ON THE MOTION OF FREE MATERIAL TEST PARTICLES IN ARBITRARY SPATIAL FLOWS
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1 ON THE MOTION OF FREE MATERIA TEST PARTICES IN ARBITRARY SPATIA FOWS Tom Martin Grait Rsarh Institt Boldr, Colorado Abstrat W sho ho th motion of fr matrial tst partils in arbitrar spatial flos is asil rmind ithin th ontt of ordinar tor alls. This ma b sfl for ron, inlding nginrs and othr non-spialists, hn thinking abot graitational problms. It alrad has alid appliation to simpl problms sh as th problms of motion in rotating and alrating frams and to th graitational problm of th singl sphriall smmtri attrator. Whn applid to th to bod graitational problm, it ma hlp s rmin th atal dirtion of th flo. Introion In a rnt pbliation [], disssd th possibilit that Natr might prfr nonstati and spatiall floing tp soltions of th graitational fild qations rathr than th sal stati and spatiall rd tp soltions. In th as of th to bod graitational problm of th Earth-Sn sstm, disord that thr is a r strong graitational tim dilation fft nar th graitational saddl point hih an b tilid, primntall, to distingish th phsial ralit of th to possibl tps of soltion. It is or intntion in th prsnt papr to sho ho th qations of motion of fr matrial tst partils in arbitrar spatial flos an b asil rmind ithin th ontt of ordinar tor alls. This obiats th nd to introd th sal ompliation of oariant diffrntiation ith its assoiatd Christoffl smbols and affin paramtrs. Th rslting Gnral Rlatiisti qations of motion alrad ha alid appliation to simpl problms sh as th problms of motion in rotating and
2 alrating frams of rfrn and to th graitational problm of th singl sphriall smmtri attrator. Whn applid to th to bod graitational problm, th ma hlp s rmin th atal dirtion of th flo.. Spa-tim ith Spatial Flo Whn spa-tim is haratrid b a flo of phsial spa, thr ists a global Galilan oordinat fram { r, t} in hih th flo is rprsntd b a 3-spa tor fild ( r, t. This flo of spa is a gnraliation of Nton's onpt of absolt spa in hih no part of spa is moing ith rspt to an othr part (Nton's absolt spa orrsponds to th as in hih thr is a Galilan fram in hih 0 rhr. In ths Galilan oordinats, th propr tim lmnt of an atomi lok is gin b hr dτ γ /, (-. (- Hr, τ is th propr tim of th lok, is th oordinat spd of light in phsial spa (a onstant, t is th oordinat tim, r is th oordinat position tor of th lok, d r / is th oordinat loit of th lok, and is th oordinat loit of th lok rlati to phsial spa. From th abo qations, s that th spa-tim lin lmnt is d τ ( + dr ( dr (-3 in ths Galilan oordinats. Frthrmor, th total tim dilation of a lok in motion ith loit ( rstrit < in an arbitrar spatial flo is gin b dτ ( + /. (-4
3 3 W ha shon in rfrn [] that ths Galilan oordinats ar prftl aptabl oordinats for stding th Gnral Rlatiisti problm of th rotating fram in flat spa-tim and th Gnral Rlatiisti problm of th singl sphriall smmtri attrator. Ths, th qations of motion obtaind in th folloing to Stions ar appliabl to ths anonial ass as ll as to man othrs.. Th Eqations of Motion Th path of a fr matrial tst partil in Gnral Rlatiit is on or hih th intgral of th partil's propr tim is an trmm: δ d τ 0. (- Th path is obtaind from th assoiatd Elr-agrang qations. In Galilan oordinats ith spatial flo, th allation of th path is simplifid b sing th oordinat tim t as th path paramtr. This rds th nmbr of Elr-agrang qations from for (inoling th spa-tim oordinats to thr (inoling th spatial oordinats. From (-, ha δ dτ δ ( / 0. (- t s prform th allation in rtanglar Galilan oordinats {,,, t} ith th orthonormal spatial basis,, }. Th Elr-agrang qations ar { hr d r, (-3 / ( r, ; t ( γ (-4 is th agrangian, and + +, (-5
4 4 r. (-6 In this Elr-agrang formalism, r and ar tratd as indpndnt ariabls. (t is a paramtri tor, hil, ( t r is a tor fild. Ths, (, (-7 hr (-8 is th tnsor haratriing th spatial inhomognit of th flo. From (-7, ha ( ( (, hn r ( ( ( /. Sin / / ( ( +, (, so
5 5 d d d d 4 3 d. Stting γ (from (-4, s that or Elr-agrang qations (-3 ar d γ d + (. (-9 Taking th salar pro of this qation ith, gt d γ d + ( ((. (-0 Sin ( γ, (-0 boms d γ ((. (- Sbstitting this bak into (-9, find that d ( + (((. (-a W an rit this mor formall as d + ( (/ ( 0, (-b hr is th idntit tnsor in 3-spa and motion of th tst partil. (/ is th smmtri tnsor of This is th qation of motion of a fr matrial tst partil in Galilan oordinats ith arbitrar spatial flo. Th qation is fll rlatiisti (in th sns that th matrial tst partil ma ha an arbitrar rlatiisti spd ( 0 < rlati to phsial spa.
6 6 3. Bond Nton is Th non-rlatiisti (slo motion approimation of th rlatiisti qation (- d (. (3-a In trms of and, this is d d ( (, (3-b hr d / is th driati of th paramtri tor ( t ( r( t, t along th path r (t spifid b th loit tor d r /. B th hain rl of diffrntiation, d. (3- t Using th tor idntitis ( ( rl (3-3 and (, (3-4 find that d + ( rl + t. (3-5 To rmind orsls that th alration partil, rit this as a d / is that of a fr matrial tst a fr + ( rl t. (3-6
7 7 Th first trm on th right-hand sid of this qation rprsnts th graitational 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 proids s ith an as a to std th problm of motion in rotating and alratd frams in ordinar Ntonian mhanis. W do this b fosing on th flos indd b th fram motions. For ampl, a rotating Galilan fram in flat spa-tim ith an aial rotational loit ω ω(t, inds th flo ( r, t r ω( t. (3-7 Mor importantl, qation (3-6 has appliation bond ordinar Ntonian mhanis, bas it holds for arbitrar flos and not jst thos indd b th motion of frams. An ampl of a flo that is not indd b th motion of a fram is th flo assoiatd ith a singl sphriall smmtri graitational attrator. In a sphrial Galilan fram ntrd on th attrator, this is gin b on or th othr of th flos ( r ± GM / r, (3-8 r hr G is th graitational onstant and M is th mass of th attrator. As disssd in som ail in rfrn [], th Prinipl of Gnral Coarian assrs s that no phsial primnt an distingish btn th to flos (3-8 as long as ar daling ith a singl isolatd attrator. Hor, hn a sond attrator is prsnt, thr ar trms in (3-6 hih ma hlp s in dsigning an primnt to masr th atal dirtion of th flo. W disss this possibilit, at last qalitatil, in th nt Stion. 4. Satllit Motion throgh th Srfa of Transition of th Earth-Sn Sstm Sppos a satllit ith a stabl atomi lok onboard is snt throgh th rgion of th Earth-Sn graitational saddl point as sggstd in rfrn []. If th frqn of
8 8 th lok aris in a a that is onsistnt ith th spatial flo tp soltion of th to bod problm, ill b jstifid in stding th ails of this flo in othr as. t s disss th Earth-Sn to bod flo from th point of i of a non-rotating Galilan fram ntrd on th Sn. In this fram of rfrn, th Earth and th graitational saddl point ar orbiting abot th Sn. (W nglt all othr plantar prtrbations. Th flo is qisnt at th saddl point [], and thr ill b a srfa of transition btn th Earth's spatial flo and th Sn's spatial flo. In th rgion of th saddl point, this hprboloid-lik srfa is spt aa from th Sn, and it is moing ith th Earth and th saddl point as a bondar btn th trrstrial flo and th solar flo. It is losst to th Earth hr it intrsts th saddl point (th saddl point is abot 60,000 km from th Earth. Considr a point P on th srfa of transition and in th orbital plan at a distan of sral thosand kilomtrs from th saddl point. At som, bt as t primntall nrmind, small distan into th trrstrial sid of th flo from P, th trrstrial flo ill b approimatl paralll to th transition srfa and th orbital plan, and it ill ha a spd on th ordr of Gm / r.75 km/s, hr m is th Earth's mass and r is th distan from th Earth to th saddl point. On th solar sid of th flo from P, th solar flo ill also b approimatl paralll to th transition srfa and th orbital plan, bt its spd ill b on th ordr of GM / R 4. km/s, hr M is th solar mass and R is th distan from th Sn to th saddl point. If snd a satllit from th Earth prpndilarl throgh th srfa of transition at P and ot into th solar flo, ma b abl to std its motion ith qation (3-6. This dpnds on th natr of th transition btn th flos. Thr ar thr possibilitis: th transition is a strit tangntial disontinit of th flo, th transition is trblnt flo, or 3 th transition is a smoothl aring flo. W an appl qation (3-6 ith onfidn onl to th third as. Sppos, for th sak of argmnt, that th flo is smoothl aring throgh th transition. In i of th diffrns of th flos on ithr sid of P as stimatd abo, th trm ( / hil th trms old gi th satllit an alration in its forard dirtion, ( rl and old gi th satllit alrations in th t
9 9 dirtion of th solar flo. So, in th as of a smoothl aring transition, it is possibl, in prinipl, to rmin th atal dirtion of th flo. [] Martin, T. (998, Tsting th Bondar Conditions of Gnral Rlatiit Nar th Earth-Sn Saddl Point,
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