Instrument for Measuring the Earth s Time-Retarded Transverse Gravitational Vector Potential

Transcription

1 -1- Instument fo Measuing the ath s Time-Retaded Tansvese Gavitational Veto Potential J. C. Hafele 1 Hee within the basi design fo a gound-based instument fo measuing the magnitude of the ath s time-etaded tansvese gavitational veto potential is desibed. The fomula fo the ath s tansvese veto potential is deived fom the known fomula fo the neolassial timeetaded tansvese gavitational field (axiv:94.383v [physis.gen-ph] 5May1). The devie senses the elativisti shift in the fequeny of lase-diode osillatos set into iula motion at the tips of a two-am oto. The instument employs fibe optis and a digital eletoni intefeomete/spetomete to measue the effet of the elativisti time dilation on the fequeny-modulated (FM) hamoni amplitudes in the beat signals between the tip-diodes and a stationay efeene diode. The FM amplitudes depend on the oientation of the oto. Fo the vetial-eastwest oientation with a oto fequeny of 73.9 Hz the pedited FM amplitudes fo ovetones at 148 Hz Hz and 96 Hz ae espetively Hz Hz and Hz. The ovetones in the beat signals an be amplified and obseved with a tunable FM digital audio amplifie. The measued values fo the hamonis of the veto potential an be detemined by bak-alulating what the amplitudes must have been at the input to the amplifie. The instument an be used to establish the speed of the ath s gavitational field and to study the stutue of the ath s mantle and oute oe. Key Wods: time-etaded gavity; tansvese gavity; speed of gavity; gavity instuments 1. INTRODUCTION The mystey of the flyby anomaly has been esolved. In May of 1 this autho published a new neolassial time-etaded theoy that explains exatly the small anomalous speed hange epoted in 8 by NASA sientists fo six spaeaft flybys of the ath. (1) The neolassial time-etaded theoy is based on the slow-speed weakfield appoximation fo geneal elativity theoy. () Any ausal inteation equies time etadation whih means that it takes a etain amount of time fo a ausal field to popagate fom a moving point-mass soue to a possibly moving distant field-point. The pointmass soues in a lage otating sphee (a simulation of the ath) ae in iula motion aound the otational axis of the sphee. This iula motion geneates a time-etaded tansvese gavitational field that is othogonal to the lassial Newtonian adial gavitational field. The tansvese field is popotional to the atio of the equatoial sufae speed of the otating sphee to the speed of S. 4 th St. Laamie WY 87 USA ahafele@besnan.net

2 -- popagation of the gavitational field. Motion of the field point may ause a time-dependent tansvese gavitational field. This time dependene geneates an indution field that is popotional to the time deivative of the tansvese field. The pedited speed-hange fo a flyby is popotional to the indution onstant. The exat numeial value fo the indution onstant is at this time unetain but the obseved speed hanges fo six ath flybys indiate that the ode-of-magnitude is 1-4 s/m. The objetive of this epot is to desibe a feasible design fo a gound-based instument that an podue an exat measue fo the indution onstant and theeby give a diet measue fo the speed of the ath s gavitational field. The poposed instument employs a high-speed two-am oto with lase diodes at the tips. If a digital intefeomete/spetomete is used to supeimpose the lase beams the esulting hamonis in the beat signals would be fequeny-modulated (FM). The beat-signals fo the vetial-east-west oientation would ontain detetable ovetones fom the fist to the thid. Ode-of-magnitude estimates fo the FM hamoni amplitudes fo the vetial-east-west oientation ae listed in Table I (page 3). In setion.1 the dependenes of the vaious gavitational aeleation fields on u/v fo a lage otating sphee in the slowspeed weak-field appoximation ae eviewed. The lassial adial aeleation field is independent of (u/) and the neolassial tansvese aeleation field depends on (u/) 1. The gavitomagneti fields being of highe ode in (u/) ae negligible. Thee ae thee elevant potentials; the sala potential fo the lassial Newtonian adial gavitational field a veto potential fo the neolassial time-etaded tansvese gavitational field and anothe veto potential fo the indution field. Fomulas fo the needed potential funtions ae deived in setion.. The elativisti pope time eoded by a lok depends on the speed and on the thee potential funtions. The fomula fo the pope time fo loks nea the ath s sufae in the slow-speed weak-field appoximation is deived in setion.3. The fomula fo the elativisti pope fequeny fo a lok s osillato whih depends on the lok s pope time is deived in setion 3.1. The pope fequenies fo lase-diode osillatos in a two-am oto depend on the oto s oientation whih means the esulting beat signals also depend on the oto s oientation. Odeof-magnitude estimates fo the FM hamonis in the vetial-east-west oientation the vetial-noth-south oientation and the hoizontaleast-west oientation ae deived in setions and 3.4. The vetial-east-west oientation is the only one that podues a ih aay of detetable hamonis. The othe oientations an be used to test the instument.

3 -3-. GRAVITATIONAL FILDS OF A LARG ROTATING SPHR IN TH SLOW- SPD WAK-FILD APPROXIMATION.1 Odes of u/ fo the Gavitational Fields of a Lage Rotating Cental Sphee Assume the ath an be simulated by a lage otating isotopi sphee of adius mass M angula speed and adial mass-density distibution. In the slow-speed weak-field appoximation geneal elativity theoy edues to lassial time-etaded eletomagneti field theoy. Fomulas fo the gavitational fields in the slow-speed weak-field appoximation an be found in W. Rindle s textbook. () The fomula that is analogous to the Loentz Foe Law is 1 a e u h (.1.1) whee a is the gavitational aeleation field of the ental sphee e is the gavitational equivalent of the eleti field h is the gavitational equivalent of the magneti indution field u is the veloity of a field point outside of the ental sphee in a nonotating (inetial) geoenti fame of efeene and is the vauum speed of light. The field e is defined by the gadient of the gavitational sala potential gad and the field h is defined by the ul of the gavitational veto potential ulv. Rindle defines the time-etaded potential funtions fo e and h by using a ypti notation [] and [u] to denote the value etaded by the light tavel time to the oigin of whee is the adial distane fom a iulating point-mass of mass dv to the field point. () The fomulas fo and V using this notation ae dv G 1 u dv V G. (.1.) Let s laify what this notation means fo a lage ental otating isotopi sphee. Let be the adial mass-density distibution and let be the mean value fo. The fomulas fo and V an be ewitten as G dv 1 dv G u V. (.1.3) Let t be the oodinated time at the field point and let t be the etaded time at the iulating soue point. Let the spae and time oodinates fo the field point be (t) whee the definition fo (t) (bold fae Couie) is now hanged to be the adial veto

4 -4- fom the ente of the sphee to the field point. Let the spae and time oodinates fo the soue point be ( t ) whee (t ) (bold fae Aial) is the adial veto fom the ente of the sphee to the soue point. The spae vaiables ae funtions of t and the spae vaiables ae funtions of t. To satisfy ausality a gavitational signal emitted at t by a pointmass at the soue point aives at the field point at a slightly late time t. Let be the etaded veto distane fom the iulating soue-point to the field-point defined by (tt ) (t)-(t ). If the speed of the signal is and is the absolute magnitude of the fomula elating t to t is t t. (.1.4) The etaded-time deivative dt/dt is dt dt 1 1 d dt. (.1.5) The fomulas fo and V (q. (.1.3)) an be ewitten expliitly in tems of (tt ) and dv(t )= os( )dd d as follows. 3 d d G os( ) d (t) (t) 3 G (t)d u d os( ) d V (t) (t). (.1.6) Afte the tiple integation ove the volume of the sphee the funtions and V depend only on the time t at the field point. The dive time t is pe-detemined by the otation of the ental sphee. The diven time t is post-detemined by the motion of the field point. The motion of the field point may o may not be onstained by nongavitational foes. The integand fo the tiple integal ove the volume is a ylial funtion of t and t is in tun a ylial funtion of t. The dependene of on t an be found by integation ove t fom to t. But fist thee is the tiple integation ove the volume of the sphee. This fist integation an be aomplished by tansfoming the integation vaiable fom t to t. By using the Jaobian fo this tansfomation dt/dt the integation ove time esults in two

5 -5- integals as follows. t t(t) dt (t)dt (t) dt dt t() t(t) t(t) 1 d (t)dt (t) dt dt t() t(). (.1.7) The fist integal all it (t) gives the lassial Newtonian instantaneous ation-at-a-distane sala potential fo e. The seond integal all it 1 (t) gives the time-etaded potential funtion that is popotional to the atio (d /dt )/. The seond integal gives the potential funtion fo a votex field. Theefoe the gadient gad 1 (t) is zeo but the ul ul 1 (t) is not zeo whih means the funtion 1 (t) (bold fae hi) is a veto potential funtion. The fomula fo the total gavitational potential fo e whee 1 (t) is the magnitude of the veto 1 (t) beomes t(t) (t) (t)dt t() t(t) 1 d 1(t) (t) dt dt t() (t) (t) 1(t) χ χ. (.1.8) The fomulas fo the sala potential and the magnitude of the veto potential χ 1 beome (f. qs. (.1.) and (.1.6)) 1 dv G 1 d dv G (.1.9) dt whee the tiple integal is defined by q. (.1.6). Notie that is of ode (u/) and 1 is of ode (u/) 1. The fomula fo the analogous gavitational eleti field beomes (f. q. (.1.1)) 1 χ 1 e e e gad ul. (.1.1) Next onside the veto potential V fo h. The instantaneous and time-etaded integals fo V beome t t(t) dt V(t)dt V(t) dt dt t() t(t) V(t) V (t)dt t() t(t) t(t) 1 d V(t)dt V (t) dt dt t() t()

6 -6- t(t) 1 d 1(t) (t) dt dt t() V V. (.1.11) The fomulas fo V and V 1 beome dv V G u u 1 d dv dt V 1 G. (.1.1) Notie that V is of ode (u/) 1 and V 1 is of ode (u/). The fomula fo the analogous gavitational magneti indution field (gavitomagneti fields) beomes () h h h ulv ulv. (.1.13) Theefoe the odes fo the analogous Loentz Foe Law beome (f. q. (.1.1)) 1 1 χ 1 a e gad ode (u/) a e ul ode (u/) 1 a 4 4 u h u ulv ode (u/) a3 4 4 u h1 u ulv 1 ode (u/) 3. (.1.14) This shows that the analogous gavitational magneti indution field elative to χ 1 is ineased slightly by the novel fato of 4 but is deeased by the huge denominatos of o. The aeleation field a is the lassial Newtonian instantaneous ation-at-adistane adial gavitational field g whee the subsipt indiates that this field is dieted adially outwad o inwad. The aeleation field a 1 is the neolassial time-etaded tansvese gavitational field g e whee the subsipt e indiates that this field is dieted towads the east o west. The emaining aeleation fields ae of highe ode in (u/) and an be negleted fo a fist ode appoximation. The neolassial tansvese gavitational field g e auses a small petubation on the lassial Newtonian field g. It has been shown that an indution field designated F that is popotional to the time ate of hange of g e explains exatly the obseved anomalous hange in the speed of six ath spaeaft flybys epoted by NASA sientists. (1)

7 -7-. Gavitational Potential Funtions fo a Lage Rotating Sphee in the Slow-Speed Weak-Field Appoximation Let e e and e be othogonal unit vetos fo the spae-time spheial system (t); e is dieted adially outwad e is dieted southwad and e is dieted eastwad. The pola angle o olatitude ineases towads +e (southwad); the latitude ineases towads -e (nothwad). The dot poduts e e = et. The oss poduts e e =e et. The self dot poduts e e =1 et. The self oss poduts e e = et. The time t is the obseved oodinated time at the field-point. The fomula fo the pope veloity of the field point u in the (t) spae-time system beomes d d d u eu eu eu e e e. (..1) dt dt dt The fomula fo the gavitational field g in the (t) system beomes g g g e g e g. (..) e e The tansvese gavitational field is pependiula to the adial gavitational field g g e =e e g g e =. By the fundamental hypothesis fo the time-etaded theoy the aeleation field F is popotional to the time deivative of g e. If k is the onstant of popotionality the fomula fo F is (the fomula fo the ul an be found in J. D. Jakson s textbook (3) ) dge 1 F ek e F. (..3) dt Solving fo (F )/ and integating fom t= to t gives t t t d dt dg dt d dt e F dt F dt k dt. (..4) This equation is satisfied fo all values of t if and only if t d dt dge F k dt dt d dt whih leads to d dt dg d F k dt dt e

8 -8- and t t dge dge (..5) k k d u F (t) dt dt dt dt dt v whee the indution speed is defined by v k =1/k. The exat value fo k is unetain at this time but it is known fom the NASA flyby data to have the ode-of-magnitude value k1 1-4 s/m (v k 1 km/s). (1) The fomula fo the lassial sala potential funtion is dv GM G A (..6) whee the fomula and numeial value fo A ae GM A m s 7. (..7) The fomula fo g beomes g GM GM. (..8) () () e The fomula fo the magnitude of the neolassial veto potential funtion 1 is 1 G 1 d dv dt. (..9) The fomula that onnets g e to 1 is g χ. (..1) e 1 Pehaps the easiest way to find the fomula fo the magnitude of the veto potential χ 1 is to invet the ul opeation using the known fomula fo g e. (1) Let e be the eastwad omponent of the sideeal angula speed fo the field point e d/dt and is the geoenti latitude fo the field point. The ath s sideeal angula speed is d /dt. In the (t) system g e =e g e and χ 1 = e 1. The fomula fo the signed magnitude of g e is e g e( ) ( 1)A g os ( )PS() whee the oeffiient A g with g = is defined by (..11) A g G. (..1)

9 -9- The powe seies PS() is defined by PS() C C C4 C6. (..13) Numeial values fo the oeffiients ae C.718 C.45 C 4.7 C (..14) Notie that PS() is not a pue invese-ube funtion. If the ath wee a homogeneous sphee i.e. if wee equal to the oeffiient C would equal 1 and C C 4 and C 6 would be zeo in whih ase PS() would be a pue invese-ube funtion. The fomula fo the ul opeation in spheial oodinates an be found in J. D. Jakson s textbook; (3) 1 e ( e 1) e ( 1). (..15) Solving fo the patial deivative gives ( 1) ge. (..16) Solving this fomula fo 1 and substituting q. (..11) fo g e gives ( 1) g ( )d 1 e e d Ag os ( ) PS(). (..17) Given a value fo the integal ove an be solved easily by using numeial integation but a powe seies epesentation will be useful. Let PS 1 () be a fou-tem powe seies defined as 4 6 PS 1() C1 C1 C14 C16. (..18) By using a least-squaes fitting outine the following oeffiients wee found to povide an exellent fit of the powe seies to the integal. C C C C Now the fomula fo 1 an be ewitten as 6 4. (..19) e e 1 Ag os ( )PS 1() A 1 os ( )PS 1() whee the fomula and numeial value fo A 1 ae (..) A 1 Ag m s. (..1)

10 -1- Let be the veto potential fo F. The fomula that onnets F to is 1 F e F χ e ( e ) e. (..) Solving fo the patial deivative gives ( ) F. (..3) Solving this fomula fo and substituting q. (..5) fo F gives (t) t ( 1) ( 1) d (t) Fd F dt dt () t t ( 1) 1 dge u dt udt dt v k ( 1) u t t dge dt udt. (..4) dt v k The fomula fo dg e /dt is given by (f. q. (..11)) dge d e ( 1)A g os ( )PS() dt dt. (..5) The ath s otational sufae speed at the equato all it v q povides a onvenient efeene speed v q = = m/s. The fomula fo an be ewitten as v t t q v q d e u u A os ( )PS() dt dt vk dt vq v q g v u d u A os ( )PS() dt dt so that (t) A I t t q e vq dt vq (..6) whee the fomula and numeial value fo A ae v v A A A 1.1 m s (..7) q q g 1 vk vk and the fomula fo the double integal I is v u d u I (t) os ( )PS() dt dt t t q e. (..8) vq dt vq Notie that is zeo if u = o if e =.

11 Wold Lines and Pope Times The wold line fo a small mass m is the path of m in spae-time the spae being thee dimensional nonotating and nonaeleating (inetial) spae and the time being one dimensional oodinated time fo a bakgound sea of hypothetial synhonized loks. Let the spae-time oodinates fo m be those fo the field-point in the (t) system. Assume the mass m is an ideal lok. Also assume the pope speed u << and the absolute magnitude of the potentials << χ 1 << and χ <<. Let d and dt be elemental hanges of path length along the wold line. Let d be the elemental hange in pope time duing the oesponding hange (ddt). Cloks eod pope time. Moving loks un slow and loks in a deepe (moe negative) gavitational potential un slow. The fomula fo pope time in the slow-speed weak-field appoximation to fist-ode in u / and / is 1u d 1 1 dt. (.3.1) To solve this fomula equies knowing the pope speed u(t) and the signed magnitudes fo (t) 1 (t) and (t) at evey instant of oodinate time t along the wold line. Let s do a simple thought expeiment. Suppose a lok is tanspoted aound the ath at the equato (=) fistly in the dietion of the ath s otation (eastwad) and seondly oppositely to the ath s otation (westwad). Suppose = fo both tips. Suppose the tip gound speed is +v q fo the eastwad tip and is -v q fo the westwad tip. Suppose the time eoded by the taveling lok is ompaed with the time eoded by anothe (efeene) lok that emains stationay on the ath s sufae. Suppose the two loks ae at the same plae and ae synhonized to ead the same time (t=) befoe eah tip aound the ath. Let t be the oodinate time inteval (sideeal time) equied fo eah tip. A good fist appoximation fo t is /v q =3.93 hous. Let e and w be the time eoded by the taveling lok duing the eastwad and westwad tips espetively. Let be the time eoded by the gound efeene lok duing eah tip. Pope time intevals ae found by integation of q. (.3.1). t 1u 1 1 dt. (.3.) The oodinate speeds potentials and pope times ae as follows. Gound efeene lok: u u v m s q

12 -1-7 A m s 1 A 1 os ()PS 1( ) m s A I m s (u = and dg e /dt=) 1 vq A 1 t astwad tip: u u v m s q 7 A m s. (.3.3) 3 1 A 1 os ()PS 1( ) A 1PS 1( ) m s A I m s (u = and dg e /dt=). The fomula fo the pope time diffeene and its numeial value ae vq A A 1PS 1( ) 1 vq A e 1 t 1 t Westwad tip: u u m s 3 vq A 1PS 1( ) t ns 7 A m s 3 1 A 1 os ()PS 1( ) A 1PS 1( ) m s A I m s (u = and dg e /dt=) A A 1PS 1( ) 1 vq A w 1 t 1 t. (.3.4) 1 vq A 1PS 1( ) t ns astwad plus thee times westwad:. (.3.5) e 3w 3 vq A 1PS 1( ) 1 vq A 1PS 1( ) t 3 t A1PS 1( ) 5 t ns. (.3.6)

13 -13- This heuisti exeise shows that the taveling lok elative to the gound efeene lok would lose about 31 nanoseonds duing the eastwad tip and would gain about 13 nanoseonds duing the westwad tip. Notie that the elatively lage effet of the sala potential A anels out but the effet of the squaed speed atio u / and the effet of the veto potential A 1PS 1( ) do not anel out. The small effet of the veto potential an be extated by alulating e +3 w whih fo this thought expeiment is vey small about 1-5 ns/day. Moden peision loks an detet a time diffeene as small as about 1 ns/day but pobably annot detet a time diffeene of about 1-5 ns/day. An instument designed to detet and measue the tansvese veto potential utilizing elativisti time dilation will need to employ a method that suppesses the elatively huge effets of u / and A. 3. INSTRUMNT FOR MASURING TH ARTH S TIM-RTARDD TRANSVRS GRAVITATIONAL VCTOR POTNTIAL 3.1 Pope Fequeny fo a Clok Reoding Pope Time What is a physial lok? By definition a lok is a devie whih eods the numbe of yles (peiods) of an osillato. If the pope time fo a lok diffes fom the oodinate time t the fequeny of the lok s osillato must diffe by a ommensuate amount. The instantaneous pope time fo a lok given by q. (.3.1) is 1 u(t) (t) 1(t) (t) d dt dt. (3.1.1) Integation ove a time inteval fom to t gives the eoded pope time afte an inteval of oodinate time t. t t (t) 1 dt d 1 u(t) (t) (t) (t) t dt t dt. (3.1.) The absolute value fo the quantity in paenthesis is vey small odes of magnitude less than one. Let f be the lok-osillato s fequeny (yles pe seond o Hetz). Let f be the fequeny fo d/dt=. The fomula fo the fequeny beomes t df f(t) f dt. (3.1.3) dt

14 -14- Compaing q. (3.1.3) with q. (3.1.) indiates that 1 df 1 u(t) (t) 1(t) (t) f dt. (3.1.4) If the lok is losing time i.e. if d/dt< the lok-osillato s elative fequeny is less than 1 i.e. df/dt<f and vie vesa. The lok s osillato may be a flywheel balane wheel eletomagneti ystal osillato optial osillato atomi osillato o some othe kind of (pefeably stable) osillato. To get a vey lage value fo f suppose the lok s osillato is a edlight lase-diode (as found in lase pointes lase ba-ode eades et.) fo whih the optial wavelength is about 7 nm and the intensity usually is about 1 mw. Fo ed light the appoximate optial fequeny is 14 f Hz m. (3.1.5) The numbe of photons pe seond in a 1 mw beam of ed light is vey lage. The photon enegy flow divided by Plank s onstant times the fequeny gives 1 mj/hf = photons pe seond. At this intensity thee is no need fo a statistial analysis of the flow of photons. 3. Pope Fequenies fo Osillatos Ciulating in a Two-Am Roto Shematis fo thee oientations of a two-am oto with lase-diode osillatos at the tips ae shown in Fig. 1. Let R be the am length let be the angula position of the oss am let d/dt be the oto s angula speed and let v be the tip gound speed v R. Suppose R=1 m. The ath s otational sufae speed at the geoenti latitude is v ()= os(). If v=v q =v ()= m/s the angula speed =v q /R= ad/s and the oto s peiod P ot =/=13.54 ms. The ps and pm would be ps=1/p ot = Hz and pm=6/p ot =4463 ounds/minute. At the tips the speed would be ultasoni and the gvalue would be vey lage v q /R=1968g whee g is the aeleation of gavity at the ath s sufae g =9.85 m/s. To safely ahieve suh high values the oto would need to be appopiately designed onstuted fom high stength mateials and opeated in a vauum hambe. Suppose the optial signal fom the tip-diodes is onduted though optial fibes to the ente of a hollow axel shaft and out though a otating fibe-opti oupling to a stationay digital eletoni intefeomete/spetomete. Suppose that in addition a thid idential lase-diode osillato is loated at a onvenient stationay point on the axis of the oto to seve as a efeene osillato. Call the thee signals: os1 os and os. When two light beams of the same intensity the same polaization and nealy the same fequeny ae supeimposed the esult is a light beam with two fequenies half the sum of the two initial fequenies and

15 -15- (a) (b) () Figue 1. Shematis fo a two-am oto. (a) Vetial-east-west oientation; the vetial plane of the oto is moving towads the east with speed v (). (b) Vetialnoth-south oientation; the vetial plane of the oto is moving into the pape with speed v (). () Hoizontal-east-west oientation; the hoizontal plane of the oto is moving towads the east with speed v ().

16 -16- half the diffeene between the two initial fequenies. The following tig identity shows this effet. 1 1 os( ) os( ) os ( ) os ( ). (3..1) The intensity (powe in Watts) of a light beam is popotional to the squae of the sum. If the squae beomes os( ) os( ) 1 4 os ( ) os ( ) beat signal aie signal. (3..) The fato equaling half the sum is alled the aie signal and the fato equaling half the diffeene is alled the modulation signal o the beat signal. If and ae ylial (hamoni) funtions of time i.e. if -os(nt) the beat signal is fequeny modulated (FM). The fequenies in white light whih ontains a ontinuous spetum of fequenies an be identified by using a spetomete. But fo monohomati light light of a single fequeny (olo) a spetomete may not be needed. The FM fequenies fo nealy monohomati light pobably an be identified by imposing amplitude modulation (AM) on the intensity of the initial lase beams at the fequeny of the desied hamoni. 3.3 Two-Am Roto in the Vetial-ast-West Oientation Conside fist the vetial-east-west oientation fo the oto (Fig. 1(a)). Let 1 up 1 n and 1 e be the vetial nothwad and eastwad omponents fo the adial distane fom the ath s ente to the os1 and let the same notation apply to the othe osillatos. Fo this ase 1 up R os( t) 1 n 1 e R sin( t) up R os( t) n e R sin( t) up n e. (3.3.1) Let 1 and be the absolute magnitude fo the adial distane fom the ath s ente to os1 os and os espetively. To fist ode in R/ 1 up n e R os( t) 1 R up n e 1 os( t) 1. (3.3.) up n e Let u1 up u1 n and u1 e be the upwad omponent the nothwad omponent and the eastwad omponent of the pope veloity of os1 and let the

17 -17- same notation apply to os and os. u1up v sin( t) u1n u1e v v os( t) uup v sin( t) un ue v v sin( t) uup un ue v. (3.3.3) The squaed magnitude fo the veloity is the sum of the squaes of the omponents. v v u1 u1up u1n u1e v 1 os( t) v v v v u uup un ue v 1 os( t) v v u v. (3.3.4) The fequenies of the osillatos ae edued by half of the squae of the speed atios. The diffeenes beome v v u1 u v os( t) v v qv os( )os( t) v v v v u u v os( t) v v qv os( )os( t) v v v u1 u v os( t) v v os( )os( t) q v. (3.3.5) Notie that thee ae no ovetones and that the beat signal fo u1 -u is popotional to v. The fequenies of the osillatos ae also hanged by the potentials. Let 1 and be the sala potential values fo os1 os and os espetively. To fist ode in R/ the sala potentials and the diffeenes edue to (f. q. (..6)) 1 A R R 1 A 1 os( t) A 1 os( t) 1 1 A R R A 1 os( t) A 1 os( t) A A 1 A os( t) R A os( t) R 1 A os( t). (3.3.6) R Notie that the beat signal fo the sala potential is independent

18 -18- of v and up to the fist ode in R/ ontains no ovetones. Let 1 e e and e be the eastwad omponent of the angula speed of os1 os and os espetively. u1e v 1e 1 sin( t) os( ) v ue v e 1 sin( t) os( ) v u e e os( ) The angula speed atios beome 1e v sin( t) v e v sin( t) v e. (3.3.7). (3.3.8) The veto potential 1 is given by q. (..). Let 11 be the 1 value fo os1 1 fo os and 1 fo os. 1e 11 A 1 os ( ) PS 1(1) v R A 1 os( ) sin( t)ps 1 1 os( t) v q v R 1 A 1 os( ) sin( t)ps 1 1 os( t) v q 1 ( e = ). (3.3.9) The powe seies fo 1 is given by q. (..18). To fist ode in R/ PS 1(1) 4 R R C1 1 os( t) C1 1 os( t) 6 8 R R C14 1 os( t) C16 1 os( t) R C1 C1 C14 C16 C1 4C1 6C1 4 8C1 6 os( t) C1 R C11 1 os( t) C11 (3.3.1) whee the sums C11 and C1 thei numeial value and the atio ae 4 C11 C1 C1 C1 4 C

19 -19-3 C1 C1 4C1 6C1 4 8C C C11. (3.3.11) The fomula fo PS 1 () is simila R PS 1() C1 C1 C14 C16 C1 4C1 6C1 4 8C1 6 os( t) C1 R C11 1 os( t) C11. (3.3.1) By using the tig identity sin(t)os(t)=sin(t) the fomulas fo 11 1 and 1 and the diffeenes edue to v 1 C1 R A 1C11 os( ) sin( t) sin( t) vq C11 v 1 C1 R A 1C11 os( ) sin( t) sin( t) vq C11 v 11 1 A 1C11 os( )sin( t) vq. (3.3.13) This shows that the amplitudes fo the FM hamonis in the beat signal fo 1 ae popotional to A 1 v (f. q. (..1)). The fist ovetone fo 11-1 and 1-1 is popotional to A 1 vc1. The fist ovetone fo 11-1 anels out. The time deivatives fo 1 and ae needed to alulate values fo the veto potential. d1 u1up v sin( t) R sin( t) dt d uup v sin( t) R sin( t) dt. (3.3.14) The fomula fo is given by q. (..4). Let 1 be the value of fo os1 and let the same notation apply to os and os. By substituting the omponents fo up of q. (3.3.1) the angula speed atios of q. (3.3.8) the values fo u up of q. (3.3.14) and the fomula v=r the fomulas fo beome v v v R 1 A os( ) I1 A os( ) I1 vq vq R I1 1 os( t) I1 sin( t)dt t t R d I1 1 os( t) sin( t) sin( t)ps(1) dt dt

20 -- v R A os( ) I vq R I 1 os( t) I sin( t)dt t t R d I 1 os( t) sin( t) sin( t)ps() dt dt (3.3.15) whee I1 and I ae the fist integals fo I 1 and I espetively. The fomula fo PS() is given by q. (..13). By substituting the tig identity sin(t)os(t)=sin(t) and to fist ode in R/ the fomula fo the agument fo the deivative in I1 edues to 3 5 R R C 1 os( t) C 1 os( t) sin( t)ps(1) sin( t) 7 9 R R C4 1 os( t) C6 1 os( t) 1 C R C1 sin( t) sin( t) C1 (3.3.16) whee the sums C1 C the atio and thei numeial values ae C1 C C C4 C C 3C 5C 7C 4 9C C C1. (3.3.17) The fomula fo the agument fo the deivative in I edues to 1 C sin( t)ps() C1 sin( t) sin( t). (3.3.18) C1 Theefoe the time deivatives edue to d dt d dt C sin( t)ps(1) C1 os( t) os( t) C1 C C1 os( t) os( t) 1 C1 C C C1 os( t) os( t) C1 C1 C sin( t)ps() C1 os( t) os( t) C1 C C C1 os( t) os( t) C1 C1. (3.3.19)

21 -1- Beause R/ <<C/C1 the fomulas fo I1 and I edue to C t sin( t) sin( t)os( t) C1 I1 C1 dt C C R 3 sin( t)os ( t) sin( t)os ( t) C1 C1 C t sin( t) sin( t)os( t) C1 I C1 dt. (3.3.) C C R 3 sin( t)os ( t) sin( t)os ( t) C1 C1 The solution fo these integals an be found by applying the following fomula fom the integal tables. t n 1 (n 1) sin( t)os ( t)dt 1 os ( t) (n 1) n-1. (3.3.1) The solutions ae I1 C 1 C1 C1 1 os ( t) 1 os ( t) C1 3 C C C 1 os( t) os ( t) 3 C1 C1 C1 C 3 1 C R 4 os ( t) os ( t) 3 C1 C1 I 1 1 C C 1 os( t) os ( t) 3 C1 C1 C1 C 3 1 C R 4 os ( t) os ( t) 3 C1 C1. (3.3.) 1 os( t) 1 os ( t) C 1 3 C R 1 4 By etaining tems to fist ode in R/ and ignoing the onstant tem the fomulas fo I 1 and I edue to I1 I 1 1 C 1 C 1 3 os( t) os ( t) os ( t) C1 3 C1 C1 6 1 C 4 1 C R 5 os ( t) os ( t) 6 C1 1 C1 1 1 C 1 C 1 3 os( t) os ( t) os ( t) C1 3 C1 C1 6 1 C 4 1 C R 5 os ( t) os ( t) 6 C1 1 C1. (3.3.3) The tig identities fo os (t) os 3 (t) os 4 (t) and os 5 (t) ae 1 1 os ( t) os( t) os ( t) os( t) os(3 t) 4 4

22 os ( t) os( t) os(4 t) os ( t) os( t) os(3 t) os(5 t). (3.3.4) By substituting these tig identities into q. (3.3.3) the fomulas fo I 1 and I edue to I1 I 3 1 C 1 C os( t) os( t) C1 8 3 C1 6 C1 1 1 C 1 C R os(3 t) os(4 t) os(5 t) 4 48 C1 16 C1 3 1 C 1 C os( t) os( t) C1 8 3 C1 6 C1 1 1 C 1 C R os(3 t) os(4 t) os(5 t) 4 48 C1 16 C1. (3.3.5) Beause = and to fist ode in R/ the beat signals edue to v R 1 1 A os( ) I1 vq 3 1 C 1 C os( t) os( t) v R 8 3 C1 6 C1 A os( ) C1 vq 1 1 C os(3 t) os(4 t) 4 48 C1 v R A os( ) I vq 3 1 C 1 C os( t) os( t) v R 8 3 C1 6 C1 A os( ) C1 vq 1 1 C os(3 t) os(4 t) 4 48 C1 v R A os( ) C1 os( t) os(3 t) vq 4 1. (3.3.6) This shows that the beat signals fo ae popotional to v. The beat signal fo 1- ontains a ih aay of ovetones ovetones at t 3t and 4t. But the ovetones at t and 4t anel out of the beat signal fo 1-. Let u 1 and be the time dilation effets of the FM hamoni amplitudes fo u 1 and in the beat signals fo the vetial-east-west oientation. 1 u1 u 1 fv v u f os( t) 1 u1 u f v v f os( t)

23 -3- Table I. Ode-of-magnitude estimates of the FM hamoni amplitudes fo a two-am oto in the vetial-east-west oientation fo os1-os with = and v=v q. The hamoni fequenies ae /=73.9 Hz /=148. Hz 3/=1.8 Hz and 4/=95.6 Hz. hamoni t t 3t 4t u Hz Hz Hz Hz Hz Hz Hz Hz 1 fa R f os( t) 1 fa R f os( t) C11 sin( t) 11 1 fa 1 v 1 f os( ) 1 R v q C1 sin( t) 11 1 fa 1 v f os( )C11 sin( t) vq 3 1 C1 C os( t) C os( t) fa v R 6 f os( ) vq 1 C1 os(3 t) 4 1 C os(4 t) 48 3 C1 os( t) 1 fa v R 4 f os( ) vq 1 C1 os(3 t) 1. (3.3.7) The expeted ode-of-magnitude FM hamoni amplitudes fo the vetial-east-west oientation at the equato (=) with v=v q ae listed in Table I. This table shows that the amplitude fo the fundamental is dominated by u but the amplitudes at t 3t and 4t ae dominated by. By allowing suffiient oto opeating time say moe than 1 hou=36 s and by using suffiient amplifiation say moe than 1 6 these ovetones pobably will be detetable. 3.4 Two-Am Roto in the Vetial-Noth-South Oientation Next onside the vetial-noth-south oientation fo the oto (Fig. 1(b)). Fo this ase e = fo all thee osillatos whih means the veto potentials equal zeo fo all thee osillatos. Also the squaed speeds anel out in the os1-os beat signal. This

24 -4- oientation povides a onvenient efeene fo the time dilation effet of only the sala potential. Again let 1 up 1 n and 1 e be the vetial nothwad and eastwad omponents fo the adial distane fom the ath s ente to the os1 and let the same notation apply to the othe osillatos. Fo this ase 1 up R os( t) 1 n R sin( t) 1 e up R os( t) n R sin( t) e up n e. (3.4.1) Let 1 and be the absolute magnitude fo the adial distane to os1 os and os espetively. To fist ode in R/ 1 up n e R os( t) 1 R up n e 1 os( t) 1. (3.4.) up n e Again let u1 up u1 n and u1 e be the upwad nothwad and eastwad omponents of the pope veloity fo os1 and let the same notation apply to os and os. u1up v sin( t) u1n v os( t) u1e v uup v sin( t) un v os( t) ue v uup un ue v. (3.4.3) The fomulas fo u ae v u1 v 1 v v u v 1 v u v. (3.4.4) The diffeenes beome u1 u v u u v u1 u. (3.4.5) Again let 1 and be the sala potential values fo os1 os and os espetively. To fist ode in R/ the sala

25 -5- potentials and the diffeenes beome A R 1 A 1 os( t) 1 A R A 1 os( t) A A R 1 A os( t) A os( t) R R 1 A os( t). (3.4.6) The numeial value fo is fa os( t) Hz os( t). (3.4.7) R This shows that the vetial-noth-south oientation povides a good way to hek on the effet of the sala potential ating alone. 3.5 Two-Am Roto in the Hoizontal-ast-West Oientation Finally onside the hoizontal-east-west oientation fo the oto (Fig. 1()). This oientation povides a onvenient hek fo time dilation effets without the effet of the sala potential. Again let 1 up 1 n and 1 e be the vetial nothwad and eastwad omponents fo the adial distane fom the ath s ente to the os1 and let the same notation apply to the othe osillatos. Fo this ase 1 up up up 1 n R sin( t) 1 e R os( t) n R sin( t) e R os( t) n e. (3.5.1) Let 1 and be the absolute magnitude fo the adial distane to os1 os and os espetively. To ode R/ 1 1 up n e R sin ( t) R os ( t) 1 up n e 1. (3.5.) up n e Again let u1 up u1 n and u1 e be the upwad omponent the nothwad omponent and the eastwad omponent of the pope veloity of os1

26 -6- and let the same notation apply to os and os. u1up u1n v os( t) u1e v v sin( t) uup un v os( t) ue v v sin( t) uup un ue v. (3.5.3) The squaed magnitude fo the veloity is the sum of the squaes of the omponents. v v u1 v os ( t) v v sin( t) v 1 sin( t) v v v v u v os ( t) v v sin( t) v 1 sin( t) v v u v. (3.5.4) The diffeenes beome u1 u v v v sin( t) u u v v v sin( t) u1 u 4v v sin( t). (3.5.5) Again let 1 and be the sala potential values fo os1 os and os espetively. To ode R/ the sala potentials and the diffeenes beome 1 A 1 R 1 A 1 A 1 1 A 1 R A 1 A A A 1 1. (3.5.6) Let 1 e e and e be the eastwad omponent of the angula speed of os1 os and os espetively. u1e v v v 1e 1 sin( t) 1 sin( t) os( ) os( ) v v ue v e 1 sin( t) os( ) v ue v e os( ) os( ). (3.5.7)

27 -7- The angula speed atios beome 1e v sin( t) v e v sin( t) v e. (3.5.8) The veto potential 1 is given by q. (..). Let 11 be the 1 value fo os1 1 fo os and 1 fo os. 1e 11 A 1 os ( ) PS 1(1) v R A 1 os ( ) sin( t)ps 1 1 os( t) vq os( ) v C1 R A 1 os( ) sin( t)c11 1 os( t) vq C11 v C1 R 1 A 1 os( ) sin( t)c11 1 os( t) vq C11 1. (3.5.9) By using the tig identity fo sin(t)os(t) the fomulas fo 11 1 and the diffeene an be ewitten as v 1 R A 1 os( ) C11 sin( t) C1 sin( t) vq v 1 R A 1 os( ) C11 sin( t) C1 sin( t) vq v 11 1 A 1 C11 os( )sin( t) vq. (3.5.1) Beause u1 up u up and u up equal zeo all the values fo ae zeo. Again let u 1 and be the time dilation effets of the FM hamoni amplitudes fo u 1 and in the beat signals fo the hoizontal-east-west oientation. 1 u1 u fv v u f os( t) 1 u1 u f v v f os( t) 1 f 1 f

28 -8-1 C11 sin( t) 11 1 fa 1 v f os( ) 1 R v q C1 sin( t) f 11 1 fa 1 v vq os( )C11 sin( t) 1 f 1 f. (3.5.11) These esults show that the hoizontal-east-west oientation povides a good way to hek fo the effet on the fist ovetone fo the veto potential 1 ating alone. CONCLUSIONS AND RCOMMNDATIONS This feasibility study indiates that a high-speed two-am oto an be designed to povide peision measuements fo the indution onstant k the oesponding indution speed v k =1/k and the adial mass density oeffiient sums C1 C C11 and C1. The instument an be made potable by installing it in a semi-tato-taile system. Then it ould be used to measue the hamoni amplitudes at diffeent latitudes and diffeent altitudes. Of ouse many engineeing details need to be woked out to pefet this basi design. That takes only time money and desie. ACKNOWLDGMNTS I thank Patik L. Ives fo eviewing the oiginal manusipt and suggesting impovements. RFRNCS 1. J. C. Hafele ffet of the ath s Time-Retaded Tansvese Gavitational Field on Spaeaft Flybys axiv:94.383v [physis.gen-ph] 5May1.. W. Rindle ssential Relativity Speial Geneal and Cosmologial (Spinge New Yok 1977). 3. J. D. Jakson Classial letodynamis (Wiley New Yok 1975).