. Dicke, R.H. (1963) Experimental Relativitythe iron atoms their gravitational in “Relativity, Groups and Topology” (Lesmasses, mgi , are changed and, Houches Lectures), p. 185.according to Eq.(13), become 5. Roppl, P.G et. al. (1964) Ann. Phys (N.Y), 26,442. 2 µiσ i ai 2 mgi = mi − 2 1 + D − 1mi [22] 6. Braginskii, V.B, Panov, V.I (1971) Zh. 4πc2 f 3 mi Eksp. Teor. Fiz, 61,873. 7. Donoghue, J.F, Holstein, B.R (1987)Substitution of (21) into (22) yields European J. of Physics, 8,105. 2 2 8. L. V. Hau, S. E. Harris, Z. Dutton, and C. µi σi ai (∆zI)4 −1mi [23 2mgi = mi − 2 1+ 6cS m ] H. Behroozi (1999) Nature 397, 594. i 9. M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, H. Welch, M. D. Note that the equation above Lukin, Y. Rostovsev, E. S. Fry, and M. O.doesnt depends on f . Scully, (1999) Phys. Rev. Lett. 82, 5229 ; Z. Thus, assuming that the Dutton, M. Budde, Ch. Slowe, and L. V.radius of the iron atom is Hau, (2001) Science 293, 663 ; A. V.riron = 1.40×10−10 m ; airon = 4πriron = 2.46×10−19m2 2 Turukhin, V. S. Sudarshanam, M. S.and miron = 55.85(1.66×10−27kg) = 9.27×10−26kg Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, (2002) Phys. Rev. Lett. 88,then the Eq.(23) can be rewritten as 02360]follows 10. Stutzman, W.L, Thiele, G.A, Antenna { } m gi = mi − 2 1 + 2.38 ×10 −4 I 4 − 1 mi [24] Theory and Design. John Wiley & Sons, p.48. The equation above shows that 11. Stutzman, W.L, Thiele, G.A, Antennathe gravitational masses of the iron Theory and Design. John Wiley & Sons,atoms can be nullified for I ≅ 8.51A . p.49.
- 5. APPENDIX: A Simple Derivation of the Correlation between Gravitational and Inertial Mass.In order to obtain the general pi = pg = 0 . Consequently, Eqs.[4] andexpression of correlation between mg [6] reduces toand mi , we will start with the definition ( δH = 2 δp2c2 + mi2c4 − mi c2 ) [7]of inertial Hamiltonian, H i , and andgravitational Hamiltonian, H g , i.e., Hi − H g = (mi − mg )c 2 [8] Hi = c pi2 + mi2c2 + Qϕ [1] Substitution of Eqs[7] and [8] into Eq.[5] yields Hg = c pg + mgc2 + Qϕ 2 2 [2] (m i ( − m g )c 2 = 2 δ p 2 c 2 + mi2 c 4 − mi c 2 )where mi and mg are respectively, From this equation we obtainthe inertial and gravitational masses at δp 2 rest ; pi is the inertial momentum and mg = mi − 2 1+ mc − 1 mi [9] i pg the gravitational momentum; Q is the electric charge and ϕ is an This is the general expression ofelectromagnetic potential. correlation between gravitational and A momentum shift , δp , on the inertial mass. Note that the term inside theparticle, produces an inertial square bracket is always positive.Hamiltonian shift, δH ,given by Thus, except for anti-matter ( mi < 0 ),δH = ( pi + δp)2 c2 + mi2c4 − pi2c2 + mi2c4 [3] the second term on the right hand Fundamentally δp is related to side of Eq.[9] is always negative.absorption or emission of energy. In particular, we can look on the In the general case of momentum shift (δp ) as due toabsorption and posterior emission, in absorption or emission ofwhich the particle acquires a δp at the electromagnetic energy by the particleabsorption and another δp at the ( by means of radiation and/or byemission, the total inertial Hamiltonian means of Lorentzs force upon theshift is charge of the particle).δH = 2 ( pi + δp) c2 + mi2c4 − pi2c2 + mi2c4 [4] 2 Note that δH is always positive. We now may define thecorrelation between H i and H g asfollows H i = H g + δH [5] If δH = 0 , H i = H g , i.e., mg = mi . In addition from the Eqs.[1] and[2], we can write:Hi − H g = pi2c 2 + mi2c4 − pg c 2 + mg c 4 [6] 2 2 For a particle at rest, V = 0 ;
- 6. Transmitter 9.9mHz 26mm coiled spring70mm r=123mm iron µ i σ i counterweight (dipole + wires) antenna Cross section counterweight scale antenna Front view Fig.1 - Schematic diagram of the System H
- 7. 7 mg (iron sphere ) I ( kg ) (A) theory experimental 0.00 60.50 60.5 1.00 60.48 60.(4) 2.00 60.27 60.(3) 3.00 59.34 59.(4) 4.00 56.87 56.(9) 5.00 51.81 51.(9) 6.00 43.09 43.(1) 7.00 29.82 29.(8) 8.00 11.46 11.(5) 8.51 0.0 0.(0) 9.00 -12.16 -12.(1) 10.00 -40.95 -40.(9) Table 1Note: The inertial mass of the iron sphere is miron sphere = 60.50kg

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