GENERAL RELATIVITY AND SPATIAL FLOWS: I. ABSOLUTE RELATIVISTIC DYNAMICS *

Transcription

1 GENERA REATIVITY AND SPATIA FOWS: I. ABSOUTE REATIVISTIC DYNAMICS * Tom Martin Gravit Rsarh Institt Boldr, Colorado martin@gravitrsarh.org Abstrat To omplmntar and qall important approahs to rlativisti phsis ar plaind. On is th standard approah, and th othr is basd on a std of th flos of an ndrling phsial sbstratm. Prvios rslts onrning th sbstratm flo approah ar rvid, pandd, and mor losl rlatd to th formalism of Gnral Rlativit. An absolt rlativisti dnamis is drivd in hih nrg and momntm tak on absolt signifian ith rspt to th sbstratm. Possibl n ffts on satllits ar dsribd.. Introdtion Thr ar to fndamntall diffrnt as to approah rlativisti phsis. Th first approah, hih as Einstin's a [], and hih is th standard a it has bn pratid in modrn tims, rognis th masrmnt ralit of th impossibilit of dtting th absolt translational motion of phsial sstms throgh th ndrling phsial sbstratm and th masrmnt ralit of th limitations imposd b th finit spd of light ith rspt to lok snhroniation prodrs. Th sond approah, hih as ornt's a [] (at last for Spial Rlativit, rognis th onptal spriorit of rtaining th phsial sbstratm as an important lmnt of th phsial thor and of sing onptall sfl frams of rfrn for th ndrstanding of ndrling phsial prinipls. Whthr on dos rlativisti phsis th Einstinian a or th orntian a rall dpnds on on's motivs. Th Einstinian approah is primaril onrnd ith *

2 bing abl to arr ot pratial spa-tim primnts and to rlat th rslts of ths primnts among variosl moving obsrvrs in as ffiint and nompliatd mannr as possibl. For this rason, th Einstinian approah is primaril basd on an pistmolog hih is onrnd ith pratial masrmnt stratgis. In th rstritd as of Spial Rlativit (inrtial frams of rfrn, it provids s ith th simplifiation of or not having to ontnd ith th ndrling phsial sbstratm (absolt spa in mhanis or th athr in ltrodnamis at all. Hovr, it pas for this simplifiation b bing somhat phsiall inomprhnsibl and b bing pron to all th misndrstandings, th onfsion, and th sming parados hih invitabl aris from th Einstin snhroniation pross (lok snhroniation b light signals and th masrmnt stratg assmption of isotropi onstan for th spd of light in all inrtial frams. Th inrtial oordinat frams of Spial Rlativit and, to a first approimation, th Sharshild-lik oordinat frams of Gnral Rlativit ar all basd on ths Einstin snhroniation prodrs. Th ar pratiall raliabl and an b st p b sing simpl and familiar radar and tim transfr thniqs. On th othr hand, th orntian approah to spa-tim phsis is basd on th onptal advantags of rtaining an ndrling phsial sbstratm in phsial thoris (hih implis anisotrop for th spd of light and of sing Galilan frams of rfrn among hih thr ists absolt tim (qivalntl, absolt simltanit to nvision and ndrstand th haratristi phnomna of th thoris. Ths Galilan frams of rfrn ar sfl for or imaginations and for or onptal ndrstanding of th phsis, bt th ar sall vr diffilt to st p as pratiall fntioning phsial frams of rfrn. This diffilt ariss, bas th ahivmnt of absolt simltanit btn rfrn frams ssntiall rqirs that snhroni th oordinat loks ith infinitl fast signals (hih ar obviosl not availabl. Bt h shold ths pratial diffiltis lad s to limit or mntal ndrstanding of th ssntials of phsial phnomna b rstriting or thoght prosss onl to frams of rfrn hih ar asil ahivabl ith th sal limitd sintifi instrmntation? This old b as ill-advisd in th onptal arna as it old b to attmpt to alas st p Galilan frams of rfrn in hih to arr ot or primnts and rlat or rslts to othr obsrvrs (ho old also b strggling to st p sh frams in th masrmnt arna. W shall disovr that thr is mh to b gaind b alloing or minds to roam bond th pistmologial limitations of or sintifi instrmnts. W ar sggsting that th Einstinian and orntian vis of spa-tim phsis

3 3 omplmnt ah othr. Th Einstinian vi is bst sd for stting p simplifid and pratial masrmnt shms ( hav a grat bod of primntal vidn for th sss of this approah in th past ntr, hil th orntian vi ma prov to b th bst a to ahiv a onptal ndrstanding of th ndrling phsis and possibl n phsial insights ( ar spiall advoating a rnal of this mthod to gravitational phsis. As a simpl ampl of this omplmntar rlationship hih has bn vidnt to som ll-knon phsiists [3,4], mntion that it is alas possibl in Spial Rlativit to mak a ontinos transformation from th sal inrtial oordinats in th Einstinian vi to Galilan oordinats in th orntian vi in hih th phsial sbstratm ists and has a tmporall onstant and spatiall homognos flo. With th s of ths Galilan oordinats and th rifiation of th sbstratm, all of th smingl mstifing aspts of Spial Rlativit in th Einstinian vi ar spt aa. Bt, an tak this a stp frthr. Sin th Prinipl of Covarian holds in Gnral Rlativit, it also holds in Spial Rlativit. Ths, th ontinos transformation rlating th Einstin snhronid inrtial oordinats and th Galilan oordinats assrs s that no phsial primnt (on arrid ot ith ordinar laborator instrmnts an distingish btn th Einstinian vi and th orntian vi in Spial Rlativit. Historiall, ornt sall thoght of th ndrling phsial sbstratm as bing th tmporall onstant and spatiall homognos sbstratm of Nton's absolt and immtabl spa. In ontrast to ornt and Nton, ar vr mh intrstd in gnraliing th sbstratm vi to ompltl arbitrar spatial flos. W nvision th possibilit that phsial spa is floing into or ot of plantar bodis, that it is sirling and spiraling in galati strtrs, and that in rtain irmstans, its flo might vn b sprlminal. In this papr, ill rvi and pand som of th impliations of th rslts obtaind in prvios ork [5,6], rlat thm a littl mor losl to th formalism of Gnral Rlativit, and skth th rdimnts of an absolt rlativisti dnamis (in hih nrg and momntm tak on an absolt signifian ith rspt to th sbstratm. Or ndrstanding of this dnamis ill b nhand b sing th phsiall intitiv notation and thniqs of ordinar vtor analsis [3,4]. W ar primaril intrstd in dtrmining ho phsial spa transfrs nrg and momntm

4 4 to matrial bodis as th mov throgh it in ford or nford motion. W ill also rvi a f of th impliations of this spatial flo dnamis for possibl n ffts in th lstial mhanis of satllits. A ord abot trminolog: hav bn rfrring to th floing phsial sbstratm as th flo of phsial spa (or mor sintl as th flo of spa. As mntiond abov, in th gis of Ntonian mhanis, this sbstratm has alas bn rfrrd to as absolt spa. In th ltromagnti ralm, it has ommonl bn alld th athr. In mor modrn tims, it has vn bn assoiatd ith th rathr vag onpt of a qantm vam. Th radr is fr to hoos hatvr trminolog h dsirs, as long as h dos not inadvrtntl bring along an of its historial baggag. For or prposs, th flo of phsial spa is ompltl haratrid b a 3-spa vtor fild ( r, t in a global Galilan oordinat fram { r, t} on th spa-tim manifold. Th onl othr dfining fatr is that th spd of light ith rspt to this sbstratm is isotropi and qal to th sal onstant.. Rlativit and Spatial Flos Th paprs [5] and [6] prsntd th ssntial stps on mst tak in ordr to disss th possibilit that all of rlativisti phsis (gravitation inldd an b ndrstood in trms of th intrations of phsial sstms ith a floing ndrling phsial sbstratm (phsial spa. This intrprtation, ith its nhanmnt of phsial ndrstanding, as alrad availabl in th rstritd as of Spial Rlativit as mntiond abov (s [,3,4]. Th paprs [5] and [6] sggst th mans b hih this nhand phsial ndrstanding might b ralid in Gnral Rlativit as ll. Absolt rotational motion ith rspt to phsial spa has bn knon sin anint tims (th ordinar prin of ntrifgal fors. Th first optial primnt dtting absolt rotation as prformd b Sagna and rportd b him in 93 [7]. asr gros ar sd rotinl in th prsnt ra to dtt th smallst rotations for navigational prposs. In ths rotational prins, ar mad aar of th phsial ralit of th ndrling sbstratm. In ontrast to th rotational as, Natr is vr sbtl and illsiv hn it oms to or dtting an translational motion throgh phsial spa. On might sa that this is

5 5 th paramont fat of all of phsis. It as rtainl an important ingrdint in th voltion of Einstin's approah to rlativisti phsis (phsial indistingishabilit of all inrtial frams, and it involvd th minds and monmntal fforts of som of th gratst primntal phsiists in th smingl fritlss attmpt to dtt it. Th nll rslts of thir primnts ford phsiists to look losr at th intrrlationship of th sbstratm flo, ltrodnamis, and th strtr of mattr. It bam apparnt that thir primnts old b intrinsiall insnsitiv to th translational flo th r attmpting to dtt. W shold not b distrbd b this inhrnt dsign of Natr, bas, in a dpr phsial sns, it is th vr ssn of or frdom to travl throgh nlimitd spa (Galilo's Prinipl. It is dobtfl, thrfor, that translational motion throgh phsial spa ill vr b dttd b a loalid laborator primnt. Nvrthlss, th paprs [5] and [6] hav som n nonloalid primnts to sggst hih might b fftiv in posing th phsial ralit of th translational motion of phsial spa. Ths primnts ar basd on th possibilit that Natr might not b so sssfl at hiding th ralit of th flo at th bondar btn to sparat flos. In [5], rmindd th radr of th fat that thr ar spatial inflo and otflo soltions of Einstin's fild qations for plantar objts in Gnral Rlativit hih might hav phsial ralit. Ths floing spa soltions ar atall a part of and atall prditd b Gnral Rlativit. If ar orrt in ding that sh flos of spa ill b along th gravitational alrativ fild lins for slol moving tst bodis (Brnolli analog, thn, in th to-bod gravitational problm, thr ill b a bondar sparating th flos of ah of th to bodis. W ar not t prisl rtain of th at onfigration th bondar ill tak in th as of to mtall orbiting bodis, bt thr ill b a stagnation point for th to flos somhr in th largr nighborhood of th gravitational saddl point of th Ntonian gravitational potntials. In [5], dsribd a simpl satllit primnt hih an dtt this rgion of stagnation ith srprising snsitivit. This might b th first primnt hih stablishs th phsial ralit of th translational flo of spa. Thn, in [6], rvald that thr might b dnamial ffts on satllits (as ll as tim dilation jmps as th pass throgh th to-bod bondar. Ths dnamial ffts might b th first to rval th atal dirtion of th translational flo. Thr ar to rasons h ths bondar ffts on satllits hav not t bn

6 6 onsiosl obsrvd (if, in fat, th flo soltions orrspond to phsial ralit. Th first rason is that ths ffts hav not bn ptd, and if th hav bn obsrvd, th ill hav most probabl bn nfoldd into othr ffts. Sondl, most satllits ar trakd b th Dopplr radar thniq. An plink signal is rivd b a transpondr onboard th satllit, and th donlink signal is phas-lokd to this plink signal so th transpondr fftivl ats as a prft mirror (ith gain. This phas-loking maks th signals insnsitiv to an tim dilation affts hih might b prind b th intrnal osillators onboard th satllit. A frl rnning and snsitiv lok mst b onboard th satllit in ordr to dtt th hpothtial tim dilation jmps at th to-bod bondar. Whthr or not th spatial flo soltion for th to-bod problm trns ot to orrspond to phsial ralit, kno from th Prinipl of Covarian that no phsial primnt an distingish th Galilan flo soltion from th rvd spa Sharshild soltion in th as of a singl isolatd attrator (th nar fild of th Earth is approimatl that of an isolatd attrator. W an alas s th flo and its assoiatd Galilan oordinat fram as a prftl valid and arat onptal modl ith hih to std th at dtails of th ffts of sh an attrator on phsial sstms. Hn, it is probabl sfl to mntion th folloing lmntar omparison btn th nonstati Galilan oordinats and th stati and spatiall rvd Sharshild oordinats of sh an attrator (s rfrn [5] for th mathmatial dtails. Whn th spa-tim manifold assoiatd ith a simpl gravitational attrator is oordinatd ith Sharshild oordinats, th oordination implis that nonmoving mtr rods laid ot in th radial dirtion ar shrnk, hil thos orintd in th horiontal position ar not. Th shrinking in th radial dirtion is intrprtd as bing "asd" b th non-elidan natr of spa. Bas of this intrprtation, it is nssar to anal vrthing from a diffrntial gomtri point of vi and mthodolog. Whn this vr sam spa-tim manifold is oordinatd ith Galilan oordinats, nonmoving mtr rods laid ot in th radial dirtion ar shrnk b th ornt ontration arising from th infloing or otfloing sbstratm. Nonmoving mtr rods in th horiontal orintation ar not shrnk, bas thr is no flo throgh thm in thir linal dirtion. This givs ris to th sam "non-elidan gomtr" (from th

7 7 vipoint of plaing th mtr rods nd to nd, bt gt to talk abot it in onptal trms as bing "asd" b th flo of phsial spa (in th Elidan oordinat spa of th Galilan fram. W an talk abot it ithot nssaril bringing in all th mahinr of diffrntial gomtr. This as on of th prposs hih motivatd th riting of rfrn [5]. For a sphriall smmtri gravitational attrator, an gt all of th phsial prditions of Gnral Rlativit ithot having to do diffrntial gomtr pr s. Rfrn [6] as similarl motivatd, and it shos s ho to dtrmin th trajtoris of tst bodis in ompltl gnral spatial flos ithot having to s th mathmatial mahinr of diffrntial gomtr pr s (no ovariant diffrntiation, no Christoffl smbols, no affin paramtrs, and so forth. So, th spatial flo point of vi has th potntial to bring trmndos onptal and mathmatial simplifiation to all of spa-tim phsis and spiall to Gnral Rlativit. Sin thr is a possibilit that th gravitational fild of th Earth is atall asd b an inflo or otflo of th phsial sbstratm, ar natrall ld to ondr abot hat old b th planation of stllar abrration hn thr is vrtial sbstratm flo. Withot going into an dtail hr, simpl stat that, if th sbstratm is floing ssntiall vrtiall into or ot of plantar objts, mst s Stoks' 845 planation of stllar abrration [8,9]. ornt's objtion [0] to Stoks' planation as misplad. ornt assmd (as did most phsiists of th da that, if th sbstratm r ntraind at th Earth's srfa, it old b horiontall ntraind. Th rl of th atal vrtial flo is ro, and so it mts th rqirmnts of Stoks' planation. Finall, as an historial riosit, mntion that kno toda that Mihlson and Morl's famos optial intrfromtr [] as inhrntl insnsitiv to spatial flo of an kind (bas of th ltrodnami phnomnon of ornt ontration. Still, thr is a bit of iron in th possibilit that th sbstratm as amiabl floing vrtiall throgh thir intrfromtr as th patintl rotatd it in th horiontal plan. Mihlson, hovr, as th on ho had th last ord. H latr masrd th small horiontal omponnt in th vrtial sbstratm flo (hih ariss in an Earth ospinning fram in th sprb Mihlson-Gal primnt [].

8 8 3. Absolt Rlativisti Dnamis W no prod to stablish th rdimnts of rlativisti dnamis for tst bodis in arbitrar spatial flos. If th spatial flo approah to spa-tim phsis is nivrsall appliabl to all phsial irmstans, vr spa-tim manifold of phsial signifian ill b apabl of bing haratrid b a flo of phsial spa. A global Galilan oordinat fram { r, t} ill alas ist in hih th flo is rprsntd b a global 3-spa vtor fild ( r, t. Th rat of an atomi lok ill dpnd on its absolt spd ith rspt to phsial spa. Ths, in ths Galilan oordinats, th propr tim lmnt of an atomi lok ill b givn b dτ γ / /, (3- hr v (3- is th absolt vloit of th lok rlativ to phsial spa. Hr, τ is th propr tim of th lok, is th spd of light ith rspt to phsial spa (a onstant, t is th oordinat tim, r is th oordinat position vtor of th lok, and oordinat vloit of th lok. v d r / is th From th abov qations, s that th spa-tim lin lmnt in Galilan oordinats ith arbitrar spatial flo alas taks th form d τ ( + dr ( dr. ( In for dimnsional rtanglar Galilan oordinats {,,, } { t,,, }, th omponnts of th orrsponding spa-tim mtri tnsor ar sn to b ( / / / / / 0 0 g κλ, (3-4 / 0 0 / 0 0

9 9 hr s th onvntion that ds dτ g κλ d κ d λ. (3-5 From (3-3 or (3-4, it is immdiatl vidnt that vr sli of spa-tim ith onstant tim ( 0 is flat 3-dimnsional Elidan spa. W also rali that th nivrsalit of th spatial flo approah to Gnral Rlativit orrsponds to th possibilit of alas bing abl to impos algbrai oordinat onditions in th gis of th spial form of th mtri tnsor in (3-3 and (3-4. From (3- and (3-, it is vidnt that th total tim dilation of an atomi lok in motion ith vloit v in an arbitrar spatial flo is givn b dτ ( v + v /. (3-6 In or sbstratm pitr, is th limiting spd of all matrial tst bodis ith rspt to absolt phsial spa. Ths, for matrial tst bodis, ill alas hav th ondition <. Th absolt rlativisti momntm of a partil ith onstant rstmass m and absolt vloit v is p γ m, (3-7 hr or γ is alas th γ assoiatd ith absolt motion in th sbstratm: γ / /. (3-8 Th total absolt rlativisti nrg of sh a partil is E γ m. (3-9 Sin th rstmass nrg is m, th absolt rlativisti kinti nrg is K ( γ m. (3-0

10 0 W kno that ths ar th orrt dfinitions, bas an infinitsimal Galilan oordinat fram omoving ith th sbstratm is th sam as an inrtial fram, and ths ar th orrt dfinitions in th inrtial frams of Spial Rlativit. On an asil vrif that ths absolt dnamial variabls satisf th fndamntal rlationship, E + 4 p m. (3- Th absolt for F ating on th partil is th tim rat of hang of th absolt momntm: F dp d( γ m dγ m d + γ m. (3- Sin dγ d(( / hav, in omplt gnralit, that / ( ( / 3 / ( / d, dγ 3 γ d. (3-3 Ths, (3- boms dp γ d d γ d F γ m ( + γ m γ m ( +, (3-4 hr is th idntit tnsor in Galilan 3-spa ( ar sing ordinar vtor-dadi notation and thniqs [3, 4]. Th tnsor ( + ( γ is non-singlar, and it has th somhat simplr invrs ( + ( γ ( /. ( 3-5 proof: ( + ( γ ( / / + γ ( /.

11 W all th tnsor ( / th tnsor of motion of th partil. Not onl dos vr simpl tst partil hav an assoiatd kinmati vloit vtor, it also has an assoiatd kinmati tnsor. From (3-4, hav, in omplt gnralit, that F dp d γ m( /. (3-6 This fndamntal rlation btn th tim rat of hang of th absolt rlativisti momntm (th for and th absolt alration gnraliation of Nton's famos Sond a, d of th partil is th F ma, in arbitraril floing absolt spa. Th qation is fll rlativisti (in th sns that th partil ma hav an phsiall alloabl spd ( 0 < ith rspt to phsial spa. Th fndamntal qation (3-6 also shos s that, for rlativisti spds, th inrtia of th partil is no longr a simpl salar m. Instad, th inrtia of th partil is a tnsor. In analog ith th sal form of th Sond a, F ma, all th tnsor γ m ( / th inrtia tnsor of th partil. Ths, th inrtia of a rlativisti partil is dirtionall dpndnt, and th for and th assoiatd alration ar no longr nssaril ating in th sam dirtion. Writing (3-6 in th form d ( / F F ( F, (3-7 γ m γ m γ m s that thr is a vloit indpndnt omponnt of th alration in th dirtion of th for as ll as a vloit dpndnt omponnt of th alration in th dirtion of th vloit. If th for is prpndilar to th partil's motion rlativ to phsial spa, (3-7 shos s that th rlativ strngth of its inrtia is transvrs inrtia γ m. (3-8 On th othr hand, if th for F is paralll to th vloit, so that ± F / F, thn 3 d ( F / γ m( / F / γ m, and th rlativ strngth of its inrtia is longitdinal inrtia 3 γ m. (3-9

12 Th tim rat of hang of th total absolt rlativisti nrg an b allatd from (3-9 and (3-3. W hav, in omplt gnralit, that de 3 d γ m. (3-0 Clarl, th nrg of th tst partil is onstant hn th absolt alration is orthogonal to th absolt vloit. No ork is don in alrating th partil transvrsl as it movs throgh th sbstratm. This is h onl th longitdinal inrtia appars. B taking th dot prodt of qation (3-7 ith, on an asil stablish that de F, (3- as ptd. Th absolt ork dw don on th partil b a for F ating throgh th dirtd sbstratm distan ds is obviosl dw Fds F. (3-4. Th Godsi Eqations of Motion Th path of a fr matrial tst partil in Gnral Rlativit is on ovr hih th intgral of th partil's propr tim is an trmm: δ d τ 0. (4- Th path is obtaind from th assoiatd Elr-agrang qations hih aris in driving th soltion to th variational problm (4-. In th fll gnralid diffrntial gomtri approah of Gnral Rlativit, th qivalnt invariant ds κ λ dτ gκλ d d (4-

13 3 is varid, and th rslting Elr-agrang qations ar th godsi qations of motion, d dτ κ + Γ κ λµ d dτ λ d dτ µ 0, (4-3 if on ss an affin paramtr sh as th propr tim τ. If on ss a non-affin 0 paramtr sh as oordinat tim t /, th godsi qations tak on a mor ompliatd form: d κ d d d d t d τ ( d τ λ µ κ κ + Γλµ. (4-4 This diffrntial gomtri approah brings in all th mathmatial ompliations of th κ mtri tnsor g κλ, th Christoffl smbols Γ λµ, th onpt of an affin paramtr, and th nssar thniqs of gnralid tnsor analsis. It is of intrst, thrfor, to disovr that hn ar daling ith a spa-tim manifold hih is haratrid b th flo of phsial spa, hav no nd to bring in all this gnralid mathmatial mahinr. Th qations of motion of fr tst partils (qivalntl, th godsi qations an b asil dtrmind ithin th ontt of ordinar vtor analsis. This as originall dmonstratd in rfrn [6], and bas th drivation is so short and simpl (and also for th sak of ompltnss, rpat it hr. Natrall, th allation is arrid ot in Galilan oordinats. Th allation is simplifid b sing th oordinat tim t as th path paramtr. This rds th nmbr of Elr-agrang qations from for (involving th spa-tim oordinats to thr (involving th spatial oordinats. From (3-, hav δ dτ δ ( / 0. (4-5 W hoos rtanglar Galilan oordinats {,,, t} ith th orthonormal spatial basis,, }. Th sal Elr-agrang qations ar { d v r, (4-6

14 4 hr / ( ;, ( γ v r t (4-7 is th agrangian, and v v v v + +, (4-8 grad r + +. (4-9 In this Elr-agrang formalism, r and v ar tratd as indpndnt variabls. v(t v is a paramtri vtor, hil, ( t r is a vtor fild. Ths, grad v grad grad (, (4-0 hr grad (4- is th tnsor haratriing th spatial inhomognit of th flo. From (4-0, hav grad grad grad ( ( (, hn grad grad grad r ( ( ( /. Sin

15 5 / / ( ( v v v +, v v (, so d d d d d v 3 4. Stting γ (from (4-7, s that or Elr-agrang qations (4-6 ar grad + ( d d γ. (4- Taking th salar prodt of this qation ith, gt grad + (( ( d d γ. (4-3 Sin ( γ, (4-3 boms grad (( d γ. (4-4 Sbstitting this bak into (4-, find that grad grad ((( ( + d, (4-5 and an rit this mor formall as 0 grad + ( / ( d. (4-6

16 6 Again, is th idntit tnsor in 3-spa, and / is th smmtri tnsor of motion of th tst partil (this has alrad arisn in or skth of rlativisti dnamis in Stion 3. (4-6 is th qation of motion of a fr matrial tst partil in Galilan oordinats ith arbitrar spatial flo. It is th godsi qation in ths oordinats. Th qation is fll rlativisti (in th sns that th matrial tst partil ma hav an arbitrar rlativisti spd ( 0 < ith rspt to phsial spa. Sin th agrangian (4-7 is Galilan invariant (invariant ndr th gnral vloit transformation q a q + hr is a onstant vtor, th qation itslf is Galilan invariant. This is also immdiatl vidnt from a prsal of th trms in (4-6. A similar prsal stablishs that th qation is tim-rvrsal invariant (invariant ndr th transformation t a t. In ontrast to th gnralid form of th godsi qations (4-3 or (4-4, all of th fators apparing in th Galilan form (4-6 ar Galilan invariant vtors or tnsors. 5. Pondrabl and Non-pondrabl Fors W ar no in a position to dtrmin ho th fr intrativ motion of a tst bod ith th sbstratm (godsi motion transfrs absolt momntm and nrg to th tst bod (think of a frl-falling satllit. From qations (3-6 and (4-6, obtain th tim rat of hang of th rlativisti momntm of a frl-falling tst bod. W dnot this F sbstratm : dp Fsbstratm ( sbstratm γ m( grad. (5- From qation (3-, obtain th tim rat of hang of its total rlativisti nrg: de ( sbstratm γ m(( grad. (5- Sin th tst bod is in fr-fall, th for sbstratm F is atall a fititios or nonpondrabl for ating on th bod (inapabl of bing dttd b transdrs of an

17 7 kind ithin th satllit. Nvrthlss, it rtains its signifian as a tim rat of hang of momntm. W obsrv that th transfr of nrg and momntm to a frl-falling tst bod onl involvs th motion of th tst bod and th inhomognitis of th flo (as rprsntd b grad. In gnral, th total for ating on a tst bod is th sm of th pondrabl (think of rokt thrst and th non-pondrabl sbstratm fors: F F pondrabl + F sbstratm. (5-3 Ths, in gnral, F pondrabl F F sbstratm d γ m( / + γ m( grad. (5-4 W s that th pondrabl for is d to th tmporal alration of th tst bod d / ith rspt to th sbstratm as ll as th motion of th tst bod throgh th inhomognitis of th sbstratm's flo (spatial alrations of th flo. Eampl : Th gravitational for on a tst bod on th srfa of a plant. In sphrial Galilan oordinats { r, θ, φ, t} ntrd on a sphriall smmtri attrator, ill first std th gravitational fild arising from th sbstratm inflo GM / r, hr G is th gravitational onstant and M is th mass of th r attrator. This inflo is on of th Galilan rprsntations of th Sharshild soltion (s rfrn [5]. Sin th bod is at rst on th srfa of th plant, v 0. From v, hav and d d / vgrad (hain rl 0. Ths, / t qation (3-6 implis F 0, and qations (5- and (5-3 impl Fpondrabl Fsbstratm γ m( grad. (5-5a Th prssion for th tnsor [5]. For th flo GM / r r, find that grad in sphrial oordinats is ommonl availabl

18 8 grad r GM r r r r 3 r. (5-5b r Sin GM / r, (5-5a and (5-5b giv s r F pondrabl GM γ m r. (5-5 r Ths, ar abl to ndrstand th printial for of gravit as stand on th srfa of a plant. It is a pondrabl for dirtd pard on th bottom of or ft. (5-5, hih is orrt for rlativisti sbstratm spds, shos s that or ight is γ m instad of jst m. This pondrabl for of gravit (5-5a and (5-5 ariss, not bas th bod is tmporall alratd ith rspt to th sbstratm, bt bas it is prsnt in a spatial inhomognit of th flo. Ths, thr rall is an ndrling distintion btn th pondrabl for of a thrsting rokt and th pondrabl for of gravit on th srfa of a plant. Th pondrabl for of th thrsting rokt dos ork on its paload, hil th pondrabl for of gravit on th srfa of a plant dos no ork (th total for F is ro, and hn, F is ro. If rvrs th dirtion of th flo, so that hav th otflo GM / r rprsnting th gravitational fild of th sphriall smmtri r attrator, s that th signs of and grad ill also b rvrsd. From (5-5a, s thr ill b no hang in th rslting gravitational for (5-5. Th inflo and otflo prod th sam gravitational ffts. It is not th dirtion of th flo hih is rsponsibl for gravitation, bt rathr, th inhomognitis of th flo. This is rlatd to th fat that th godsi qations of motion (4-6 ar tim-rvrsal invariant. Eampl : A tst bod in a tmporall onstant homognos sbstratm flo. Hr, th flo is onstant in tim and th sam at vr point of th Galilan fram. Ths, grad 0. (5-6a

19 9 From (5-, F sbstratm 0, and hn F pondrabl F. Sin d / vgrad + t 0 and d / dv / a, qation (3-6 tlls s th Galilan alration ill b a ( / F γ m pondrabl. (5-6b In th rlativisti as, rmmbr that th pondrabl for (rokt thrst and th rslting alration ar not nssaril ollinar. Eampl 3: A tst bod on th innr srfa of a rotating irlar lindr. If th rotating lindr is rotating ith onstant rotational vloit ω arond its ais of smmtr, thn, in a lindrial Galilan fram { ρ, φ,, t} hih is rotating ith th lindr abot its -ais, thr is an indd sbstratm flo ω ρ. Using th form of th tnsor grad in lindrial oordinats [5], find that φ grad ω ω. (5-7a φ ρ ρ φ For a tst bod at rst on th innr srfa of th lindr, v 0, a 0, and d / vgrad + 0. Hn, and d / 0. Ths, b (5-4, thr t is th pondrabl ntriptal for, F pondrabl γ mω ρ (/ ω ρ / mω ρ, (5-7b ρ ρ ating on th bod. Th rlativisti ight fator is again γ m. If rvrs th dirtion of th flo b spinning th lindr in th opposit dirtion, ill obviosl obtain atl th sam for. Whras this indpndn of th phsis on th dirtion of th flo as not so obvios in th gravitational as, it is familiar from vrda prin in th rotational as. Th thr simpl ampls hav givn ar anonial, bas th an b sd to point ot th ssntial distintiv haratristis of gravitational flos, inrtial flos, and non-inrtial flos. For th prpos of larit in or disssion, ill all th vtor

20 0 grad th inhomognit vtor of th flo. It is indpndnt of th dirtion of th flo, and it points in th dirtion of gratst inras in th spd of th flo. A prl gravitational flo is irrotational ( rl 0, and its inhomognit vtor is paralll to th flo. In Eampl abov, th gravitational inflo and otflo soltions both hav th sam inhomognit vtor, and so th giv ris to th sam pondrabl gravitational for. An inrtial flo is tmporall onstant and spatiall homognos. Ths, its inhomognit vtor is vrhr ro (ths ar th flos of Spial Rlativit. A prl non-inrtial flo is solnoidal ( div 0, and its inhomognit vtor is prpndilar to th flo. In Eampl 3 abov, th lokis and ontrlokis rotations hav th sam inhomognit vtor, and so th prod th sam pondrabl ntriptal for. 6. Clstial Mhanis of Artifiial Satllits Th fndamntal distintions btn a lstial mhanis hih is basd on Ntonian ation-at-a-distan gravitational thor and a lstial mhanis hih is basd on a thor of sbstratm flo ar mad vidnt in a std of th non-rlativisti (slo motion approimation to th fll rlativisti godsi qations of motion (4-6. Th non-rlativisti approimation of th rlativisti qation in th form (4-5 is d ( grad. (6- In trms of v and, this is dv d ( grad ( v, (6- hr d / is th drivativ of th paramtri vtor ( t ( r( t, t along th path r (t spifid b th vloit vtor v d r /. B th hain rl of diffrntiation,

21 d v grad +. (6-3 t Using th vtor idntitis v grad ( grad v ( rl v (6-4 and ( grad grad, (6-5 find that dv grad + ( rl v + t. (6-6 To rmind orslvs that th Galilan alration (think of a satllit, rit this as a d v / is that of a fr tst bod a fr grad + ( rl v + t. (6-7 Th first trm on th right-hand sid of this qation rprsnts th gravitational and gnralid ntrifgal alrations, th sond trm rprsnts th gnralid Coriolis alration, and th third trm is th alration arising from an pliit dpndn of th flo on tim. This qation provids s ith an as a to std th problm of motion of tst bodis in rotating and alratd frams of rfrn in ordinar Ntonian mhanis. Ordinar Ntonian mhanis assms that th ndrling sbstratm is that of absolt immtabl spa. In or flo pitr, this amonts to making th assmption that 0 vrhr in som spial non-rotating and non-alrating Galilan fram. With this sbstratm of ordinar mhanis, an std th problm of motion of tst bodis b fosing on th flos that ar indd b th motion of th rotating and alrating frams. In othr ords, in th rotating and alrating frams of ordinar Ntonian mhanis, absolt spa ill appar to b floing bas of th rlativ

22 motions of th frams and absolt spa. As an ampl, a rotating Galilan fram ith a tim dpndnt aial rotational vloit ω ω(t ill ind th flo ( r, t r ω( t. (6-8 This flo an b sbstittd into qation (6-7 to larn vrthing on nds to kno abot th ffts of th rotational motion of th Galilan fram on th motion of tst bodis as th ar obsrvd in th rotating Galilan fram. Thr is an ntir ralm of non-rlativisti mhanis availabl for ploration bond th ralm of ordinar Ntonian mhanis hn on allos th sbstratm to flo in ompliatd as. Eqation (6-7 holds for arbitrar flos of phsial spa and not jst for th flos indd b th motion of frams in absolt immtabl spa. Th anonial ampl of a flo that is not indd b th motion of a fram is th flo assoiatd ith an isolatd sphriall smmtri gravitational attrator. As mntiond in Stion 5, in a sphrial Galilan fram ntrd on th attrator, this is givn b on or th othr of th flos ( r ± GM / r, (6-9 r hr G is th gravitational onstant and M is th mass of th attrator (s also rfrns [5] and [6]. Th onntion btn ths possibl flo soltions and th ordinar Ntonian gravitational thor is mad throgh th hristi qation, onst. ψ, (6-0 hih all th Brnolli analog. This rlats th gravitational potntial ψ of ordinar Ntonian gravitational thor to th sqar of th spd of th flo in th approimatl orrsponding spatial flo vrsion. It has to b approimatl orrt, bas ordinar Ntonian gravitational thor is approimatl orrt. Whn sbstitt (6-0 into qation (6-7, gt a fr gradψ for a tim-indpndnt gravitational flo, and this is th basi qation of ordinar Ntonian gravitational thor. Th onstant in (6-0 is dtrmind b th bondar onditions for th spd of th flo nar th sors. In othr ords, hn thr ar svral point sors prsnt, th bondar onditions ar that th spd of th flo nar ah sor of strngth M

23 3 mst approah ψ GM / r GM / r. In th as of a singl attrator, th onstant is ro, and. In gnral, th onstant ill var from stramlin to stramlin. Th Brnolli analog an also b sd to lidat th onntion btn rotational flos and th ntrifgal potntial. In alling (6-0 th Brnolli analog, do not intnd to sggst that th flo of th sbstratm obs Elr's qation as it is normall drivd b appling Nton's Sond a to a prft flid having matrial dnsit (rfr to an ttbook on Flid Mhanis. On th ontrar, ar attmpting to lidat th gnraliation of Nton's Sond a b mans of a floing sbstratm. Th sbstratm mst not b givn inrtia, bas this is hat it is bing sd to plain. In this papr, ar not making an assrtions abot hat ar th appropriat qations of motion for th sbstratm, itslf. Within th ontt of Gnral Rlativit, on an prod dirtl to qations of motion for b sbstitting th flo mtri (3-4 into Einstin's qations, 8π G Tµν. (6- Rµν gµν R 4 Hovr, a disssion of th orrt qations of motion for th sbstratm mst b rsrvd for a ftr pbliation. Withot going into grat dtail hr, an d that th flo in th simpl nonrotating to-bod problm ill b approimatl along th sal gravitational fild lins hih ar obtaind b adding th Ntonian potntials of th to bodis. Whn appl th Brnolli analog to th sprposition of th potntials and also appl th appropriat bondar onditions for th flo spds nar th sors, find that th flo strtr divids into to sparat flos, ah sparat flo bing assoiatd ith on of th to bodis. As a rslt, thr ill b a bondar or a srfa of transition sparating th flos. This is shmatiall rprsntd in figr (6- b th dottd lin. Th spd of th flo is shmatiall rprsntd b th dnsit of th stramlins.

24 4 (6- Whn th lss massiv bod is orbiting th mor massiv bod, it is possibl that th stagnation point is trnd somhat in th dirtion of th orbit. In this to-bod soltion, thr ill b a signifiant spding p of an atomi lok at th stagnation point ( 0 (s th tim dilation qation (3-6. This is qit diffrnt from hat on old normall pt in th non-floing to-bod soltion, and it is th basis of th satllit primnt hih as sggstd in rfrn [5]. Thr ill also b tim dilation jmps for a frl rnning lok hn th satllit rosss th srfa of transition anhr in th gnral rgion shon in figr (6-. What happns to th flo along th srfa of transition rall nds to b rvald b primnt. Thr ma b a strit disontinit btn th sparat flos, thr ma b trblnt flo thr, or thr might b a smooth transition btn th flos (in hih as, thr ill b non-vanishing rl in th transition srfa. In th as of a smooth transition, th ( / grad trm on th right hand sid of qation (6-7 tlls s thr ill b a forard alration of a tst satllit as it passs throgh th transition srfa from th rgion of slor flo into th rgion of fastr flo (this is shon shmatiall in figr (6-3. In rtain to-bod onfigrations, this might provid a sbstitt for proplsion. It is a kind of slingshot fft. Contrariis, motion from th rgion of fastr flo into th rgion of slor flo ill as a dlration.

25 5 ( rl v (6-3 grad v As is also shon shmatiall in figr (6-3, th gnralid Coriolis alration ( rl v of qation (6-7 ill at on a satllit moving from a rgion of slor flo to a rgion of fastr flo so as to rval th gnral dirtion of th flo. In th as hn v is prpndilar to th flo, th Coriolis alration ill b atl in th dirtion of flo. This is th basis of th satllit primnt hih as sggstd in rfrn [6]. Aknoldgmnt This pbliation as mad possibl b a gift from Harold and Hln MMastr. Rfrns [] A. Einstin, Th Maning of Rlativit, Printon Univrsit Prss, Printon (956. [] H. A. ornt, Th Thor of Eltrons, Sond Edition, Dovr Pbliations, N.Y. (95. [3] J. S. Bll, Spakabl and Unspakabl in Qantm Mhanis, Cambridg Univrsit Prss, Cambridg, Chaptr 9 (987. [4] D. Bohm, Th Spial Thor of Rlativit, Addison-Wsl, N.Y. (989.

26 6 [5] T. Martin, Tsting th Bondar Conditions of Gnral Rlativit Nar th Earth-Sn Saddl Point, (998. [6] T. Martin, On th Motion of Fr Matrial Tst Partils in Arbitrar Spatial Flos, (998. [7] G. Sagna, Compts Rnds d l'aadmi ds Sins (Paris, 57, 708, 40 (93. Translatd in R. Haltt and D. Trnr, Th Einstin Mth and th Ivs Paprs, Dvin-Adair, Old Grnih, Conntit, pp (979. [8] G. G. Stoks, Philosophial Magain, 7, 9 (845. Rprintd in Mathmatial and Phsial Paprs,, pp [9] E. Whittakr, A Histor of th Thoris of Athr and Eltriit,, Dovr Pbliations, N.Y., p. 386 (989. [0] Rfrn 9, p [] A. A. Mihlson and E. W. Morl, Amrian Jornal of Sin, 34, 333 (887. [] A. A. Mihlson and H. G. Gal, Astrophsial Jornal, 6, 40 (95. [3]. Pag, Introdtion to Thortial Phsis, D. Van Nostrand, N.Y. (935 [4] J. W. Gibbs, Vtor Analsis, Dovr Pbliations, N.Y. (960 [5] R. S. Brodk, Th Phnomna of Flid Motions, Dovr Pbliations, N.Y., p. 49 (995