Relativistic Pendulum and the Weak Equivalence Principle Jarslav Hynecek * Isete, Inc. ABSTRACT This paper derives equatins fr the relativistic prper perid f scillatins f a pendulum driven by the electrical frces and fr a pendulum driven by the gravitatinal frces. The derivatins are based n the Einstein s Special Relativity Thery and in particular n the Lrentz crdinate transfrmatin, which has been eperimentally verified many times and which is a well-recgnized principle fr all the mdern physics. Since the pendulum prper perid f scillatins is an abslute inertial mtin invariant the derived frmulas may be used t study the mtin dependence f the inertial and gravitatinal masses. It is fund that the well-publicized equivalence between these tw masses, which is assumed independent f any inertial mtin, cannt be sustained and a new mass equivalence principle must be cnsidered where the equivalence f these tw masses hlds nly at rest. INTRODUCTION The pendulum is an ages prven device that has attracted attentin f many researchers in the past fr its simplicity f peratin, its accuracy t measure time, and fr its ability t study the gravitatinal r electrical fields. One can nly wnder why it was nt studied in mdern times in mre detail, since it ffers sme clues fr reslving the mystery f the inertial and gravitatinal mass equivalence, the s called Einstein s week equivalence principle [1]. Recently an interesting article was published [] where the authr derived relativistic equatins f mtin fr the pendulum starting frm a simple relativistic Lagrangian and the frmula fr the relativistic cnservatin f energy. This paper will als fcus its attentin f the relativistic equatins f mtin f the simple pendula, ne that is driven by electrical frces, and the secnd ne that is driven by gravitatinal frces and will cmpare hw these pendula behave when they underg an inertial mtin relative t the labratry crdinate system. The key idea f this wrk is t derive frmulas fr the prper perid f scillatins f the particular pendulum in terms f the pendulum physical parameters such as the mass f the bb, the length f the pendulum string, and the remaining parameters f the eperimental setup. Since the prper perid f scillatins is an inertial mtin invariant, the derived frmulas must thus als be inertial mtin invariants and * jhynecek@netscape.net 4/4/006 1
this will frce sme interesting and perhaps unepected inter-relatins between the parameters f the eperimental setup. ELECTRICAL PENDULUM The eperimental setup fr the pendula eperiments is shwn in Fig 1. Fr the electrical pendulum case it is assumed that the bb is nearly an ideal mass pint, has a negligible diameter, has an inertial rest mass m, and is charged by charge -q. The string has the rest length l, it is nt cnductive, has a negligible mass and a very large stiffness, s it des nt change its length during the swing f the bb. The string is anchred at the pint A withut any frictin r any ther mechanical resistance. Finally, the pendulum mtin will be cnsidered sufficiently slw that n radiatin effects frm the charge r mass acceleratin will have t be cnsidered. z A l α B m, -q W M, Q y L Fig.1. Pendulum psitined in a labratry crdinate system,y,z that is anchred abve a very large base plate t make the vertical electrical r gravitatinal fields abve the plate unifrm and where the fringing field effects frm the edges can be neglected. The base plate has the area A L W and may have a gravitatinal mass M, r charge Q. The gravitatinal filed f the Earth is nt cnsidered here. The bb has inertial mass m and may have charge q. The base plate which supprts the pendulum via a very stiff, mass-less and nn cnductive beam arrangement is sufficiently large, s that the electrical field abve the plate can be cnsidered
vertical and unifrm within the pendulum swing path. The area f the base plate is A L W. The base plate is als nncnductive and charged by a ttal charge Q that is embedded int the plate s matri and cannt mve. The angle f the pendulum swing is determined by the angle α that will be cnsidered very small in these eperiments. With these assumptins it is then pssible t write fr the electrical filed the fllwing: E Q. (1) ε A This result fllws directly frm the Mawell s electrical field divergence equatin and the Gauss integral frmula. If the bb were statinary then the vertical frce n the bb wuld simply be the field multiplied by charge q. Hwever, the bb is mving and therefre prducing a current, which carries with it a magnetic filed. This filed then interacts with the charge f the base plate and generates additinal frce accrding t the Lrentz frce equatin. The cmputatin f this frce may nt be s simple fr an arbitrary velcity f the bb and fr an arbitrary angle, s it is desirable t make sme plausible simplificatins. Since the charge is a universal invariant it is reasnable t epect that the frce n the charge will be the same if we cnsider the bb steady and the base plate mving in ppsite directin r when we cnsider the base plate steady and the bb mving, at least fr the case when the angle α is zer r near zer. It is thus easy t see that the mving plate prduces a magnetic filed alng the y directin with the intensity as fllws: B y µ Q v A v / c. () The vertical cmpnent f the frce acting n the bb will then be accrding t Lrentz equatin: r r r r F q E + v B, (3) equal t: F z ( ) q Q v 1 ε A v / c c. (4) In these equatins it was cnsidered that the length L underges the Lrentz cntractin, v is the bb instant velcity, ε µ c, and c is the speed f light. This frce can nw be made equal t frce referenced t the labratry crdinate system, which is nt mving even thugh the result was btained when it was mmentarily cnsidered that the crdinate system is referenced t the bb, which is mving, since charge is a mtin invariant as was already mentin abve. In the net step the pendulum mtin equatin will be derived using Newtn s secnd law written in the relativistic frm. This can be epressed as fllws: 3
d r m v v / c r F. (5) By cnsidering that the tangential velcity f the bb is: v l dα/ it is pssible using Eq.4 t write fr a small α the fllwing result: d 1 v / c dα q Q ε l m v A / c α. (6) By intrducing the prper time dτ v / c, Eq.6 is simplified as fllws: d α qq + α 0. (7) dτ ε l m A Frm this result then directly fllws that the prper perid f the pendulum scillatins fr small deflectins α is equal t: τ ε l π. (8) q m Q A Since the left hand side f Eq.8 is an invariant, the right hand side must als be an invariant and we can thus cnstruct the pendulum invariant I l m A. This quantity must be independent f any inertial mtin. Fr eample: when the whle pendulum cntraptin, including the base plate, mves unifrmly in the directin with the speed u the result fr the pendulum invariant will be: m ( ) 1 / I. (9) I u l A u c u / c Indeed the pendulum invariant will remain unchanged even when the labratry bserver will measure a different bb inertial mass and a different base plate length L. The pendulum string will be bserved having the same length as in the rest. The same result is btained fr the mtin in the z directin where the plate area is nt changed but the string will be bserved with a cntracted length. This result thus applies t any unifrm mtin withut rtatin in any directin. This result is nt surprising, it is epected, and it als fllws directly frm the classical Newtnian physics by replacing the standard time with the prper time in the equatin fr the scillatin perid. This result can be further generalized t any clck pwered by electrical fields and having bbs with an inertial mass. Fr eample a mechanical wristwatch, r quartz watch will all wrk the same way, since they use the mass inertia and the electrical fields (cnverted t a mechanical spring actin) smewhere in their system. Finally this result can be cnsidered as yet anther way t shw that the inertial mass must depend n velcity as fllws: 4
m i m. (10) v / c GRAVITATIONAL PENDULUM The eperimental setup fr the gravitatinal pendulum is the same as fr the electrical pendulum. The nly difference is that the bb and the base plate are nt charged; instead the base plate has nw a large gravitatinal mass M. The gravity f the Earth will nt be cnsidered here. It will be assumed that the plate supplies all the gravitatinal frce fr the pendulum. T be cnsistent in the derivatins, similar equatins fr the gravitatinal filed as the Mawell s field equatins will be used fr this case. M.L. Ruggier and A. Tartaglia [3] have published linearized Einstein s field equatins fr the small velcities and btain the fllwing system: r r E g 4π ρ g, (11) r 1 r 1 r Bg 4π j g, (1) c where the E g and B g are the gravitstatic and the gravitmagnetic fields respectively and the ρ and j are the mass density and the mass current density. Frm these equatins it is again simple t find the slutins fr the cnfiguratin that is used in this eperiment and arrive at the frce equatin relative t the bb as fllws: π m M v Fz 1. (13) A (1 v / c ) c In the derivatin it was cnsidered that the gravitatinal mass M beys the same velcity dependence as the inertial mass accrding t the weak equivalence principle. The inertial mass is nt an invariant, as in the previus case f the electrical pendulum, where the charge was, and cnsequently, when the transfrm f this frce is made back t the labratry crdinate system, it is necessary t divide this result by anther Lrentzian factr mtin fr the gravitatinal pendulum will thus finally becme: v / c. The equatin f d 1 v / c dα π M l A ( v / c ) By intrducing again the prper time int this equatin the result will be: 3 v 1 c α. (14) 5
v 1 d α π M c + α 0. (15) dτ l A v 1 c It is wrth nting that the result des nt depend n the mass f the bb. This is imprtant fr the cnfirmatin f the fact that the result agrees with the Galile s free fall eperiments. Since the linearizatin f the Einstein s field equatins was perfrmed nly fr relatively small velcities, it is withut prblem t cnsider that the brackets cntaining the velcity factrs actually cancel. This is reasnable, since the prper perid f pendulum scillatins shuld nt depend n the pendulum velcity. This culd als be cnsidered as a cnsistency check fr the derivatin as well as fr the linearizatin f the Einstein s field equatins. The equatin fr the prper pendulum scillatin perid can thus be simply written as: τ l A π. (16) π M This is again an epected result that can be btained directly frm the Newtnian physics by replacing the standard time with the prper time. The gravitatinal pendulum invariant then becmes: l A I. (17) M Unfrtunately, when a test is made t see if this is really an invariant, a serius prblem is encuntered. Fr eample, fr an inertial mtin in the directin this will be: l A u / c u 1 I( u ) I. (18) M c u / c The gravitatinal pendulum invariant is nt an invariant!? The nly reasnable cnclusin that can be drawn frm this result, as it was drawn previusly fr the inertial mass, is that the gravitatinal mass must depend n velcity as fllws: M g M v / c. (19) This leads t a new mass equivalence principle m m m. The Einstein s weak mass i g equivalence principle hlds nly at rest. T cmplete the prf it is necessary, hwever, t make certain that the derivatin abve is self-cnsistent. It is necessary t repeat the derivatin again using the new mass equivalence principle, since the ld weak mass equivalence principle was 6
used in the derivatin f Eq.13. It shuld als be nted that the abve-derived dependency f the gravitatinal mass n velcity des nt alter the fact that the massive bdies fllw the gedetic curves in a curved space-time as is assumed in General Relativity Thery. PENDULUM USING THE NEW MASS EQUIVALENCE PRINCIPLE The eperimental setup fr this case is, f curse, identical with the previus ne. Hwever, n gravitstatic and gravitmagnetic fields will be cnsidered. The nly assumptin that will be used is that the gravitating mass depends n velcity accrding t Eq.19. With this assumptin it is nt necessary t cnsider the crdinate system in reference t the bb. It is sufficient t stay in the labratry crdinate system and write fr the equatin f mtin simply: d 1 dα π M 1 v / c α. (0) v / c l A The square rt term n the right hand side f Eq.0 remains there after the bb s gravitatinal mass dependence n velcity was substituted there accrding t Eq.19. By intrducing the prper time again the result becmes: d α π M + α 0. (1) dτ l A N simplificatins and n apprimatins, ecept the small angle scillatins assumptin are necessary t btain this result. The gravitatin pendulum invariant is the same as befre: l A I, () M ecept that it is nw the true invariant independent f any inertial mtin similarly as in the electrical pendulum case. It is cmfrting t knw that any clck being electrical r grandfather s type will keep the same time in any mving inertial crdinate system. It wuld certainly be eceedingly strange if the electrical clck and the gravity driven clck displayed a different time when in mtin. Finally, frm Eq.10 it als directly fllws that: m i c mi v m c. (3) This frmula can be cnverted t a well knw fur-vectr energy-mmentum equatin: E c using the famus Einstein s mass-energy equatin: p m c, (4) 7
E mic. (5) Hwever, frm the abve derivatin it is clear that this equatin is strictly valid nly fr the inertial masses and nt fr the gravitatinal masses, since it is derived nly fr that case. The etensin t gravitatinal masses may nt be fully justified, r the validity f it is just a cincidence since Eq.4 is nt satisfied by the gravitatinal masses. Sme ther physical phenmenn may be respnsible fr the validity f Einstein s mass-energy equatin in nuclear mass t energy cnversins. An indicatin that such a phenmenn may be related t gravity can be fund in the relativistic Lagrangian. It is well knw that Lagrangian fr a free massive particle has the frm: v 0c c L m. (6) By identifying the right hand side f Eq.6 with the gravitatinal mass-energy the Lagrangian is simply: L mgc. (7) Since the Lagrangian is a difference f the kinetic and ptential energy in the classical limit it is clear that at rest the Lagrangian will cntain nly a ptential energy f the system thus crrespnding t the gravitatinal rest mass generated by the prcess, hence als the minus sign. In mtin, hwever, after applying the Lrentz crdinate transfrmatin, the same Lagrangian will als cntain a kinetic energy. The equatin f mtin is, f curse, easily btained as an Euler-Lagrange equatin crrespnding t the actin integral: This will lead t: which finally implies that: d t δ mgc 0. (8) t 1 c m v g d m v v / c 0, (9) v mg mi. (30) c v c This equatin prvides a clear link f the gravitatinal mass t the inertial mass and cnfirms again that the inertial mass cannt be equal t the gravitatinal mass when in mtin. 8
DISCUSSION The immediate bjectin t this result, even thugh the lgic f the derivatin is indisputable, is that the weak mass equivalence principle has already been eperimentally prven. But let s see if this is really s. It seems that all the eperiments and the theretical arguments prve nly Eq.10 and Eq.4 as fr eample in the Cmptn effect r in particle acceleratrs. There is nly ne eperimental evidence, which may be cnsidered a gd cunter argument against the validity f Eq.19, and these are the atmic weights f the elements. This is the nly case where the masses are really weighed in a gravitatinal filed f the Earth. Hwever, it seems that t assume the validity f Eq.10 autmatically fr the gravitatinal mass n the basis f the atmic weights is jumping t a cnclusin. The prcesses in the nucleus are cmple and ther frces are in play. What if the nuclear frces present in the nucleus prvide the gravitatinal mass t the nucleus when the energy and the inertial mass are supplied in a cllisin. Fr eample, the well-knwn search fr the Higgs bsn might be just the right answer fr this prblem. The nucleus is nt in an inertial mtin, s this des nt cntradict Eq.19. The arguments in supprt f Eq.19 are, n the ther hand, many: Accrding t Eg.19 phtns d nt have gravitatinal mass, s they cannt be cnfined by a black hle. Mre ver, if they had the gravitatinal mass, they wuld certainly attract each ther during the lng trip frm the distant galaies and cllapse int a single clump. There wuld be n galaies t bserve. The same applies t gravitns, if they eist, and neutrins, which are assumed t have a small rest mass. The famus frmula fr the deflectin f light by a gravitating mass is calculated assuming that the relativistic line element f the space-time metric ds 0. This als permits the assumptin f zer gravitatinal mass. The Newtn s classical calculatin f the light deflectin by a gravitating mass gives ne half f the bserved value. This is very strange and nt lgical, the result shuld be zer, since the Newtn s thery cnsiders nly a flat space-time. It is well knwn that the light bending by gravity is a curved space-time phenmenn with a crrect result btained nly when a curved space-time metric is cnsidered. This discrepancy disappears when it is cnsidered that light has n gravitatinal mass als in Newtn s classical thery. The pulsating stars such as the Cepheid variables can be cnsidered as a natural gravity driven clcks, althugh nt a purely gravity driven. The astrnmers use them as a measuring yardstick fr distance measurements in the universe. It is assumed that they keep a crrect time even when they mve away at sme, nt negligible, receding velcities. If Eq. were nt an inertial invariant large errrs in the estimatin f the universe distances wuld be btained. 9
The epansin f the universe, fr the distant galaies that recede with high velcities, is less and less affected by the gravity, s the galaies will nt slw dwn. There is n need t cnsider any strange dark matter r a csmlgical cnstant that generates a repulsive frce. The initial etremely fast epansin f the universe was nt slwed dwn by gravity, since all the matter mved at a very high velcity as we can bserve tday n the distant galaies that still carry it. The Universe wuld nt be able t eplde, since the etreme gravity wuld pull it immediately back t a small subatmic regin. The quantum vacuum generates many virtual particles that shuld have a gravitatinal mass and cllapse the universe. This is nt happening, because the fast mving virtual particles have n gravitatinal mass. CONCLUSIONS In this article it was shwn that the weak equivalence principle is nt valid fr mving bdies. This was clearly illustrated by deriving the relativistic epressin fr the prper perid f scillatins f a gravity driven pendulum. Based n these results a new mass equivalence principle was therefre prpsed where the gravitatinal mass depends n the velcity differently than the inertial mass. Several arguments against and fr the validity f the new equivalence principle were als briefly discussed. REFERENCES 1. W. Rindler, Relativity, Special, General, and Csmlgical, Ofrd University Press, Ofrd 001.. C. Erkal, The simple pendulum: a relativistic revisit: Eur. J. Phys. 1 (000) 377-384. 3. M.L. Ruggier, A. Tartaglia, Gravitmagnetic effects, published in: arxiv:gr-qc/007065 v, 3 Jul 00. 10