Preferred Spatial Directions in the Universe: a General Relativity Approach

Transcription

1 Otobe, 006 PROGRESS IN PHYSICS Volume 4 Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah Laissa Boissova lboissova@yahoo.om Heein is onstute, using Geneal Relativity, the spae meti along the Eath s tajetoy in the Galaxy, whee the Eath taes outs a ompliate spial in its obital motion aoun the Sun an its onomitant motion with the sola system aoun the ente of the Galaxy. It is eue heein that this spae is inhomogeneous an anisotopi. The obsevable popeties of the spae, haateizing its gavitation, otation, efomation, an uvatue, ae obtaine. The theoy peits that the obsevable veloity of light is anisotopi, ue to the anisotopy an inhomogeneity of spae ause by the pesene of gavitation an the spae otation, espite the wol-invaiane of the veloity of light emaining unhange. It is alulate that two pais of synhonise loks shoul eo a iffeent spee of light fo light beams tavelling towas the Sun an othogonal to this ietion, of about i. e. 10 km/se, 0.04% of the measue veloity of light. This effet shoul have osillations with a 1-hou peio ue to the aily otation of the Eath an 6 month peio ue to the motion of the Eath aoun the Sun. The best equipment fo eteting the effet is that being use by R. T. Cahill Flines Univesity, Austalia in his uent expeiments measuing the veloity of light in an RF oaxial-able equippe with a pai of high peision synhonize Rb atomi loks. The geniality of geomety, its appliability to ou eal wol, an be veifie by obsevation o expeiment, not logial eution. N. A. Kozyev 1 Intoution We onstut heein, by Geneal Relativity, a mathematial moel fo a spae boy moving aoun anothe boy the ente of attation, both moving in an obseve s efeene spae. The Eath otates aoun the Sun, an obits in ommon with it aoun the ente of the Galaxy; the Sun otates aoun the ente of the Galaxy an obits in ommon with the Galaxy aoun the ente of the Loal Goup of galaxies; et. As a esult thee ae pefee ietions etemine by obital motions, so the eal Univese is anisotopi inequivalene of ietions. Beause thee ae billions of entes of gavitational attation, the Univese is also inhomogeneous inequivalene of points. Hene, fo the eal Univese, we annot ignoe the anisotopy of spae an gavitation. On the othe han, most osmologists use the onept of a homogeneous isotopi Univese wheein all points an ietions ae equivalent. Suh a moel an be built only by an obseve who, obseving matte in the Univese fom afa, oesn t see suh etails as stas an galaxies. Suh oneptions lea to a viious ile most osmologists ae sue that ou Univese is a homogeneous isotopi ball expaning fom an initial point-like state singulaity; they ignoe the anisotopy of spae an gavitation in suh moels. Relativisti moels of a homogeneous isotopi univese whih inlue the Fiemann solutions ae only a few patial solutions to Einstein s equations. Besies, as shown uing the last eae, many popula osmologial metis inluing the Fiemann solutions ae inamissible, beause the iffeene between the aial ooinate an the pope aius isn t taken into aount thee see [1, ] an Refeenes theein. An so foth, we shall show that the homogeneous isotopi meti spaes ontain no otation an gavitation, an that they an only unego efomation: no stas, galaxies o othe spae boies exist in suh a univese. Why o the sientists use suh solutions? The answe is lealy evient: suh solutions ae simple, an theeby easie to stuy. We shall onsie anothe poblem statement, the ase of an inhomogeneous anisotopi univese as fist set up in 1944 by A. Zelmanov [4, 5]. Suh a onsieation is appliable to any loal pat of the Univese. We show in this pape that along suh a pefee ietion, ause by the obital motion of a spae boy, an anisotopy of the obsevable veloity of light an be eue, espite the wol-invaiane of the veloity of light emaining unhange. Using this esult as a basis, we will show in a subsequent pape now in pepaation that not only is the anisotopy of the veloity of light expete along a satellite s tajetoy, but even its motion is pemitte only in a non-empty spae fille by a istibution of matte an a λ-fiel both eive fom the ight sie of Einstein s equations. This onlusion leas to This situation is simila to the stana solution of the gavitational wave poblem, whih onsies them as spae efomation waves in a spae fee of otation an gavitation [3]. The obsevable veloity of light is iffeent to the wol-invaiant veloity of light if onsiee by means of the mathematial appaatus of physially obseve quantities in Geneal Relativity so-alle honometi invaiants [4, 5]. L. Boissova. Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah 51

2 Volume 4 PROGRESS IN PHYSICS Otobe, 006 the possibility of a new soue of enegy woking in a otating non-holonomi spae, an has a iet link to the onlusion that stas poue enegy ue to the bakgoun spae non-holonomity as eently eive by means of Geneal Relativity in [6, 7]. Obseve haateistis of spae in the Eath s motion in the Galaxy How o the Eath an the planets move in spae? The Eath otates aoun its own axis at 465 m/se at the equato, with an appoximately 4-hou peio, an moves at 30 km/se aoun the Sun with a ay peio astonomial yea. The Sun, in ommon with the planets, moves at 50 km/se aoun the ente of the Galaxy with an 00 million yea peio. An so the Eath s obit taes a yline, the axis of whih is the galati tajetoy of the Sun. As a esult, the loal spae of the Eath aws a vey stethe spial, spanne ove the galati yline of the Eath s obit. Eah planet taes a simila spial in the Galaxy. We aim to buil a meti fo the spae along the Eath s tansit in the Galaxy. We o this in two steps. Fist, the meti along the Eath s tansit in the gavitational fiel of the Sun. Seon, using the Loentz tansfomation to hange to the efeene fame moving with espet to the fist fame along the axis oiniing with the ietion in whih the Eath moves in the Galaxy. We use a efeene fame whih otates an moves fowas in a weak gavitational fiel. We theefoe use ylinial ooinates. Then the meti along the Eath s tansit in the gavitational fiel of the Sun has the fom s = 1 GM ω 1 + GM t ω tϕ ϕ z, whee ω is the angula veloity of the Eath s otation aoun the Sun: ω = vob = 10 7 se 1. We now hange to a efeene fame that otates in a weak gavitational fiel an moves unifomly with a veloity v assoiate with the motion of the Sun in the Galaxy along the z-axis. We apply the Loentz tansfomations z = 1 z + vt t +, t vz =, 1 1 v v whee z an t ae oesponing ooinates in the new ef- See any textbook on elativity. Note that the gavitational fiel is inlue in the omponents of the funamental meti tenso g αβ as GM. The mass of the Sun is M = g, the mass of the Eath is M = = g; the istane between the Sun an the Eath is m, the Eath s aius is m. We obtain GM = 10 8, GM = So, in this onsieation we mean the aily otation of the Eath an its gavitational fiel neglete quasi-newtonian appoximation. eene fame. We iffeentiate z an t, then substitute the esulting z, t an t into. Fo v = 50 km/se we have v / = , hene 1+ 1 v / v /. We ig- noe tems in powes highe than 1. As a esult we obtain the meti along the Eath s tajetoy in the Galaxy opping the tile fom the fomulae s = 1 GM t ω tϕ 1+ GM ω ϕ ωv ϕz z. This meti iffes fom 1, beause of a spatial tem ω v/ epening upon the linea veloity v. In oe to obtain eally obsevable effets expete in the meti 3, we use the mathematial metho of physial obseve quantities [4, 5], whih onsies a fixe spatial setion onnete to a eal efeene fame of an obseve. Fo suh an obseve the funamental metial tenso has the thee-imensional invaiant fom 3 h ik = g ik + 1 v i v k, i, k = 1,, 3, 4 epenent upon the linea veloity of the spae otation v i = = g 0i g00. In 3 the meti tenso has the omponents h 11 = 1 + GM, h = h 3 = ω v, h 33 = 1, while its ontavaiant omponents ae 1 + ω h 11 = 1 GM, h = 1 ω, h 3 = ωv, h33 = 1., Aoing to the theoy [4, 5], any efeene spae has pinipal obsevable honometially invaiant haateistis: the h.inv.-veto of gavitational inetial foe F i = 1 1 w w x i v i t 5 6 ; 7 the h.inv.-tenso of the angula veloity of the spae otation A ik = 1 vk x i v i x k + 1 F iv k F k v i ; 8 an the h.inv.-tenso of the ates of the spae efomation D ik = 1 h ik t, 9 The spatial inies 1,, 3 ae enote by Roman lettes, while the spae-time inies 0, 1,, 3 ae enote by Geek lettes. 5 L. Boissova. Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah

3 Otobe, 006 PROGRESS IN PHYSICS Volume 4 whee w = 1 g 00, while = 1 t g00 is the so-alle t honometially invaiant time eivative. Calulating these fo the meti spae 3, we obtain A 1 = ω F 1 = ω GM 1 GM + ω 1 + ω ; 10, A 31 = ω v. 11 All omponents of D ik equal zeo. Hene the efeene boy gavitates, otates, an moves fowa at a onstant veloity. Appopiate haateistis of the metis 1 an 3 oinie, asie fo A 31 : A 31 = 0 in 3. The obsevable time inteval ontains v i [4, 5]: = 1 w t 1 v i x i. 1 Within an aea wheein A ik = 0 holonomi spae the time ooinate x 0 = t an be tansfome so that all v i = 0. In othe wos, the time inteval between two events at iffeent points oes not epen on the path of integation: time is integable, so a global synhonization of loks is possible. In suh a spae the spatial setion x 0 = onst is eveywhee othogonal to time lines x i = onst. If A ik 0 non-holonomi spae, it is impossible fo all v i to be zeo: the spatial setion is not othogonal to the time lines, an the time inteval between two events at iffeent points epens on the path of integation time is non-integable. Zelmanov also intoue the h.inv.-pseuoveto of the angula veloity of the spae otation [4] Ω i = 1 ε ijk A jk, 13 whee ε ijk = e ijk h is the thee-imensional isiminant tenso, e ijk is the ompletely antisymmeti thee-imensional tenso, h = et h ik. Hene, Ω 1 =A 3, Ω =A 31, Ω 3 =A 1. In ou statement we have two boies, both otating an gavitating. The fist boy is at est with espet to the obseve, whilst the seon boy moves with a linea veloity. As seen fom 11, fo the est boy only Ω 3 0. Fo the moving boy we also obtain Ω 0 an Ω 3 0. In othe wos, any linea motion of an obseve with espet to his efeene boy povies an aitional egee of feeom to otations of his efeene spae. Besies the afoementione obsevable physial haateistis F i, A ik, an D ik, evey efeene spae also has an obsevable geometi haateisti [4]: the h.inv.-tenso of the thee-imensional spae uvatue C lkij = H lkij 1 Aki D jl + A ij D kl + + A jk D il + A kl D ij + A li D kj, 14 This is beause any linea motion leas to an aitional tem in the obsevable meti tenso h ik : see fomulae 5 an 6. whih possesses all the popeties of the Riemann-Chistoffel uvatue tenso R αβγδ in the spatial setion. Hee H lkij = = h jm Hlki m, whee H m lki is the h.inv-tenso simila to Shouten s tenso [8]: H j lki = Δ j il x k Δ j kl x i + Δ m il Δ j km Δm klδ j im. 15 If all A ik o D ik ae zeo in a spae, C iklj = H iklj. Zelmanov also intoue H ik = h mn H imkn, H = h ik H ik, C ik = h mn C imkn an C = h ik C ik. The h.inv.-chistoffel symbols of the fist an seon kins, by Zelmanov, ae Δ k ij = h km Δ ij, m = 1 t h im h jm x j + x i h ij x m, 16 whee = 1 is the so-alle h.inv.-spatial eivative. x i x i Calulating the omponents of Δ k ij fo the meti 3, we obtain Δ 1 = 1 GM + ω, Δ 1 11 = GM, Δ 1 = ω Δ1 3 = ωv,, Δ 13 = ωv, while non-zeo omponents of C iklj, C ik an C ae C 11 = GM + 3ω, C 11 = GM 3 + 3ω, C = GM + 3ω, C = GM 3 + 3ω We have thus alulate by the theoy of obsevable quantities, that: The obsevable spae along the Eath s tajetoy in the Galaxy is non-holonomi, inhomogeneous, an uve ue to the spae otation an/o Newtonian attation. This shoul be tue fo any othe planet o its satellite as well, o any othe boy onsiee within the famewok this analysis. 3 Deviation of light in the fiel of the Galati otation We stuy how a light ay behaves in a efeene boy spae esibe by the meti 3. Light moves along isotopi geoesi lines. Suh geoesis ae tajetoies of the paallel tansfe of the fou-imensional isotopi wave veto K α = Ω x α σ, g αβ K α K β = 0, 19 L. Boissova. Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah 53

4 Volume 4 PROGRESS IN PHYSICS Otobe, 006 whee Ω is the pope fequeny of the aiation, σ = = h ik x i x k is the thee-imensional obsevable inteval. The equations of geoesi lines in h.inv.-fom ae [4, 5] Ω Ω F i i + Ω D ik i k = 0, Ω i + ωd i k + A i k k ΩF i + ΩΔ i kn k n = 0, 0 whee i = xi is the obsevable h.inv.-veloity of light its squae is invaiant i i = h ik i k =. Substituting the h.inv.-haateistis of the efeene spae 3 into equations 0, we obtain 1 Ω Ω 1 Ω Ω ω GM Ωω 1 GM ω GM Ω ϕ Ω 1 GM + Ωω 1 + ω = 0, 1 + 3ω ϕ 1+ ω Ωωv ϕ z + ω ϕ = 0, 1 + GM + + ω ϕ + Ωωv z = 0, 3 + ω Ω z Ωω v = 0. 4 Integating 1 we obtain the obsevable pope fequeny of the light beam at the moment of obsevation Ω =, 5 Ω 0 Ω 0 1+ GM 1 GM ω + ω whee Ω 0 is its initial pope fequeny in the absene of extenal affets. We integate 4 with the use of 5. Rewite 4 as Ω z integation of whih gives = Ωω v, 6 Ω z = Ωω v + Q, Q = onst, 7 whee ż 0 = z z is the initial value of 0, while the integation onstant is Q = Ω 0 ż 0 ω v 0. So the spae-time inteval s = g αβ x α x β in h.inv.-fom is s = σ = 0. Theefoe, beause s = 0 along isotopi tajetoies by efinition, thee σ =. Substituting 7 into 3 an 4 an using Ω fom 5, we obtain the system of equations with espet to an ϕ, Ω ϕ + ω Ω Ω + Ωω 1 + ω Ωω ω GM Ω 1 + GM + ω ϕ + Ω 0ωvż 0 1 GM + + 3ω 1+ ω Ω 0ωvż 0 1 GM + ω ϕ = 0, ϕ ϕ = 0. 8 We ae looking fo an appoximate solution to this system. The last tem has the imensionless fato vż0. Fo a light beam, ż 0 the initial value of the light veloity along the z-axis is. Hene vż0 = v. At 50 km/se, attibute to v the Eath moving in the Galaxy, = The tems GM an ω, elate to the obital motion of the Eath, ae in oe of We theefoe op these tems fom onsieation, so equations 8 beome Ω Ωω ϕ Ω ω GM Ω Ω ϕ + Ωω +ω We ewite 30 as ϕ Ω 0ωvż 0 ϕ, 9 ϕ +Ω 0ωvż 0 = ϕ + ϕ + ω ṙ = 0, 31 whee ω = ω 1+ vż 0, ϕ = ϕ, ϕ = ϕ. This is an equation with sepaable vaiables, so its fist integal is ϕ = B ω, B = onst = ϕ 0 + ω 0, 3 whee ϕ 0 an 0 ae the initial values of ϕ an. We ewite 9 as ω ϕ + GM ω ϕ = 0, 33 whee ṙ =, = GM. In ou onsieation, ω is zeo, so the motion of the Eath aoun the Sun satisfies the weightlessness onition [9, 10] a balane between the Eah planet, in its obital motion, shoul satisfy the weightlessness onition w = v i u i, whee w is the potential of the fiel attating the planet to a boy aoun whih this planet is obiting, v i is the linea veloity of the boy s spae otation in this obit, an u i = x i /t is the ooinate veloity of the planet in its obit. The obital veloity is the same as the spae otation veloity. Hene the weightiness onition an be witten as GM/ = v = v i v i [9, 10]. 54 L. Boissova. Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah

5 Otobe, 006 PROGRESS IN PHYSICS Volume 4 ating foes of gavity GM an inetia ω. Taking this into aount, an substituting 3 into 33, we obtain + ω B 3 = We eplae the vaiables as ṙ = p. So = p p equation 36 takes the fom an the p p = B 3 ω, 35 whih an be easily integate: p = = B ω + K, K = onst, 36 whee the integation onstant is K = ṙ0 + ϕ 0 + ω ω 0, so we obtain = ± K ω B. 37 Looking fo τ as a funtion of, we integate 37 taking the positive time flow into aount positive values of τ. We obtain τ = ω 4 + K B Intouing a new vaiable u = we ewite 38 as τ = 1 u u 0 u ω u + Ku B, 39 whih integates to τ = 1 [ ω + K asin ω K 4 ω B ω ] 40 0 asin K 4 ω B whee K 4 ω B Q = = ṙ0 + 0 ϕ 0[ṙ ω ω + ϕ ϕ 00 ] 41, so we obtain an = Q ω sin ωτ + 0, = Q ω sin ωτ + 0, 4 whee 0 is the initial isplaement in the -ietion. Substituting 4 into 3 we obtain ϕ, τ B ϕ = 0 ω ωb = ωτ + Q 4ω Q + Q 4 ω tan ωτ + ω 0 ln Q Q 4 ω tan ωτ + ω 0 + ϕ 0, 43 whee ϕ 0 is the initial isplaement in the ϕ-ietion. Substituting Ω fom 5 into 7, an eliminating the tems ontaining GM an ω, we obtain the obsevable veloity of the light beam in the z-ietion ż = ω v + ż 0 ω v 0, 44 the integation of whih gives its obsevable isplaement z = ż 0 τ + ω Qv 4 ω 3 1 os ωτ + z 0, 45 whih, taking into aount that ω = ω 1+ vż 0, is z = ż 0 τ + vq 1 os ωτ 1 vż 0 4 ω + z We have obtaine solutions fo ṙ, ϕ, ż an, ϕ, z. We see the galati veloity of the Eath in only ż an z. Let s fin oetions to the isplaement of the light ż an its isplaement z ause by the motion of the Eath in the otating an gavitating spae of the Galaxy. As follows fom fomula 41, Q oesn t inlue the initial veloity an isplaement of the light beam in the z-ietion. Besies, Q = 0 if ṙ 0 = 0 an 0 = 0. In a eal situation ṙ 0 0, beause the light beam is emitte fom the Eath so 0 is the istane between the Sun an the Eath. Hene, in ou onsieation, Q 0 always. If ϕ 0 = 0, the light beam is iete stitly towas the Sun. We alulate the oetion to the light veloity in the -ietion Δż 0 we mean ϕ 0 = 0, ż 0 = 0. Eliminating the tem 1 vż 0 we obtain Δż = Qv sin ωτ, Q = ṙ 0 ṙ0 + 4 ω We see that the oetion Δż 0 is a peioial funtion, the fequeny of whih is twie the angula veloity of the Eath s otation aoun the Sun; ω = se 1. Beause the initial value of the light veloity is ṙ 0 =, an also 4 ω 0, we obtain the amplitue of the hamoni osillation Qv = ṙ 0 then the oetion to the light veloity in the -ietion Δż 0 is, ω 0 v, 48 Δż = v sin ωτ = sin ωτ. 49 Fom this esulting key fomula we have obtaine we onlue that: The omponent of the obsevable veto of the light veloity iete towas the Sun the -ietion gains an aition oetion in the z-ietion, beause the Eath moves in ommon with the Sun in the L. Boissova. Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah 55

6 Volume 4 PROGRESS IN PHYSICS Otobe, 006 Galaxy. The obtaine oetion manifests as a hamoni osillation ae to the wol-invaiant of the light veloity. The expete amplitue of the osillation is , i. e. 10 km/se; the peio T = 1 ω is half the astonomial yea. So the theoy peits an anisotopy of the obsevable veloity of light ue to the inhomogeneity an anisotopy of spae, ause by its otation an the pesene of gavitation. In ou statement the anisotopy of the veloity of light manifests in the z-ietion. We theefoe, in this statement, all the z-ietion the pefee ietion. We an veify the anisotopy of the veloity of light by expeiment. By the theoy of obsevable quantities [4, 5], the invaiant is the length = h ik i k = hik x i x k = σ 50 of the h.inv.-veto i = xi of the obsevable light veloity. Let a light beam be iete towas the Sun, i. e. in the -ietion. Aoing to ou theoy, the Eath s motion in the Galaxy eviates the beam away fom the -axis so that we shoul obseve an aitional z-omponent to the light veloity invaiant. Let s set up two pais of etetos synhonise loks along the -ietion an z-ietion in oe to measue time intevals uing whih the light beams tavel in these ietions. Beause the istanes Δσ between the loks ae fixe, an is onstant, the measue time in the z-ietion is expete to have a ilation with espet to that measue in the -ietion: by fomula 49 the light veloity measue in both ietions is expete to be iffe by 10 km/se at the maximum of the effet. The most suitable equipment fo suh an expeiment is that use by R. T. Cahill Flines Univesity, Austalia in his uent expeiments on the measuement of the veloity of light in an RF oaxial-able equippe with a pai of high peision synhonize Rb atomi loks [11]. This effet pobably ha a goo hane of being etete in simila expeiments by D. G. To an P. Colen Utah State Univesity, USA in the 1980 s [1] an, espeially, by Rolan De Witte Belgaom Laboatoy of Stanas, Belgium in the 1990 s [13]. Howeve even De Witte s equipment ha a measuement peision a thousan times lowe than that uently use by Cahill. Beause the Eath otates aoun its own axis we shoul obseve a weak aily vaiation of this effet. In oe to egiste the omplete vaiation of this value, we shoul measue it at least uing half the astonomial yea one peio of its vaiation. 4 Inhomogeneity an anisotopy of spae along the Eath s tansit in the Galaxy We just applie the meti 3 to the Eath s motion in the Galaxy. Following this appoah, we an also employ this meti to othe pefee ietions in the Univese, onnete to the motion of anothe spae boy, fo instane the motion of ou Galaxy in the Loal Goup of galaxies. Astonomial obsevations show that the Sun moves in ommon with ou Galaxy in the Loal Goup of galaxies at the veloity 700 km/se. The meti 3 an take into aount this aspet of the Eath s motion as well. In suh a ase we shoul expet two weak maximums in the time ilation measue in the above esibe expeimental system uing the 4-hou peio, when the z-ietion oinies with the ietion of the apex of the Sun. The amplitue of the vaiation of the obsevable light veloity shoul be.8 times the vaiation ause by the Eath s motion in the Galaxy. Sweish astonomes in the 1950 s isovee that the Loal Goup of galaxies is a pat of an ompat lou alle the Supeluste of galaxies, onsisting of galaxies, small goups of galaxies, an two lous of galaxies. The Supeluste has a iamete of 98 million light yeas, while ou Galaxy is loate at 6 million light yeas fom the ente. The Supeluste otates with a peio of 100 billion yeas in the ental aea an 00 billion yeas at the peiphey. As suppose by the Swees, ou Galaxy, loate at /3 of the Supeluste s aius, fom its ente, otates aoun the ente at a veloity of 700 km/se. See Chapte VII, 6 in [14] fo the etails. In any ase, in any lage sale ou meti 3 gives the same esult, beause any of the spaes is non-holonomi otates aoun its own ente of gavity. All the spaes ae inlue, one into the othe, an ause bizae spials in thei motions. The geate the numbe of the spae stutues taken onto aount by ou meti 3, the moe ompliate is the spial tae out by the Eath obseve in the spae the spial is plaite into othe spae spials the fatal stutue of the Univese [15]. This analysis of ou theoetial esults, obtaine by Geneal Relativity, an the well-known ata of obsevational astonomy leas us to the obvious onlusion: The main fatos foming the obsevable stutue of the spae of the Univese ae gavitational fiels of bulky boies an thei otations, not the spae efomations as peviously thought. Many sientists onsie homogeneous isotopi moels as moels of the eal Univese. A homogeneous isotopi spae-time is esibe by Fiemann s meti s = t R x + y + z [ 1 + k 4 x + y + z ], 51 whee R = Rt; k = 0, ±1. Fo suh a spae, the main obsevable haateistis ae F i =0, A ik =0, D ik 0. In othe wos, suh a spae an unego efomation expansion, The ietion of this motion is pointe out in the sky as the apex of the Sun. Inteestingly, the Sun has a slow ift of 0 km/se in the same ietion as the apex, but within the Galaxy with espet to its plane. 56 L. Boissova. Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah

7 Otobe, 006 PROGRESS IN PHYSICS Volume 4 ompession, o osillation, but it is fee of otation an ontains no gavitating boies fiels. So the meti 51 is the neessay an suffiient onition fo homogeneity an isotopy. This is a moel onstute by an imaginay obseve who is loate so fa away fom matte in the eal Univese that he sees no suh etails as stas an galaxies. In ontast to them, we onsie a osmologial moel onstute by an Eath obseve, who is aie away by all motions of ou planet. Zelmanov, the pionee of inhomogeneous anisotopi elativisti moels, pointe out the mathematial onitions of a spae s homogeneity an isotopy, expesse with the tems of physially obsevable haateistis of the spae [4]. The onitions of isotopy ae F i = 0, A ik = 0, Π ik = 0, Σ ik = 0, 5 whee Π ik = D ik 1 3 Dh ik an Σ ik = C ik 1 3 Ch ik ae the fatos of anisotopy of the spae efomation an the theeimensional obsevable uvatue. In a spae of the meti 3 we have D ik = 0, hene thee Π ik = 0. Howeve F i an A ik ae not zeo in suh a spae see fomulae 10 an 11. Besies these thee ae the non-zeo quantities, Σ 11 = 1 GM ω ; Σ = 1 GM 3 + ω ; Σ 33 = GM 3 3 ω. 53 We see that a spae of the meti 3 is anisotopi ue to its otation an gavitation. The onitions of homogeneity, by Zelmanov [4], ae j F i = 0, j A ik = 0, j D ik = 0, j C ik = Calulating the onitions fo the meti 3, we obtain 1 C 11 = 3GM 4, 1 C = 3GM, 1 F 1 = ω 1 + 3ω + GM 3 1 A 1 = ω + ω + 3GM GM., 55 This means, a spae of the meti 3 is inhomogeneous ue to its otation an gavitation. The esults we have obtaine manifest thus: The eal spae of ou Univese, whee spae boies move, is inhomogeneous an anisotopi. Moeove, the spae inhomogeneity an anisotopy etemine the bizae stutue of the Univese whih we obseve: the pefee ietions along whih the spae boies move, an the hieahial istibution of the motions. 5 Conlusions By means of Geneal Relativity we have shown that the spae meti 3 along the Eath s tajetoy in the Galaxy, whee the Eath follows a ompliate spial tae out by its obital motion aoun the Sun an its onomitant motion with the whole sola system aoun the ente of the Galaxy. We have shown that this meti spae is: a globally nonholonomi ue to its otation an the pesene of gavitation, as manifeste by the non-holonomi h.inv.-tenso A ik 11 alulate in the meti spae ; b inhomogeneous, beause the h.inv.-chistoffel symbols Δ k ij iniating inhomogeneity of spae, being alulate in the meti spae as shown by 17, ontain gavitation an spae otation; uve ue to gavitation an spae otation, epesente in the fomulae fo the thee-imensional h.inv.-uvatue C iklj alulate in the meti spae as shown by 18. Consequently, in eal spae thee exist pefee spatial ietions along whih spae boies unego thei obital motions. We have eue that the obsevable veloity of light shoul be anisotopi in spae ue to the anisotopy an inhomogeneity of spae, ause by the afoementione fatos of gavitation an spae otation, espite the wol-invaiane of the veloity of light. It has been alulate that two pais of synhonise loks shoul eo iffeent values fo the spee of light in light beams iete towas the Sun an othogonal to this ietion, at about % of the measue veloity of light, i. e. 10 km/se. This effets shoul unego osillations with a 1-hou peio ue to the aily otation of the Eath an with a 6-month peio ue to the motion of the Eath aoun the Sun. Equipment most suitable fo eteting the effet is that use by R. T. Cahill Flines Univesity, Austalia in his uent expeiment on the measuement of the veloity of light in a one-way RF oaxial-able equippe with a pai of high peision synhonize Rb atomi loks. The peite anisotopy of the obsevable veloity of light has been eue as a iet onsequene of the geometial stutue of fou-imensional spae-time. Theefoe, if the peite anisotopy is etete by expeiment, it will be one moe fat in suppot of Einstein s Geneal Theoy of Relativity. The anisotopy of the obsevable veloity of light as a onsequene of Geneal Relativity was fist pointe out by D. Rabounski in the eitoial pefae to [13], his papes [6, 7], an many pivate ommuniations with the autho, whih ommene in Autumn, 005. He has state that the anisotopy esults fom the non-holonomity otation of the Gavitation is epesente by the mass of the Sun M, while the spae otation is epesente by two fatos: the angula veloity ω of the sola spae otation in the Eath s obit equal to the angula veloity of the Eath s otation aoun the Sun, an also the linea veloity v of the otation of the Sun in ommon with the whole sola system aoun the ente in the Galaxy. L. Boissova. Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah 57

8 Volume 4 PROGRESS IN PHYSICS Otobe, 006 loal spae of a eal obseve an/o the non-holonomity of the bakgoun spae of the whole Univese. Moeove, the non-holonomi fiel of the spae bakgoun an poue enegy, if petube by a loal otation o osillation as this was theoetially foun fo stas [6, 7]. Detaile alulations povie in the pesent pape show not only that the non-holonomity otation of spae is the soue of the anisotopy of the obsevable veloity of light, but also gavitational fiels. This pape will be followe by a seies of papes wheein we stuy the inteation between the fiels of the spae nonholonomity, an also onsie these fiels as new soues of enegy. This means that we onsie open systems. Natually, given the ase of an inhomogeneous anisotopi univese, it is impossible to stuy it as a lose system sine suh systems on t physially exist owing to the pesene of spae non-holonomity an gavitation. In a subsequent pape we will onsie the non-holonomi fiels in a spae of the meti 3 with the use of Einstein s equations. It is well known that the equations an be applie to a wie vaiety istibutions of matte, even insie atomi nulei. We an theefoe, with the use of the Einstein equations, stuy the non-holonomi fiels an thei inteations in any sale pat of the Univese fom atomi nulei to lustes of galaxies the poblem statement emains the same in all the onsieations. 9. Rabounski D., Boissova L. Patiles hee an beyon the Mio. Eitoial URSS, Mosow, 001; axiv: g-q/ Boissova L., Rabounski D. Fiels, vauum, an the mio Univese. Eitoial URSS, Mosow, 001; CERN, EXT Cahill R. T. A new light-spee anisotopy expeiment: absolute motion an gavitational waves etete. Pogess in Physis, 006, v. 4, To D. G., Kolen P. An expeiment to measue elative vaiations in the one-way veloity of light. Peision Measuements an Funamental Constants, Natl. Bu. Stan. U. S., Spe. Publ., 1984, v. 617, Cahill R. T. The Rolan De Witte 1991 expeiment to the memoy of Rolan De Witte. Pogess in Physis, 006, v. 3, Voontsov-Velyaminov B. A. Extagalati astonomy. Hawoo Aaemi Publishes, N.Y., Manelbot B. The fatal geomety of natue. W. H. Feeman, San Faniso, Rabounski D. A theoy of gavity like eletoynamis. Pogess in Physis, 005, v., Refeenes 1. Cothes S. J. On the geneal solution to Einstein s vauum fiel fo the point-mass when λ 0 an its impliations fo elativisti osmology. Pogess in Physis, 005, v. 3, Cothes S. J. A bief histoy of blak holes. Pogess in Physis, 006, v., Boissova L. Gavitational waves an gavitational inetial waves in the Geneal Theoy of Relativity: a theoy an expeiments. Pogess in Physis, 005, v., Zelmanov A. L. Chonometi invaiants. Dissetation thesis, Ameian Reseah Pess, Rehoboth NM, Zelmanov A. L. Chonometi invaiants an o-moving ooinates in the geneal elativity theoy. Doklay Aa. Nauk USSR, 1956, v. 1076, Rabounski D. Thomson ispesion of light in stas as a geneato of stella enegy. Pogess in Physis, 006, v. 4, Rabounski D. A soue of enegy fo any kin of sta. Pogess in Physis, 006, v. 4, Shouten J. A., Stuik D. J. Einfühung in ie neuen Methoen e Diffeentialgeometie. Zentalblatt fü Mathematik, 1935, B. 11 un B. 19. Aoing to the Copenian stanpoint, the sola system shoul be a lose system, beause that pespetive oesn t take into aount the fat that the sola system moves aoun the ente of the Galaxy, whih aies it into othe, moe ompliate motions. 58 L. Boissova. Pefee Spatial Dietions in the Univese: a Geneal Relativity Appoah