Relativistic orbits and gravitational waves from gravitomagnetic corrections

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1 Mem. S.A.I. Vol. 81, 93 c SAI Memoie della Relaivisic obis and gaviaional waves fom gaviomagneic coecions S. Caozziello, M. De Lauenis, L. Foe, F. Gaufi, and L. Milano Diaimeno di Scienze fisiche, Univesià di Naoli Fedeico II INFN Sez. di Naoli, Coml. Univ. di Mone S. Angelo, Edificio G, Via Cinhia, I-8126, Naoli, Ialy, salvaoe.caozziello@na.infn.i Absac. Coecions o he elaivisic heoy of obis ae discussed consideing highe ode aoximaions induced by gaviomagneic effecs. Beside he sandad eiason effec of Geneal Relaiviy (GR), a new nuaion effec was found due o he c 3 obial coecion. Accoding o he esence of ha new nuaion effec we sudied he gaviaional wavefoms emied hough he caue in a gaviaional field of a massive black hole (MBH) of a comac objec (neuon sa (NS) o BH) via he quaduole aoximaion. We made a numeical sudy o obain he emied gaviaional wave (GW) amliudes. We conclude ha he effecs we sudied could be of inees fo he fuue sace lase inefeomeic GW anenna LISA. Key wods. heoy of obis gaviomagneic effecs sabiliy heoy gaviaional waves. 1. Inoducion The magneic field is oduced by he moion of elecic-chage, i.e. he elecic cuen. The analogy wih gaviy consiss in he fac ha a mass-cuen can oduce a gaviomagneic field. The fomal analogy beween elecomagneic and gaviaional fields was exloed by A. Einsein (1913), in he famewok of GR, and hen by H.Thiing (1918). I was shown, by J. Lense and H.Thiing (1918), ha a oaing mass geneaes a gaviomagneic field, which in un, causes he ecession of laneay obis. We wan o sudy how he elaivisic heoy of obis and he oducion of GW is affeced by gaviomagneic coecions. The coecions, which ae off-diagonal ems Send offin equess o: S. Caozziello in he meic, can be seen as fuhe owes in he exansion in c 1 (u o c 3 ). Neveheless, he effecs on he obi behavio involve no only he ecession a ei-ason bu also nuaion coecions. Ou aoach suggess ha, in he weak field aoximaion, when consideing highe ode coecions in he equaions of moion, he gaviomagneic effecs can be aiculaly significan. In sysems aoaching o song field egimes, hese coecions give ise o chaoic behavios in he ansiens dividing sable fom unsable obis S. Caozziello e al. (9). In geneal, such conibuions ae discaded since hey ae assumed o be oo small bu hey have o be aken ino accoun as soon as he v c aio is significan. I is ossible o ake ino accoun wo yes of mass-cuen in gaviy. The fome is induced by mae souce oaions aound he cene of

2 94 S. Caozziello: Relaivisic obis and GW fom gaviomagneic coecions mass: i geneaes he ininsic gaviomagneic field which is closely elaed o he angula momenum (sin) of a oaing body. The lae is due o he anslaional moion of souces. 2. Gaviomagneic coecions Saing fom he Einsein field equaions in he weak field aoximaion, one obains he gaviomagneic equaions and hen he coecions in he meic C. W. Misne e al. (1973); S. Caozziello e al. (1): ( ds 2 = 1 + 2Φ ) c 2 d 2 8δ l jv l cddx j c 2 c 3 ( 1 2Φ ) δ c 2 l j dx i dx j. (1) By calculaing he affine connecion elaed o he meic (1), one also obain he geodesic equaions ẍ α + Γ α µνẋ µ ẋ ν =, (2) whee he do indicae diffeeniaion wih esec o he affine aamee. In ode o u in evidence he gaviomagneic conibuions, le us exlicily calculae he Chisoffel symbols a lowe odes. By some saighfowad calculaions, one ges Γ = Γ j = 1 Φ c 2 x j Γ i j Γ k Γ k j Γ k i j = 2 c 3 ( V i x j = 1 c 2 Φ x k = 2 c 3 ( V k x j + V j x i ) ) V j x k = 1 c 2 ( Φ x j δ k i + Φ x i δ k j Φ x k δ i j ) (3) In he aoximaion which we ae going o conside, we ae eaining ems u o he odes Φ/c 2 and V j /c 3. I is imoan o oin ou ha we ae discading ems like (Φ/c 4 ) Φ/ x k, (V j /c 5 ) Φ/ x k, (Φ/c 5 ) V k / x j, (V k /c 6 ) V j / x i and of highe odes. Ou aim is o show ha, in seveal cases like in igh binay sas, i is no coec o discad highe ode ems in v/c since hysically ineesing effecs could come ou. The geodesic equaions u o c 3 coecions ae hen c 2 d2 dσ + 2 Φ 2 c 2 x j c d dx j dσ dσ 2 ( V m V m δ c 3 im x j + δ jm x i fo he ime comonen, and d 2 x k dσ + 1 ( Φ 2 c 2 x j c d ) 2 + dσ ) dx i dσ 1 Φ c 2 x δ dx i dx j k i j dσ dσ 2 Φ dx l c 2 x l dσ ( 4 V k V m ) c 3 x j δ jm c d x k dσ dx j dσ =, (4) dx k dσ + dx i dσ =, (5) fo he saial comonens. Consideing only he saial comonens, we obain he obi equaions. Calling dl euclid = δ i j dx i dx j, and e k = dx k dl euclid we have, in veco fom, de = 2 [ Φ e(e Φ)] + dl euclid c2 4 [e ( V)]. (6) c3 The gaviomagneic em is he second one in Eq.(6) and i is usually discaded since consideed no elevan. This is no ue if v/c is quie lage as in he cases of igh binay sysems o oin masses aoaching o BH. Fom he above equaions we can wie he Lagangian and deive he obial equaions of moions saing fom he Eule-Lagange equaions (see S. Caozziello e al. (9). Ou aim is o sudy how gaviomagneic effecs modify he obial shaes and wha ae he aamees deemining he sabiliy of he oblem. The enegy, he mass and he angula momenum, essenially, deemine he sabiliy. Beside he sandad eiason ecession of GR, a nuaion effec is induced by gaviomagneism and sabiliy deends on i. The soluion of he sysem of diffeenial equaions (ODE) of moion esens some difficulies since he equaions ae siff. Fo ou uoses, we have found soluions by using he so called Siffness Swiching Mehod o ovide an auomaic mean of swiching beween a non-siff

3 S. Caozziello: Relaivisic obis and GW fom gaviomagneic coecions 95 and a siff solve couled wih a moe convenional exlici Runge-Kua mehod fo he non-siff a of diffeenial equaions. Time seies of boh ṙ() and () ogehe wih he hase oai (), ṙ() ae shown assuming as iniial values of he angula ecession and nuaion velociies inege aios wih he adial velociy. In Fig. 1 we show as examle he one wih: ϕ = 1 ṙ and θ = 1 ϕ En=.95 = =.1 φ =.1 θ =.1 M 1 En.95 Θ Φ 1 Φ 1 M 1 En.95. Θ Φ 1 Φ Fig. 2. Plos of he basic obis wih he associaed field velociies in false colous Obi wih Gaviomagneic effec(red) and wihou Gaviomagneic(black) =21 v θ = θ=π /2 4 NO Gaviomagneic Fig. 1. Plos along he anel lines of: () (ue lef), hase oai of () vesus ṙ() (boom lef), ṙ() (ue igh) and () (boom igh) fo a sa of 1M. The examles we ae showing wee obained solving he sysem fo he following aamees and iniial condiions: µ 1M, E =.95,φ =, θ = π, θ 2 = 1 φ, φ = 1 ṙ and ṙ = 1 and = µ. The siffness is eviden fom he end of ṙ() and () Gaviomagneic Obis wih gaviomagneic coecions In his secion we show some examles of obis. In Fig.2 ae loed some basic obis wih he associaed field velociies in false colous. Then in Fig. 3 we show he obis wih gaviomagneic coecion (ed-line) and wihou gaviomagneic coecion (black-line). Finally in he Fig. 4 hee is he hase oai wih (ed-line) and wihou (blue-line) gaviomagneic obial coecion esecively. The examle we ae showing was obained solving he sysem fo he following aamees and iniial condiions: µ 1M, E =.95, φ =, θ = π 2, θ = 1 φ, φ = 1 ṙ and ṙ = 1 and = µ. In figue 5 we show some beak- Fig. 3. The obi wih gaviomagneic coecion (ed-line) and wihou gaviomagneic coecion (black-line). ing oins of he obial moion wih gaviomagneic coecions. 3. Gaviaional wave in he quaduole aoximaion Now, consideing he obial equaions (see S. Caozziello e al. (9)), we know ha diec signaues of gaviaional adiaion ae is amliude and is wave-fom (C. W. Misne e al. 1973). In ohe wods, he idenificaion of a GW signal is sicly elaed o he accuae selecion of he shae of

4 96 S. Caozziello: Relaivisic obis and GW fom gaviomagneic coecions En=.95 M= 1 NO= Wih= 21 φ=.1 θ= En=.5 = 13.5 θ = v v.5.1 Fis Beaking oin Fig. 4. The hase oai wih (ed-line) and wihou (blue-line) gaviomagneic obial coecion. The examle we ae showing was obained solving he sysem fo he following aamees and iniial condiions: µ 1M, E =.95,φ =, θ = π2, θ = φ,φ = and = and = µ. Fig. 6. In his figue i is shown he iniial obi wih he iniial(squaes) and final(cicles) oins maked in black. En=.95 M= 1.4 = µ =.1 φ =.1 θ = 4 En=.5 = 13.5 θ= I II III v.5 IV Beaking oins Fig. 7. Plo of field velociies of he obis Fig. 5. We show he fis fou obis in he hase lane: he ed one is labelled I, he geen is II, he black is III and he blue is IV. As i is ossible o see, he obis in he hase lane ae no closed and hey do no ovela he obial closue oin; we called hese feaues beaking oins.in his dynamical siuaion, a small eubaion can lead he sysem o a ansiion o chaos as escibed by he Kolmogoov-Anold-Mose (KAM) heoem. adiaion. I is well known ha he amliude of GWs can be evaluaed by 2G jk Q, (7) Rc4 R being he disance beween he souce and he obseve and { j, k} = 1, 2, whee Qi j is he quaduole mass enso X Qi j = ma (3xai xaj δi j a2 ). (8) h jk (, R) = a wave-foms by inefeomees o any ossible deecion ool. Such an achievemen could give infomaion on he naue of he GW souce, on he oagaing medium, and, in incile, on he gaviaional heoy oducing such a Hee G is he Newon consan, a he modulus of he veco adius of he a h aicle and he sum unning ove all masses ma in he sysem. A his oin we comued he amliude comonens wih gaviomagneic coecions in geomeized unis

5 S. Caozziello: Relaivisic obis and GW fom gaviomagneic coecions 97 Table 1. Daa fo GW fo a NS of 1.4M obiing aound a Sue-MBH µ e f (mhz) h h + h x 19 En=.95 M= 1.4 = µ =.1 φ =.1 θ = Lisa sensiiviy µ=1.4 M o µ= M o h Gav. Wave. Aml h sain Fig. 8. Toal gaviaional emission wavefom h fo a neuon sa of 1.4M h GW olaizaions h + GW olaizaions x x x x 4 Fig. 9. The gaviaional wavefom olaizaions h + and h fo a neuon sa (NS) of 1.4M x Fequency Fig.. Plo of esimaed mean values of gaviaional emission in ems of sain h fo wo binay souces fom he galacic cene wih educed mass aio µ 1.4M (ed diamonds) and µ M (geen cicles). The blue line is he foeseen LISA sensiiviy cuve. The wavefoms wee comued fo he Eah-disance o Sagiaius A (cenal Galacic Black Hole). (S. Caozziello e al. (9)). We efomed he numeical simulaions in wo cases: i) a NS of 1.4M obiing aound a Sue-MBH ( e.g. Sagiaius A* 6 M ) ii) a BH of M obiing aound a Sue-MBH. We consideed he educed mass µ = m 1M 2 m 1 +M 2. Comuaions ae efomed wih obial adii measued in mass unis. Iniial disances ae samled o show obis fom high ecceniciy u o ciculaiy (e = max min max + min ). In Fig. 7-9 we show esecively he field velociies of he obis along he axes of maximum covaiances, he oal gaviaional emission wavefom h and he gaviaional wavefom olaizaions h + and h fo a NS of 1.4M. The wavefom wee comued fo he Eah-disance fom Sagiaius

6 98 S. Caozziello: Relaivisic obis and GW fom gaviomagneic coecions A (cenal Galacic Black Hole). We obained he numeical examles solving he ODE sysem fo he following aamees and iniial condiions: µ 1.4M, E =.95,φ =, θ = π 2, θ =, φ = 1 ṙ and ṙ = 1 and = ( )µ. See also Fig. 7-9 and Table 1. Finally we show in Fig. he lo of he esimaed h GW-sain-amliudes fo he consideed binay souces a Galacic Cene disance. The blue-line is he foeseen LISA sensiiviy (one yea inegaion + whie dwaf backgound noise). The ed diamonds (1.4M ) and he geen cicles ( M ) ae he h values fo he sysems we have sudied. 4. Concluding Remaks The gaviomagneic effec could give ise o ineesing henomena in igh binding sysems such as binaies of evolved objecs (neuon sas o black holes). The effecs eveal aiculaly ineesing if v/c is in he ange ( 1 3 )c. They could be imoan fo objecs caued and falling owad exemely massive black holes such as hose a he Galacic Cene. Gaviomagneic obial coecions, afe long inegaion ime, induce ecession and nuaion effecs caable of affecing he sabiliy basin of he obis. The global sucue of such a basin is exemely sensiive o he iniial adial velociies and angula velociies, he iniial enegy and masses which can deemine ossible ansiions o chaoic behavio. In incile, GW emission could esen signaues of gaviomagneic coecions afe suiable inegaion imes in aicula fo he on going LISA sace lase inefeomeic GW anenna. Refeences Caozziello, S., De Lauenis, M., Foe, L., Gaufi, F., Milano, L. submied o Physica Scia, (9) Caozziello, S., De Lauenis, M., Gaufi, F., Milano, L., Physica Scia vol. 79,. 2591, (9) Caozziello, S., Re, V., Phys. Le. A 29 (1) 115 Einsein, A., Phys. Z., 14 (1913) 1261 Misne, C. W., Thone, K.S., Wheele, J.A., Gaviaion, Feeman, New. Yok (1973) Thiing, H., Phys. Z., 19 (1918) 4 Thiing, H., Phys. Z., 19 (1918) 33; Lense, J. and Thiing, H., Phys. Z., 19 (1918) 156; Mashhoon, B., Hehl, F.W., and Theiss, D.S., Gen. Rel. Gav., 16 (1984) 711

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