Solar system effects in Schwarzschild de Sitter space time

Transcription

1 Physics Lettes B ) Sola system effects in Schwazschild de Sitte space time Valeia Kagamanova a, Jutta Kunz b, Claus Lämmezahl c, a Institute of Nuclea Physics and Ulugh Beg Astonomical Institute, Astonomicheskaya 33, Tashkent , Uzbekistan b Institut fü Physik, Univesität Oldenbug, Postfach 2503, D Oldenbug, Gemany c ZAR, Univesität Bemen, Am Falltum, D Bemen, Gemany Received 24 Octobe 2005; eceived in evised fom 30 Januay 2006; accepted 30 Januay 2006 Available online 9 Febuay 2006 Edito:. Cvetič Abstact The Schwazschild de Sitte space time descibes the gavitational field of a spheically symmetic mass in a univese with cosmological constant Λ. Based on this space time we calculate Sola system effects like gavitational edshift, light deflection, gavitational time delay, peihelion shift, geodetic o de Sitte pecession, as well as the influence of Λ on a Dopple measuement, used to detemine the velocity of the Pionee 10 and 11 spacecaft. Fo Λ = Λ m 2 the cosmological constant plays no ole fo all of these effects, while a value of Λ m 2, if hypothetically held esponsible fo the Pionee anomaly, is not compatible with the peihelion shift Elsevie B.V. All ights eseved. 1. Intoduction Until now all gavitational effects in the Sola system and in binay systems ae well descibed by means of the Einstein geneal theoy of elativity [1,2]. 1 Cosmology, in contast, is cuently confonted with two mysteies: these ae dak matte and dak enegy. Dak matte is intoduced to obtain the gavitational field needed to descibe obsevations like the galactic otation cuves, gavitational lensing, o the stuctue of the cosmic micowave backgound. Dak enegy is invoked to explain the obseved acceleated expansion of the univese by means of an additional enegy momentum component. Thus standad cosmology addesses the obseved imbalance in the gavitational field equations by postulating unknown foms of matte and enegy. In geneal elativity, the epulsion necessay to obtain an acceleated expansion of the univese can be povided by the inclusion of a vacuum enegy. This coesponds to the wellknown modification of the Einstein equations consisting of * Coesponding autho. addesses: kavageo5@amble.u V. Kagamanova), kunz@theoie.physik.uni-oldenbug.de J. Kunz), laemmezahl@zam.uni-bemen.de C. Lämmezahl). 1 A possible exception is the Pionee anomaly [3]. the addition of a cosmological tem Λg μν. Obsevational data suggest fo the cosmological constant Λ a value of Λ m 2 [4]. Altenatively, vacuum enegy can be modeled by a dynamical field, such as quintessence [4,5]. Instead of postulating unknown foms of matte and enegy anothe line of esolving these cosmological issues consists of a modification of the gavitational field. The galactic otation cuves fo a wide class of galaxies could be explained by using an additional gavitational Yukawa potential [6], fo example. 2 oe ecently bane-wold models ae consideed as modifiedgavity theoies in this context [8]. We hee conside the gavitational field of a spheically symmetic mass in a univese with cosmological constant Λ, descibed by the Schwazschild de Sitte space time. We calculate how the pesence of such a cosmological constant would affect Sola system effects like gavitational edshift, light deflection, gavitational time delay, geodetic o de Sitte pecession. Compaing the theoetical esults with the obsevations gives Sola system estimates on the magnitude of such an additional cosmological tem in the Einstein equations. Ou investigation indicates that no pesent o futue obsevation within the Sola system has the capability to eveal ef- 2 Anothe idea is to modify the Newtonian dynamics [7] /$ see font matte 2006 Elsevie B.V. All ights eseved. doi: /j.physletb

2 466 V. Kagamanova et al. / Physics Lettes B ) fects due to a cosmological constant with the cuent value of Λ m 2 [4]. We conclude that the cosmic acceleation theefoe does not lead to obsevable Sola system effects. We then addess the influence of a cosmological constant Λ, as pesent in the Schwazschild de Sitte space time, on a Dopple measuement, used to detemine the velocity of the Pionee 10 and 11 spacecaft. By intepeting this Λ not necessaily as the cosmological constant with value Λ 0, we note that a value of Λ m 2 may lead to an anomalous acceleation on the ode of the Pionee acceleation. Howeve, this value is not compatible with the peihelion shift and, futhemoe, leads to a position-dependent acceleation. We like to emphasize that the obsevational basis of the Pionee anomaly is by fa not of the same quality as the Sola system effects. It is still possible that the Pionee anomaly is just a systematic effect, though no convincing explanation fo this possibility has been found [3]. Theefoe, it is easonable to take into account the possibility that this anomaly is due to some until now missed effect within standad physics, o that it even signals some kind of non-standad physics. Theefoe we ae looking fo effects elated to a space time metic modified by a cosmological constant. This may also be of elevance fo the design of a new space mission to test the long ange gavitational field [9]. 2. Schwazschild de Sitte space time We calculate Sola system effects fo the spheically symmetic Schwazschild de Sitte metic [10] ds 2 = α) 2 α 1 ) d 2 2 dθ 2 + sin 2 θdφ 2), whee α) = ) 3 Λ2, and Λ is the cosmological constant, and the mass of the souce. It is an exteio solution of the Einstein field equations fo a spheical mass distibution R μν 1 2 g μνr + Λg μν = 8πGT μν. In the following we take Λ to be a fee paamete and ty to obtain constaints on the magnitude of this constant fom Sola system effects. Theefoe ou aim is twofold: Fist to investigate whethe thee might be some influence of the cosmological constant Λ 0 on Sola system effects, and, second, to pesent estimates on the magnitude of such a constant Λ, when it would not coespond to the cosmological constant, but instead would be of, e.g., galactic oigin. The peihelion shift fo the Schwazschild de Sitte space time can be found in [10] and [11]. Based on the Ke de Sitte metic in the latte pape [11] also gavitomagnetic effects have been calculated. 3. Gavitational edshift Since Schwazschild de Sitte is a stationay space time thee is a time-like Killing vecto so that in coodinates adapted 1) 3) to the symmety the atio of the measued fequency of a light ay cossing diffeent positions is given by ν g 00 ) = 4) g 00 0 ). Fo expeiments in the vicinity of the cental body which in this case will be taken to be the Eath), we have Λ 2 / so that to fist ode in Λ ν = g0) 00 ) g 0) 00 0) whee g 0) 00 = 1 2 ν = Λ Λ g 0) 00 0) 2 g 0) 00 ) )),. Fo small masses this simplifies to ), whee we neglected poducts of Λ and. Clock compaison can be pefomed with an accuacy of 10 15, the H-mase in the GP-A edshift expeiment [12] eached a accuacy. Since all obsevations ae well descibed within Einstein theoy, we conclude Λ m 2, whee we assumed a clock compaison between the Eath and a satellite at km height. 4. Deflection of light The motion of point paticles and light ays in the Schwazschild de Sitte space time is descibed by the Lagangian 2L = α)ṫ 2 α 1 )ṙ 2 2 φ 2, 8) whee the dot stands fo diffeentiation with espect to the affine paamete λ, and whee fom the vey beginning the motion is esticted to the θ = π/2 plane. Since the metic 1) is independent of t and φ the quantities E α) 9) dλ, dφ L 2 dλ, ae conseved along the obit of the paticle and define its enegy and angula momentum, espectively. Fo light, the Lagangian 8) vanishes. ultiplication of 8) by α/l 2 and substitution of the angula momentum yields [ dφ 1 10) d =±1 2 b 2 α) ] 1/2 2, whee b L/E which in Schwazschild space time can be intepeted as impact paamete. The ± is elated to inceasing/deceasing. The distance of closest appoach 0 defined by d 11) dφ = 1 =0 b 2 α 0) = elates b to 0. When we eplace b by 0 in 10) the constant Λ dops out [ ) dφ ) d =± )] 1/ ) 6) 7)

3 V. Kagamanova et al. / Physics Lettes B ) Theefoe the constant Λ has no influence on the light deflection. This has aleady been ecognized in [13]. 5. Gavitational time delay We conside the standad measuement of the time delay of light whee a ada signal is sent fom the Eath to pass close to the Sun and eflect off anothe planet o a spacecaft. The time inteval between the emission of the fist pulse and the eception of the eflected pulse is measued. Let and R be the distance between the Sun and the Eath and the eflecto, espectively. The time inteval between emission and etun of a pulse as measued by a clock on the Eath is T = 2t, 0 ) + 2t R, 0 ), 13) whee t 1, 2 ) is the elapsed coodinate time along the signal s tajectoy between distances 1 and 2 to the Sun. The calculation of t 1, 2 ) equies to epesent t as a function of along the path of the pulse. Fom 9) and 10) we obtain 14) d =± 1 1 αb b 2 α) ) 1/2 2 and, thus, t, 0 ) = 0 d 1 bα) b 2 α) ) 1/2 2, 15) whee we can eplace b by 0 using 11). The esult of this integation in the Schwazschild space time can be found, e.g., in [14]. The fist ode coection to 15) due to Λ is t Λ, 0 ) = Λ [ ) ))] , 16) + 0 whee tems of the ode 2 have been neglected. Since fo Sola system obsevations 0 / R 1 and 0 / 1 we obtain as Λ-induced modification of the time delay of a ound tip of a signal t Λ = 2Λ 3 17) 9 + R ) ) 2 R In the ecent Cassini expeiment [15] one has not measued the time delay but, instead, the elative change in the fequency y = νt) = d 18) t, whee is the fequency of the adio waves emitted on the Eath. These adio waves each Cassini with a fequency ν and ae sent back with the same fequency ν. The fequency of this signal eaching the Eath is νt). The contibution to the fequency change fom the Λ-tem is y Λ = d t Λ = 4 3 Λ 0t) d 0t), 19) whee fo the Cassini setup d 0 / v is appoximately the velocity of the Eath obiting the Sun. That means that the Λ- contibution to the signal inceases with distance. The Schwazschild contibution to y is and has been veified by the Cassini measuements within an accuacy of y of Cassini measued the time delay fo appoximately 25 days, that is, duing the peiod of 12 days befoe and 12 days afte conjunction [15]. Duing one day the distance of closest appoach of the signal changed by 1.5 Sola adii. Theefoe, the measued signal would be y Λ 12d) y Λ 0) = 16Λ R v. 20) Since the Cassini measuement was in accodance to the Einstein pediction, we conclude fom y Λ 12d) y Λ 0) that Λ c 3 16G R v m Peihelion shift 21) The peihelion shift is based on the modification of the geodesic equation fo a massive paticle which can be deived in a standad manne. In fist ode in Λ and with u = 1/ one obtains d 2 u dϕ 2 = L 2 u + 3u2 Λ 3L 2 u 2 which leads to an additional shift [11] 22) δ = πλa 2 a 1 e 23) 2 = Λa4 1 e 2 ) 3/2 6 2, whee e is the eccenticity of the obit and the standad Einstein expession fo the peihelion shift, see also [10,25,26]. Obviously, fo a smalle cental mass, implying a smalle peihelion shift, the influence of the Λ-tem becomes moe impotant. Its effect should be moe ponounced fo the oute planets and fo comets with lage a.) The peihelion shift of ecuy, 43 pe centuy, is in full ageement with Einstein theoy within the accuacy of 430 µas [16]. Fom that we conclude 3 Λ 430 µas 7. Geodetic pecession 6 2 a 4 1 4e 2 ) m 2. 24) We calculate the geodetic pecession by intoducing a coodinate system otating with angula velocity ω. The new angula coodinate ϕ is defined though dϕ = dφ ω, and gives in the equatoial plane θ = π/2 the metic 25) 3 Afte submission of this Lette a simila analysis by L. Ioio has been appeaed [17].

4 468 V. Kagamanova et al. / Physics Lettes B ) ds 2 = α 2 ω 2) α 1 d 2 ) 2 ω 2 α 2 ω 2 dϕ 2 α α 2 ω 2 dϕ2. The canonical fom of the metic is ds 2 = e 2Ψ w i dx i ) 2 h ij dx i dx j, with 26) 27) e 2Ψ = α 2 ω 2, w 1 = w 2 = 0, w 3 = 2 ω α 2 ω 2, h 11 = α 1, h 33 = 2 α 28) α 2 ω 2, with the indices 1, 2, 3 efeing to, θ, ϕ, espectively. Fo a given = 0 we can choose an ω so that Ψ/ = 0. Since Ψ is elated to the acceleation, this means that the line = 0 is geodesic. This coesponds to a modified Keple fequency ω 2 = 3 Λ 3, also obtained in [11], and implies 29) e 2Ψ = 1 3, w 2 ω 1 = w 2 = 0, w 3 = 1 3/, h 11 = α 1 2 α, h 33 = 30) 1 3/. The otation ate of a gyoscope at a fixed position in the otating coodinate system is given by [10] Ω 2 = e2ψ 31) 8 hik h jl w i,j w j,i )w k,l w l,k ) and points in the negative diection elative to the obit. We obtain Ω = 3 Λ 3 = Ω 0 1 Λ3 6 ), Ω 0 = 3. 32) Fo an altenative deivation of this geodetic pecession, see Appendix A. The mission GP-B will measue the geodetic pecession Ω 0 = mas/y with an expected accuacy of 0.1 mas/y. If the value pedicted fom Einstein theoy will be confimed and no additional tem is seen, we ae led to an estimate fo Λ Λ m Dopple tacking of satellites on escape tajectoies 33) We now calculate the two-way Dopple tacking of a satellite moving on a geodesic adially away fom the Sun. This amounts to an investigation of whethe the pesence of a cosmological constant might be made esponsible fo the anomalous acceleation obseved fo the Pionee spacecaft [3]. Fo simplicity, we take an obseve at a fixed adial distance fom the Sun and a spacecaft moving adially so that the Sun, the Eath and spacecaft lie on one line. The measuement is caied out as follows: On the Eath a adio signal with fequency is sent to the spacecaft at position, whee it will aive edshifted accoding to α) ν = 34) α ). Due to the velocity of the spacecaft the measued fequency is in addition Dopple shifted ν = 1 1 v ) 2 1 v ) ν, 35) whee v = θ v) is the measued adial velocity v is the 4-velocity of the spacecaft and θ is the nomalized basis 1-fom in -diection, g μν θμ θ ν = 1; in coodinates, θ μ = α 1/2 0, 1, 0, 0)). This is also the fequency sent back to the Eath, whee it aives with the fequency ν 1 = 1 v ) α ) 36) 1 v ) 2 α) ν. The two-way Dopple tacking will then yield ν = 2 v 1 + v 2v. 37) As expected, the gavitational edshift dops out. Only the motion of the spacecaft influences the esult. Theefoe, what is finally needed is the solution fo the adial velocity v obtained fom the geodesic equation { } { 0 = d μν μν d = d α α } ) d dx μ dx ν ) α α, 38) 39) whee { μ ρσ } ae the Chistoffel symbols. We can safely neglect the tem quadatic in the velocities. Then, to fist ode in and Λ we obtain d 2 2 = 2 + Λ 3. This leads to ) 1 d 2 = E Λ2, 40) 41) whee E is an integation constant. The Dopple measuement in tems of the measued velocity is ν = 2 v 1 + v = 2 d/ α) + d/. 42) We assume Λ 2,/ E and make an expansion with espect to Λ and with espect to. Futhemoe,

5 V. Kagamanova et al. / Physics Lettes B ) ν 2E = E E 1 + 2E) 2 2E E 3 Λ E) 2 2E + O Λ 2, 2). Since 2E coesponds to v 2 0 /c2, the non-elativistic limit is ν = 2 2E 2 + O Λ 2, 2) E 3 Λ2 2E 43) 44) The fist two tems ae the standad Newtonian contibutions. The position in the Λ-tem induces an additional timedependence of the two-way Dopple signal. Appoximating = 0 + v 0 t and E = v0 2 /2, the time-dependent signal elated to the Λ-tem is d ν ) Λ = 2 3 Λṙ 1 2E = 2 3 Λ 0 + v 0 t). 45) If the fist tem is assumed to be esponsible fo the obseved Pionee anomaly, then with 0 being on the ode of 100 AU the constant Λ should be on the ode of m 2. This value is not compatible with the peihelion shift. That means that thee is no way to simulate the Pionee anomaly with a tem of the fom of a cosmological constant without being in conflict with obsevations. The influence of the cosmological constant on the semimajo axis is negligible. Fo any non-newtonian gavitational foce the distance of the planetay obits as function of the angle ϕ is given by ɛ) ϕ) = 46) 1 + e cos1 + ɛ)ϕ), whee ɛ is the paamete elated to the non-newtonian pat of the gavitational inteaction descibing the peihelion shift and the change in the planetay obital paamete 0 which is elated to the semi-majo axis). As a consequence, we can elate the peihelion shift to the change in the obital paamete: If, due to some non-newtonian o non-einsteinian gavitational foces, thee is an additional peihelion shift of the ode 400 µas pe centuy which is the pesent day accuacy [16]), then ɛ so that ɛ 0 15 cm. This is clealy beyond any obsevational access fo any planet in the Sola system. Theefoe, a value of Λ which may give fo the Pionees an acceleation of the equied ode has pactically no influence on the planetay obits. 9. Conclusion and outlook We have calculated the Sola system effects in the fame of the Schwazschild de Sitte space time, teating the cosmological constant Λ in the metic as a fee paamete. Compaison with obsevations then leads to the following constaints on the magnitude of the constant Λ see Table 1). It is emakable that the peihelion shift is by fa the most sensitive tool to obseve an influence of the cosmological constant on the scale of the Sola system which is pobably due Table 1 Estimates on Λ fom Sola system obsevations Obseved effect gavitational edshift peihelion shift light deflection gavitational time delay geodetic pecession Pionee anomaly Estimate on Λ Λ m 2 Λ m 2 no effect Λ m 2 Λ m 2 Λ m 2 to the fact that it constitutes a cumulative long tem effect. Fo light deflection the pesence of Λ has no effect. Assuming that Λ epesents the cosmological constant with the cuent value of Λ m 2, we conclude, that no pesent o futue obsevation of these Sola system effects has the capability to eveal the pesence of such a cosmological constant. This also extends to Sola system effects like the gavitomagnetic clock effect o time delay, based on the influence of an additional gavitomagnetic field [11]. Addessing the influence of a cosmological constant Λ on a Dopple measuement, used to detemine the velocity of the Pionee 10 and 11 spacecaft, we conclude that a value of Λ m 2 may lead to an acceleation of the ode of the obseved anomalous Pionee acceleation. Howeve, the acceleation caused by Λ changes with distance and is in conflict with the obseved peihelion shift. Theefoe the hypothetical pesence of a negative cosmological constant Λ m 2 as a possible explanation fo the Pionee anomaly has to be excluded. We conside the calculation of Sola system effects in the Schwazschild de Sitte space time only as a fist step towads the geneal goal, of obtaining obsevational constaints fom Sola system effects fo modified theoies of gavity. We envisage to epeat this kind of calculation in moe geneal contexts like quintessence [4,5] fo a fist step in this diection see [18]), in vaying G scenaios [19], in dilaton scenaios [20,21], and in banewold models [8,22] see [23,24] fo the discussion of one elated effect). Acknowledgements We would like to thank H. Dittus and B. ashhoon fo valuable discussions. V.K. thanks the Deutsche Foschungsmeneinschaft DFG fo financial suppot. C.L. acknowledges financial suppot of the Geman Space Agency DLR. Appendix A. Altenative deivation of geodetic pecession A gyoscope is a spin fou-vecto s μ which moves along a timelike geodesic with fou-velocity u μ. The spin is space-like g μν u μ s ν = 0 and is paalelly popagated along u μ ds μ { } μ dλ + s ν u ρ = 0. νρ A.1) A.2)

6 470 V. Kagamanova et al. / Physics Lettes B ) An obseve moving with the gyoscope on a cicula obit in the equatoial plane will see its spin pecess in the equatoial plane, see e.g. [14]. The components of the fou-velocity of the gyoscope in the θ = π/2 plane ae given by u μ = u t 1, 0, 0,ω) with ω fom 29) and u t is detemined by g μν u μ u μ = 1. Suppose that the spin initially points in the equatoial plane along the unit vecto e μˆ = 0,α1/2, 0, 0) in -diection. Fom A.2) ds θ /dλ = 0so that an initial s θ = 0 emains zeo. Fom A.1) we obtain s t = 2 ωα 1 s φ, A.3) and A.2) gives ds + 3)ωs φ = 0, A.4) ds φ ω A.5) s = 0. Choosing the initial condition s φ 0) = 0 and intoducing Ω = 1 3 ) 1/2 ω = 1 3 ) A.6) 3 Λ ) 3 3 Λ 3 the solution of A.4) and A.5) is s = s α 1/2 cosωt), ) ω s φ = s α 1/2 sinωt), Ω whee s = s μ s μ ) 1/2. A.7) A.8) Refeences [1] C.. Will, Theoy and Expeiment in Gavitational Physics, Cambidge Univ. Pess, Cambidge, 1993, evised edition). [2] C.. Will, The confontation between geneal elativity and expeiment, Living Rev. Relativ ) 4, [3] J.D. Andeson, P.A. Laing, E.L. Lau, A.S. Liu,.. Nieto, S.G. Tuyshev, Phys. Rev. D ) [4] P.J.E. Peebles, B. Rata, Rev. od. Phys ) 559. [5] C. Wetteich, Nucl. Phys. B ) 645. [6] R.H. Sandes, Aston. Astophys ) L21. [7]. ilgom, New Aston. Rev ) 741. [8] D. Dvali, G. Gabadadze,. Poati, Phys. Lett. B ) 208. [9] H. Dittus, Pionee Exploe Collaboation, A ission to Exploe the Pionee Anomaly, ESA, Noodwijk, 2005, g-qc/ [10] W. Rindle, Relativity, Oxfod Univ. Pess, Oxfod, [11] A.W. Ke, J.C. Hauck, B. ashhoon, Class. Quantum Gav ) [12] R.F.C. Vessot,.W. Levine, E.. attison, E.L. Blombeg, T.E. Hoffmann, G.U. Nystom, B.F. Fael, R. Deche, P.B. Eby, C.R. Baughte, J.W. Watts, D.L. Teube, F.D. Wills, Phys. Rev. Lett ) [13] K. Lake, Phys. Rev. D ) [14] J.B. Hatle, Gavity: An Intoduction to Einstein s Geneal Relativity, Addison Wesley, San Fancisco, [15] B. Betotti, L. Iess, P. Totoa, Natue ) 374. [16] K. Nove, Phys. Rev. D ) [17] L. Ioio, g-qc/ , Int. J. od. Phys. D, in pess. [18] J.P. belek, g-qc/ [19]. Reute, H. Weye. Phys. Rev. D ) [20] T. Damou, F. Piazza, G. Veneziano, Phys. Rev. Lett ) [21] T. Damou, F. Piazza, G. Veneziano, Phys. Rev. D ) [22] A. Lue, Phys. Rep ) 1. [23] L. Ioio, Class. Quantum Gav ) [24] L. Ioio, JCAP ) 006. [25] J.N. Islam, Phys. Lett. A ) 239. [26] G.V. Kaniotis, S.B. Whitehouse, Class. Quantum Gav ) 4817.