SIMPLE THOUGHT EXPERIMENTS THAT FALSIFY THE EINSTEIN S WEAK EQUIVALENCE PRINCIPLE. Jaroslav Hynecek 1

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1 SIPLE THOUGHT EXPERIENTS THAT FALSIFY THE EINSTEIN S WEAK EQUIVALENCE PRINCIPLE Jarslav Hynecek 1 ABSTRACT In this article it is shwn by simple thuht experiments that the Einstein Weak Equivalence Principle, defined here as the equivalence f inertial and ravitatinal masses, is nt cmpatible with the Special Relativity Thery, the Lrentz Crdinate Transfrmatin, and the axwell Thery f Electrmanetic Field. Since the General Relativity Thery is firmly based n this principle the thuht experiments and the aruments described in this article present a sinificant prblem fr the thery and thus questin its fundamental crrectness. INTRODUCTION There are several definitins f varius equivalence principles that miht intrduce cnfusin int the discussin. It is therefre desirable t first define which equivalence principle will be discussed in this paper. In ne frm the Einstein Weak Equivalence Principle (WEP) is defined as the equivalence f ravitatinal and inertial masses that is abslute and independent f any mtin. This shuld nt be cnfused with the statement that all masses fall the same way in a ravitatinal field. If the ravitatinal mass dependence n velcity is universal, applicable t all masses, then all masses wuld fall the same way even if their inertial mass depended n velcity differently than the ravitatinal mass. The Galile free fall experiments wuld still be satisfied. Hwever, the General Relativity Thery (GRT) is based n the abslute equivalence f the inertial and ravitatinal masses, reardless f their relative mtin, s it is this equivalence that will be investiated in this paper by way f thuht experiments. In this wrk it will be als assumed that the Special Relativity Thery (SRT) and in particular the Lrentz Crdinate Transfrmatin (LCT) tether with the axwell Electrmanetic Field Thery (ET) are valid theries that clsely reflect reality. 1 jhynecek@netscape.net

2 E FIELD EXPERIENT In this sectin ET and LCT will be used and tested by way f cnductin a simple thuht experiment as fllws: tw nncnductive parallel plates chared by chare + mq and mq respectively, with chare embedded int the plate s matrix s that it cannt mve, will be placed in the labratry crdinate system. The plates will mve hrizntally with a unifrm velcity v as shwn in Fi.1. The plates will be separated by a distance d, and their mass will be nelected in this first experiment. The plates will reflect liht that will bunce between them. The impact anle α f the phtns will be such that the hrizntal drift speed f the phtns will match the hrizntal speed f the plates in rder t always keep the same amunt f phtns cnfined between the plates. +mq v h f α h f d L -mq A = L W Fi.1. ET test experimental setup: tw plates built frm an insulatin material have chare + mq and mq embedded in them, lare area A, and are psitined at a small distance d apart frm each ther. The plate s inner surfaces are reflective and certain number f n phtns is cnfined between the plates. The phtn incidence anle with the plates is such tat the phtn hrizntal drift velcity matches the plate s velcity. As is well knwn frm ET the chared plates are attracted t each ther by the electrstatic frce and since they are als mvin relative t the labratry crdinate system there is additinal frce cmpnent between the plates due t the Lrentz frce. The resultin frce actin n the plates, assumin that the peripheral field effects can be nelected, is thus described by the fllwin equatin: F q m q v ε A 1 v / c c =, (1)

3 where A is the pate s area when the plates are at rest relative t the labratry, ε is the dielectric cnstant, and c the vacuum speed f liht. In Eq.1 it was als cnsidered that the plate s lenth in the directin f mtin underes the Lrentz lenth cntractin accrdin t SRT. In rder t prevent the plates frm mvin clser t each ther and cllidin, certain number f phtns is injected between the plates under the anle α that exert the necessary cunter pressure n the plates. The resultin phtn frce is exactly equal t the plate s attractive frce t keep the system in equilibrium. The balancin phtn frce f n phtns is fund frm the frmula: p n h f F p = n = csα. () t t c In derivin this frmula it was cnsidered that durin the phtn reflectins between the plates the phtn enery and mmentum are cnserved and n mmentum is transferred frm the phtns t the plates. The time between the phtn impacts n the plates and the csine f the impact anle α are fund frm the fllwin frmulas: d t =, (3) c v cs After substitutin int Eq. the frmula fr the phtn frce n the plates becmes: n h f v F p =. (5) d c It is reasnable t assume that the plates can always be kept apart at a cnstant distance α = 1 v / c. (4) d reardless f their hrizntal mtin. If the plates were nt mvin relative t the liht surce, but the anle f incidence f phtns n the plates was chaned, the frequency f the phtns wuld have t increase accrdin t frmula f = f / csα, where f is the frequency fr the incidence anle α = 0, r the frequency f the liht surce that mves with the plates. The final frmula fr the phtn frce can thus be written as fllws: n h f Fp = 1 v / c. (6) d In rder t prevent the plates frm cllidin the E frce and the phtn frce must balance F = F. By equatin these frces it is apparent that the velcity dependent terms q p 3

4 f the resultin equatin cancel, which indicates that the balance is maintained reardless f the velcity as expected. The remainin parameters must then satisfy the fllwin: hε c A A m = n = nα, (7) q λ λ d d where f was replaced by f = c / λ. The parameter α is the famus Smmerfeld fine structure cnstant α = This is an interestin result suestin that the electrns, which chared the plates, miht be sme srt f tplical dynamic structures where the E field is balanced by the phtn inertial frce. Hwever, this is nt a tpic f this paper and will nt be discussed here any further. Since this thuht experiment yielded the expected result usin ET, LCT, and SRT, the same cnsideratins will be used next fr the case where the E field is replaced by the ravitatinal filed t test the crrectness f GRT. GRAVITATIONAL FIELD EXPERIENT In this sectin the insulatin plates in the E experiment are replaced by the massive plates that will nt be chared but will attract each ther by the ravitatinal frce. It is als imprtant t nte that there is n ravitatinal field r any static electrical r manetic field between the plates; s that the space-time between the plates is nt distrted in any way. The Earth s ravitatinal field is als nt cnsidered in these experiments. The standard SRT and LCT can thus be used here withut any prblems. The phtns will mve the same way as they wuld in any ther inkwski flat spacetime aln the straiht lines withut any bendin. The repulsive phtn frce is calculated the same way as in the previus case. The ravitatinal frce is calculated frm the fllwin frmula: where F π κ =, (8) A 1 v / c is the ravitatinal mass f the plates and κ is the ravitatinal cnstant. Equatin nw the ravitatinal frce with the phtn frce t keep the plates aain in equilibrium the fllwin result is btained: 4

5 hc A v n πκ λ d c = n A v λ d c = p, (9) where is the Planck mass. This result is very imprtant fr judin the crrectness and p crrespndence with reality f varius theries f ravity. When the standard GRT is cnsidered and linearized fr the weak fields [1] the linearizatin result resembles ET equatins: E r = 4πκρ, (10) B r = 0, (11) r r 1 B E =, c t (1) r 1 v 1 E 4πκ r B = j. c t c (13) In this apprach, where the ravitmanetic B and ravitelectric E fields were intrduced, the similar frce equatin as the Lrentz frce equatin f the E thery is als derived: v v r v r F = E + B. (14) c When these equatins are used in the abve described experiment, the ravitatinal frce equatin between the plates, Eq.8, becmes: = π κ v F, (15) A 1 v / c c and frm that the dependency f the ravitatinal mass n velcity is as fllws [] : = n p v A c λ d v c. (16) It is apparent that Eq.16 will shw the inertial mass type f dependency n velcity fr small velcities but the dependency will nt be exact and will fail fr larer velcities due t the well knwn factr f tw fiurin in the denminatr velcity bracket. This is 5

6 a prblem. Since Eq.9 was derived usin SRT thery the result shuld perfectly match Eq.16, r Eq.16 shuld perfectly match the inertial mass dependency n velcity since the plates can be chsen with arbitrarily small masses t satisfy the GRT linearizatin assumptin. The trublesme denminatr bracket is the cnsequence f the ravitelectrmanetic equivalent f the E Lrentz frce that has nt been experimentally directly bserved yet. If such a frce des nt exist, then the ravitatinal mass must depend n velcity accrdin t Eq.9. It is peculiar that GRT cannt prvide an accurate linear equatin even fr weak fields fr the ravitatinal mass dependency n velcity (Lrentz velcity factr) fr such a simple experimental arranement as is described in Fi.1. This fact prves that GRT is nt cnsistent with SRT and casts a reat suspicin n its crrectness. It seems unreasnable t derive the linearized vectr thery frm GRT r use a tensr thery t describe the ravity since the ravity has nly ne type f the ravitatinal chare and nly ne type f the attractive frce between masses. Sme authrs [3] suest that the factr f tw, which causes the prblem in Eq.16, is the carryver frm the fact that the classic ravitn is a spin tw particle. It is therefre reasnable t expect a vectr thery fr the E fields, since there are bth attractive and repulsive frces, with the phtn havin the spin equal t 1. Hwever, fr the ravity a simple scalar thery shuld suffice with the ravitn havin the spin equal t zer. As a cnsequence f the scalar thery f ravity the Lrentz factr bracket in the denminatr f Eq.16 wuld be equal t unity. In such a thery, r in a thery withut the ravitmanetic effect, the ravitatinal mass and the inertial mass must have the fllwin dependencies n velcity: c = 1 v /, (17) i( v) =, (18) 1 v / c where is the rest mass, in rder nt t cntradict SRT. This seems mre reasnable than the dependency described by Eq.16. Of curse in such a scalar thery f ravity the phtns d nt have any ravitatinal mass, nly the inertial mass, which aain cntradicts ne f the key assumptins f GRT and directly challenes all the theries and muntains f publicatins cnstantly bein written abut Black Hles. 6

7 Perhaps in the future an experiment can be devised that can determine the ravitatinal mass dependence n velcity with certainty and finally put t rest the questinable crrectness f GRT. One pssibility f a testin setup arranement is shwn belw in Fi.: Trsin balance enclsure Trsin balance pivt pint L F tr1 tr R Flywheel crss sectin Fi.. Gravitatinal mass dependence n velcity measurement setup (tp view): tw flywheels with masses and radius R are pwered by mtrs tr1 and tr. trs are munted hrizntally n a vibratin insulated table in prximity t an enclsed trsin balance. trs, which can be interal with the flywheels such as in as turbines, can be independently turned n and ff as well as their RP adjusted. L is adjusted fr the ptimum sensitivity and is apprximately equal t L = R /. Tw mtrs that have massive flywheels with radius R and mass attached t them are psitined with their turnin axels in a hrizntal directin n a vibratin islated table in a labratry. Tw test masses are suspended vertically in a trsin balance arranement in a shieldin enclsure in such a way that the masses are in an ptimum distance L frm the centers f the flywheels. The trsin balance arranement is in equilibrium when bth mtrs are turned ff. After ne mtr r the ther mtr are turned n, the deflectin f the balance is read ut dependin n the RP f the mtrs. The expected frce F ruhly estimated as fllws: actin n the trsin balance fr the ptimum distance can be 7

8 ω κ F = ±. (19) 3 3 c The sin depends f the thery. It is interestin t nte that this result des nt depend n the radius f the flywheel, just n the RP. This may be helpful in the cnstructin f the test set up and in develpin the measurement technique. Fr masses 100k and the anular velcity f 100,000 RP the frce will be: 1 F = femt N. This value is indeed very small, but perhaps sme mtr RP mdulatin technique and the trsin balance resnance tunin can be emplyed t imprve the sensitivity. CONCLUSIONS In this article it was shwn, usin simple thuht experiments that the ravitatinal mass must depend n velcity differently than the inertial mass. This falsifies the Einstein s WEP. Furthermre it was shwn that the abslute equivalence f the inertial and ravitatinal masses, as is pstulated in GRT, is nt cmpatible with SRT and ET theries. There have been already several similar prfs published previusly [4], but these prfs are mre cmplicated with additinal tacit assumptins that usually fail t cnvince skeptics stubbrnly adherin t the prevailin dma. Hpefully, the apprach selected in this article is simple enuh and sufficiently clear with minimum hidden assumptins that this prblem is nw avided. An experimental verificatin f the ravitatinal mass dependency n velcity has als been prpsed and the expected manitudes f frces that need t be detected estimated. REFERENCES 1..L. Ruier, A. Tartalia, Gravitmanetic effects, published in: arxiv:rqc/ v, 3 Jul 00.. J. Hynecek, Remarks n the Equivalence f Inertial and Gravitatinal asses and n the Accuracy f Einstein s Thery f Gravity, Physics Essays, v 18, N., Jun 005, p B.ashhn, Gravitelectrmanetism: A Brief Review, published in: arxiv:r-qc/ v, 17 Apr