A Theory of the Podkletnov Effect based on General Relativity: Anti-Gravity Force due to the Perturbed Non-Holonomic Background of Space

Transcription

1 July, 007 PROGRESS IN PHYSICS Volume 3 SPECIAL REPORT A Theoy of the Podkletnov Effet based on Geneal Relativity: Anti-Gavity Foe due to the Petubed Non-Holonomi Bakgound of Spae Dmiti Rabounski and Laissa Boissova abounski@yahoo.om; lboissova@yahoo.om We onside the Podkletnov effet the weight loss of an objet loated ove a supeonduting dis in ai due to suppot by an altenating magneti field. We onside this poblem using the mathematial methods of Geneal Relativity. We show via Einstein s equations and the geodesi equations in a spae petubed by a dis undegoing osillatoy bounes othogonal to its own plane, that thee is no ôle of supeondutivity; the Podkletnov effet is due to the fat that the field of the bakgound spae non-holonomity the basi non-othogonality of time lines to the spatial setion, being petubed by suh an osillating dis podues enegy and momentum flow in ode to ompensate the petubation in itself. Suh a momentum flow is dieted above the dis in Podkletnov s expeiment, so it woks like negative gavity anti-gavity. We popose a simple mehanial system whih, simulating the Podkletnov effet, is an expeimental test of the whole theoy. The theoy allows fo othe anti-gavity devies, whih simulate the Podkletnov effet without use of vey ostly supeonduto tehnology. Suh devies ould be applied to be used as a heap soue of new enegy, and ould have impliations to ai and spae tavel. Contents 1 Intoduing Podkletnov s expeiment The non-holonomi bakgound spae Pelinimay data fom topology The spae meti whih inludes a non-holonomi bakgound 60.3 Study of the bakgound meti. The main haateistis of the bakgound spae Petubation of the non-holonomi bakgound The bakgound meti petubed by a gavity field The bakgound meti petubed by a loal osillation and gavity field The spae of a suspended, vetially osillating dis The main haateistis of the spae Einstein s equations in the spae. Fist onlusion about the oigin of the Podkletnov effet Complete geometization of matte The onsevation law The geodesi equations in the spae. Final onlusion about the foes diving the Podkletnov effet A new expeiment poposed on the basis of the theoy A simple test of the theoy of the Podkletnov effet altenative to supeonduto tehnology Appliation fo new enegy and spae tavel Appendix 1 The spae non-holonomity as otation Appendix A shot tou into the honometi invaiants Intoduing Podkletnov s expeiment In 199, Eugene Podkletnov and his team at the Tampee Institute of Tehnology Finland tested the unifomity of a unique bulky supeonduto dis, otating at high speed via a magneti field [1]. The mm supeonduto dis was hoizontally oeinted in a yostat and suounded by liquid helium. A small uent was initiated in the dis by oute eletomagnets, afte whih the medium was ooled to 0 70 K. As the dis ahieved supeondutivity, and the state beame stable, anothe eletomagnet loated unde the yostat was swithed on. Due to the Meissne-Ohsenfeld effet the magneti field lifted the dis into the ai. The dis was then diven by the oute eletomagnets to 5000 pm. A small non-onduting and non-magneti sample was suspended ove the yostat whee the otating dis was ontained. The weight of the sample was measued with high peision by an eleto-optial balane system. The sample with the initial weight of g was found to lose about 0.05% of its weight when plaed ove the levitating dis without any otation. When the otation speed of the dis ineased, the weight of the sample beame unstable and gave flutuations fom.5 to +5.4% of the initial value. [... ] The levitating supeonduting dis was found to ise by up to 7 mm when its otation moment ineased. Test measuements without the supeonduting shielding dis but with all opeating solenoids onneted to the powe supply, had no effet on the weight of the sample [1]. Additional esults wee obtained by Podkletnov in 1997, with a lage dis a 75/80 10-mm tooid un unde D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 57

2 Volume 3 PROGRESS IN PHYSICS July, 007 Fig. 1: Cyogeni system in Podkletnov s expeiment []. Coutesy of E. Podkletnov. Used by pemission. Fig. 3: Weight and pessue measuement in Podkletnov s expeiment []. Coutesy of E. Podkletnov. Used by pemission. Fig. : Suppoting and otating solenoids in Podkletnov s expeiment []. Coutesy of E. Podkletnov. Used by pemission. simila onditions []: The levitating dis evealed a lealy measuable shielding effet against the gavitational foe even without otation. In this situation, the weight-loss values fo vaious samples anged fom 0.05 to 0.07%. [...] Samples made fom the same mateial and of ompaable size, but with diffeent masses, lost the same fation of thei weight. [...] Samples plaed ove the otating dis initially demonstated a weight loss of %. When the otation speed was slowly edued by hanging the uent in the solenoids, the shielding effet beame onsideably highe and eahed 1.9.1% at maximum []. Two goups of eseahes suppoted by Boeing and NASA, and also a few othe eseah teams, have attempted to epliate the Podkletnov expeiment in eent yeas [3 7]. The main poblem they enounteed was the epodution of the tehnology used by Podkletnov in his laboatoy to podue suffiiently lage supeondutive eamis. The tehnology is vey ostly: aoding to Podkletnov [8] this e- quies tens of millions of dollas. Theefoe the afoementioned oganisations tested diss of muh smalle size, about 1 diamete; so they podued ontovesial esults at the bounday of peision measuement. As was pointed out by Podkletnov in his eent inteview Apil, 006, the NASA team, afte yeas of unsuessful attempts, made a 1 dis of supeondutive eami. Howeve, due to the ude intenal stutue this is one of the main poblems in making suh diss, they wee unable to use the dis to epliate his expeiment [8]. Podkletnov also eently epoted on a gavity field geneato [8, 9] onstuted in his laboatoy in eent yeas, on the basis of the ealie obseved phenomenon. In a nutshell, the afoementioned phenomenon is as follows. We will efe to this as the Podkletnov effet: When a dis of supeondutive eami, being in the state of supeondutivity, is suspended in ai by an altenating magneti field due to an eletomagnet loated unde the dis, the dis is the soue of a adiation. This adiation, taveling like a plane wave above the dis, ats on othe bodies like a negative gavity. The adiation beomes stonge with lage diss, so it depends on the dis s mass and adius. When the dis otates unifomly, the adiation emains the same. Duing aeleation/baking of the dis s otation, the adiation essentially ineases. Podkletnov laimed many times that he disoveed the effet by hane, not by any theoetial pedition. Being 58 D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity

3 July, 007 PROGRESS IN PHYSICS Volume 3 an expeimentalist who pioneeed this field of eseah, he ontinued his expeiments blindfolded: in the absene of a theoetial eason, the ause of the obseved weight loss was unlea. This is why neithe Podkletnov no his followes at Boeing and NASA didn t develop a new expeiment by whih the weight loss effet substantially ineased. Fo instane, Podkletnov still believes that the key to his expeiment is that speial state whih is speifi to the eleton gas inside supeondutive mateials in the state of supeondutivity [8]. He and all the othes theefoe foused attention on low tempeatue supeondutive eamis, podution of whih, taking the lage size of the diss into aount, is a highly ompliated and vey ostly poess, beyond most laboatoies. In fat, duing the last 15 yeas only Podkletnov s laboatoy has had the ability to podue suh the diss with suffiient quality. We popose a puely theoetial appoah to this poblem. We onside Podkletnov s expeiment using the mathematial methods of Geneal Relativity, in the Einstenian sense: we epesent all essential omponents of the expeiment as a esult of the geometial popeties of the laboatoy spae suh as the spae non-unifomity, otation, defomation, and uvatue. We build a omplete theoy of the Podkletnov effet on the basis of Geneal Relativity. By this we will see that thee is no ôle fo supeondutivity; Podkletnov s effet has a puely mehanial oigin due in that the vetial osillation of the dis, podued by the suppoting altenating magneti field, and the angula aeleation/baking of the dis s otation, petub a homogeneous field of the basi non-holonomity of the spae the basi non-othogonality of time lines to the spatial setion, known fom the theoy of non-holonomi manifolds. As a esult the non-holonomity field, initially homogeneous, is loally stessed, whih is expessed by a hange of the left side of Einstein s equations geomety and, though the onsevation law, a oesponding hange of the ight side the enegy-momentum tenso fo distibuted matte the altenating magneti field, in this ase. In othe wods, the petubed field of the spae non-holonomity podues enegy-momentum in ode to ompensate fo the loal petubation in itself. As we will see, the spatial momentum is dieted above the dis in Podkletnov s expeiment, so it woks like negative gavity. Owing to ou theoy we know definitely the key paametes uling the weight loss effet. Theefoe, following ou alulation, it is easy to popose an expeiment wheein the weight loss substantially ineases. Fo example, we desibe a new expeiment whee the Podkletnov effet manifests via simple eleto-mehanial equipment, without ostly supeonduto tehnology. This new expeiment an be epliated in any physis laboatoy. We theefoe laim thta with ou mathematial theoy of the Podkletnov effet, within the famewok of Geneal Relativity, we an alulate the fatos uling the weight loss. This gives us an oppotunity to onstut atual woking devies whih ould evolutionize ai and spae tavel. Suh new tehnology, whih uses high fequeny eletomagneti geneatos and mehanial equipment instead of ostly supeondutos, an be the subjet of futhe eseah on a ommeial basis due to the fat that applied eseah is outside aademia. Besides, additional enegy-momentum podued by the spae non-holonomity field in ode to ompensate fo a loal petubation in itself, means that the Podkletnov effet an be used to podue new enegy. By ou advaned study not inluded in this pape, of ou mathematial theoy, that heein gives the key fatos whih ule the new enegy, lends itself to the onstution of devies whih geneate the new enegy, poweed by stong eletomagneti fields, not nulea eations and atomi fuel. Theefoe this tehnology, fee of adioative waste, an be a soue of lean enegy. The non-holonomi bakgound spae.1 Peliminay data fom topology In this Setion we onstut a spae meti whih inludes a basi pimodial non-holonomity, i.e. a basi field of the non-othogonality of the time lines to the thee-dimensional spatial setion. Hee is some infomation fom topology Eah axis of a Eulidean spae an be epesented as the element of a ile with infinite adius [10]. An n-dimensional tous is the topologial podut of n iles. The volume of an n- dimensional tous is ompletely equivalent to the sufae of an n+1-dimensional sphee. Any ompat meti spae of n dimensions an be mapped homeomophily into a subset of a Eulidean spae of n+1 dimensions. Sequenes of stohasti tansitions between onfiguations of diffeent dimensions an be onsideed as stohasti veto quantities fields. The extemum of a distibution funtion fo fequenies of the stohasti tansitions dependent on n gives the most pobable numbe of the dimensions, and, taking the mapping n n+1 into aount, the most pobable onfiguation of the spae. This funtion was fist studied in the 1960 s by di Batini [11, 1, 13]. He found that the funtion has extema at n+1 =±7 that is equivalent to a 3-dimensional votial tous oaxial with anothe, the same votial tous, mioed with the fist one. Eah of the toii is equivalent to a 3+1-dimensional sphee. Its onfiguation an be easy alulated, beause suh fomations wee studied by Lewis and Lamoe. A votial tous has no beaks if the uent lines oinide with the tajetoy of the votex oe. Poeeding fom the ontinuity ondition, di Batini found the most pobable onfiguation of the votial tous is it haateized by the atio E = D = = 1 4 e = between the tous diamete D and the adius of tous iulation. D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 59

4 Volume 3 PROGRESS IN PHYSICS July, 007 We apply di Batini s esult fom topology to Geneal Relativity. The time axis is epesented as the element of the ile of adius R = 1 D, while the spatial axes ae the elements of thee small iles of adii the topologial podut of whih is the 3-dimensional votial tous. In a meti epesentation by a Minkowski diagam, the tous is a 3-dimensional spatial setion of the given 3+1-spae while the time lines have some inlination to the the spatial setion. In ode fo the tous the 3-dimensional spae of ou wold to be unifom without beak, all the time lines have the same inlination to the spatial setion at eah point of the setion. Cosines of the angles between the oodinate axes, in Riemannian geomety, ae epesent by the omponents of the fundamental meti tenso g αβ [14]. If the time lines ae eveywhee othogonal to the spatial setion, all g 0i ae zeo: g 0i = 0. Suh a spae is alled holonomi. If not g 0i 0, the spae is said to be non-holonomi. As was shown in the 1940 s by Zelmanov [15, 16, 17], a field of the spae non-holonomity inlinations of the time lines to the spatial setion manifests as a otation of the spae with a 3-dimensional veloity v i = g 0i g00. The mathematial poof is given in Appendix 1. So a field with the same inlination of the time lines to the spatial setion is haateized, in the absene of gavitational fields, by v i = g 0i = onst at eah point of the spatial setion. In othe wods, this is a field of the homogeneous non-holonomity otation of the whole spae. It is had to explain suh a field by eveyday analogy, beause it has zeo angula speed, and also no ente of otation. Howeve owing to the spae-time epesentation by a Minkowski diagam, it appeas vey simply as a field of whih the time aows piee the hype-sufae of the spatial setion with the same inlination at eah point. Afte di Batini s esult, we theefoe onlude that the most pobable onfiguation of the basi spae spae-time of Geneal Relativity is epesented by a pimodially nonholonomi 3+1-dimensional pseudo-riemannian spae, whee the non-holonomi bakgound field is homogeneous, whih manifests in the spatial setion 3-dimensional spae as the pesene of two fundamental dift-fields: 1. A homogeneous field of the onstant linea veloity of the bakgound spae otation ˉv = R = E = onst = m/se 1 whih oiginates fom the fat that, given the nonholonomi spae, the time-like spead R depends on the spatial-like spead as R = 1 E = The bakgound spae otation, with ˉv =, km/se at eah point of the spae, is due to the ontinuity ondition eveywhee inside the tous;. A homogeneous dift-field of the onstant dipole-fit linea veloity ˉv = ˉv π = onst = m/se whih haateizes a spatial linea dift of the nonholonomi bakgound elative to any given obseve. The field of the spatial dift with ˉv = km/se is also pesent at eah point of the spae. In the spatial setion the bakgound spae otation with ˉv =, km/se is obseved as absolute motion. This is due to the fat that a otation due to the spae nonholonomity is elative to time, not the spatial oodinates. Despite this, as poven by Zelmanov [15, 16, 17], suh a otation elates to spatial otation, if any.. The spae meti whih inludes a non-holonomi bakgound We ae going to deive the meti of a non-holonomi spae, whih has the afoementioned most pobable onfiguation fo the 3+1-spae of Geneal Relativity. To do this we onside an element of volume of the spae the elementay volume. We onside the pseudo-riemannian 3+1-spae of Geneal Relativity. Let it be non-holonomi so that the nonholonomity field is homogeneous, i.e. manifests as a homogeneous spae-time otation with a linea veloity v, whih has the same numeial value along all thee spatial axes at eah point of the spae. The elementay 4-dimensional inteval in suh a spae is ds = dt + v dt dx + dy + dz dx dy dz, whee the seond tem is not edued, fo laity. We denote the numeial oeffiient, whih haateize the spae otation see the seond tem on the ight side, as α = v/. We mean, onside the most pobable onfiguation of the 3+1-spae, v = ˉv =, km/se and also α = ˉv/ = 1/ The atio α = ˉv/ speifi to the spae it haateizes the bakgound non-holonomity of the spae, oinides with the analytial value of Sommefeld s fine-stutue onstant [11, 1, 13], onneted to eletomagneti inteations. Given the most pobable onfiguation of the spae, eah 3-dimensional volume element otates with the linea veloity ˉv =, km/se and moves with the veloity ˉv = = ˉv π = km/se elative towad any obseve loated in the spae. The meti 3 ontains the spae otation only. To modify the meti fo the most pobable onfigua- Tests based on the quantum Hall effet and the anomalous magneti moment of the eleton, give diffeent expeimental values fo Sommefeld s onstant, lose to the analytial value. Fo instane, the latest tests 006 gave α 1/ [18] D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity

5 July, 007 PROGRESS IN PHYSICS Volume 3 ds = dt + d + v os ϕ + sin ϕ dtd + v os ϕ sin ϕ dtdϕ + v dtdz vv os ϕ + sin ϕ d dz dϕ vv os ϕ sin ϕ + dϕdz dz 7 tion, we should apply Loentz tansfomation along the dietion of the spae motion. We hoose the z-axis fo the dietion of spae motion. Fo laity of futhe alulation, we use the ylindial oodinates, ϕ, z x = os ϕ, y = sin ϕ, z = z, 4 so the meti 3 in the new oodinates takes the fom ds = dt + v os ϕ + sin ϕ dtd + + v os ϕ sin ϕ dtdϕ + v dtdz d dϕ dz. Substituting the quantities t and z of Loentz tansfomations t = t + vz z + vt, z =, 6 1 v 1 v fo t and z in the meti 5, we obtain the meti fo a volume element whih otates with the onstant veloity ˉv = α and appoahes with the onstant veloity v = ˉv with espet to any obseve loated in the spae. This is fomula 7 shown on the top of this page. In that fomula 1 1 = = onst 1, 8 1 v 1 ˉv due to that fat that, in the famewok of this poblem, v. Besides thee is also v, so that the seond ode tems edue eah othe. We still do not edue the numeial oeffiient of the non-diagonal spae-time tems so that they ae easily eognized in the meti. Beause the non-holonomi meti 7 satisfies the most pobable onfiguation fo suh a 3+1-spae, we egad it as the bakgound meti of the wold..3 Study of the bakgound meti. The main haateistis of the bakgound spae We now alulate the main haateistis of the spae whih ae invaiant within a fixed thee-dimensional spatial setion, onneted to an obseve. Suh quantities ae elated to the honometi invaiants, whih ae the physial obsevable quantities in Geneal Relativity [15, 16, 17] see Appendix. Afte the omponents of the fundamental meti tenso g αβ ae obtained fom the bakgound meti 7, we alulate the main obsevable haateistis of the spae see Appendix. It follows that in the spae: v = ˉv = α = onst, vv 5 = αˉv = ˉv = onst, 9 π the gavitational potential w is zeo g 00 = 1, w = 1 g 00 = 0, 10 the linea veloity of the spae otation v i = g0i g00 v 1 = ˉv os ϕ + sin ϕ v = ˉv os ϕ sin ϕ v 3 = ˉv is 11 the elativisti multiplie is unity within the numbe of signifiant digits 1 1 = = 1, 1 ˉv the gavitational inetial foe F i, the angula veloity of the spae otation A ik, the spae defomation D ik, and the spae uvatue C ik ae zeo F i = 0, A ik = 0, D ik = 0, C ik = 0, 13 while of all the h.inv.-chistoffel symbols Δ i km, only two omponents ae non-zeo, Δ 1 =, Δ 1 = The non-holonomi bakgound spae is fee of distibuted matte, so the enegy-momentum tenso is zeo theein. Hene, as seen fom the h.inv.-einstein equations see Appendix, the bakgound spae neessaily has λ = 0, 15 i.e. it is also fee of physial vauum λ-field. In othe wods, the non-holonomi bakgound spae is empty. We onlude fo the bakgound spae exposed by the non-holonomi bakgound meti 7, that The non-holonomi bakgound spae satisfying the most pobable onfiguation of the 3+1-spae of Geneal Relativity is a flat pseudo-riemannian spae with the 3-dimensional Eulidean meti and a onstant spae-time otation. The bakgound spae is empty; it pemits neithe distibuted matte o vauum λ-field. The bakgound spae is not one an Einstein spae whee R αβ = k g αβ, k = onst due to the fat that Einstein s equations have k=0 in the bakgound spae. To be an Einstein spae, the bakgound spae should be petubed. Read about Einstein spaes and thei fomal detemination in Einstein Spaes by A. Z. Petov [19]. It should be noted that of the fat that the 3-dimensional Eulidean meti means only F i = 0, A ik = 0, D ik = 0 and C ik = 0. The Chistoffel symbols an be Δ i mn 0 due to the uvilinea oodinates. D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 61

6 Volume 3 PROGRESS IN PHYSICS July, 007 ds = 1 GM dt + z d + v os ϕ + sin ϕ dtd + vv os ϕ + sin ϕ d dz dϕ + v os ϕ sin ϕ dtdϕ + v vv os ϕ sin ϕ dϕdz 1 + GM z dtdz 0 dz.4 Petubation of the non-holonomi bakgound How does a gavitational field and loal otation the gavitational field of the Eath and the otation of a dis, fo instane affet the meti? This we now desibe. The atio v/, aoding to the ontinuity ondition in the spae see, equals Sommefeld s fine-stutue onstant α = ˉv/ = 1/ only if the non-holonomi bakgound meti is unpetubed by a loal otation, so the spae non-holonomity appeas as a homogeneous field of the onstant linea veloity of the spae otation ˉv, whih is, km/se. The gavitational potential w appeas in Geneal Relativity as w = 1 g 00, i.e. onneted to g 00. So the pesene of a gavity field hanges the linea veloity of the spae otation v i = g 0i g00. Fo an Eathbound laboatoy, we have w = GM z This numeial value is so small that petubations of the non-holonomi bakgound though g 00, by the Eath s gavitational field, ae weak. Anothe ase loal otations. A loal otation with a linea veloity ṽ o a gavitational potential w petubs the homogeneous field of the spae non-holonomity, the atio v/ in that aea hanges fom the initial value α = ˉv/ = 1/ to a new, petubed value v = ˉv + ṽ = α + ṽ ˉv α. 16 This fat should be taken into aount in all fomulae whih inlude v o the deivatives. Conside a high speed gyo used in aviation navigation: a 50 g oto of 1.65 diamete, otating with an angula speed of 4,000 pm. With moden equipment this is almost the uppemost speed fo suh a mehanially otating system. In suh a ase the bakgound field of the spae non-holonomity is petubed nea the gio as ṽ 53 m/se, that is of the bakgound ˉv =, km/se. Lage effets ae expeted fo a submaine gyo, whee the oto and, hene, the linea veloity of the otation is lage. In othe wods, the non-holonomi bakgound an be substantially petubed nea suh a mehanially otating system..5 The bakgound meti petubed by a gavitational field The fomula fo the linea veloity of the spae otation v i = g 0i, 17 g00 Mehanial gyos used in aviation and submaine navigation tehnology have otations in the ange 6,000 30,000 pm. The uppe speed is limited by poblems due to fition. was deived by Zelmanov [15, 16, 17], due to the spae nonholonomity, and oiginating in it. It is evident that if the same numeial value v i = onst emains unhanged eveywhee in the spatial setion i.e. i v i = 0 } v i = onst 18 i v i = 0 thee is a homogeneous field of the spae non-holonomity. By the fomula 17, given a homogeneous field of the spae non-holonomity, any loal otation of the spae expessed with g 0i and also a gavitational potential ontained in g 00 petub the homogeneous non-holonomi bakgound. We modify the bakgound meti 7 to that ase whee the homogeneous non-holonomi bakgound is petubed by a weak gavitational field, podued by a bulky point mass M, that is usual fo obsevations in a laboatoy loated on the Eath s sufae o nea obit. The gavitational potential in Geneal Relativity is w = 1 g 00. We assume gavity ating in the z-dietion, i.e. w = GM z, and we omit tems of highe than the seond ode in, following the usual appoximation in Geneal Relativity see Landau and Lifshitz [0] fo instane. We substitute g 00 = 1 w = 1 GM z 1 GM z 1 19 into the fist tem of the initial meti 5. Afte Loentz tansfomations, we obtain a fomula fo the non-holonomi bakgound meti 7 petubed by suh a field of gavity. This is fomula 0 displayed on the top of this page..6 The bakgound meti petubed by a loal osillation and gavitational field A supeonduting dis in ai unde the influene of an altenating magneti field of an eletomagnet loated beneath it, undegoes osillatoy bounes with the fequeny of the uent, in a vetial dietion the same that of the Eath s gavity the z-dietion in ou ylindial oodinates. We set up a hamoni tansfomation of the z-oodinate z = z + z 0 os Ω u, u = t + z, 1 whee z 0 is the initial deviation the amplitude of the osillation, while Ω is the fequeny. Afte alulating d z and d z, and using these instead of dz and dz in the nonholonomi bakgound meti 7, we obtain the bakgound meti 7 petubed by the loal osillation and gavitational field. This is fomula 3 shown above. See Appendix fo the h.inv.-diffeentiation symbol. 6 D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity

7 July, 007 PROGRESS IN PHYSICS Volume 3 d z = d z = 1 Ωz 0 sin Ω u Ωz0 dz sin Ω u dt 1 Ωz 0 sin Ω u dz Ωz 0 sin Ω u 1 Ωz 0 sin Ω u dt dz + Ω z 0 [ ] ds GM = 1 z + z 0 os Ω u vωz 0 sin Ω u Ω z0 sin Ω u dt + v os ϕ + sin ϕ + 1 Ωz 0v sin Ω u v os ϕ sin ϕ dt d + 1 Ωz 0v + 1 Ωz 0v sin Ω { [ ]} u v + Ωz 0 sin Ω u GM 1 + z + z 0 os Ω u dt dz d + vv os ϕ + sin ϕ + 1 Ωz 0 sin Ω u d dz dϕ vv os ϕ sin ϕ + [ ] GM 1 + z + z 0 os Ω u 1 Ωz 0 sin Ω u dz sin Ω u dt sin Ω u dt dϕ + 1 Ωz 0 sin Ω u dϕ dz 3 ds = 1 GM z Ωz 0v sin Ω u dt v os ϕ + sin ϕ v os ϕ sin ϕ + dt d + dt dϕ + + v + Ωz 0 sin Ω 5 u dt dz d dϕ dz 3 The spae of a suspended, vetially osillating dis 3.1 The main haateistis of the spae Meti 3 is vey diffiult in use. Howeve, unde the physial onditions of a eal expeiment, many tems vanish so that the meti edues to a simple fom. We show how. Conside a system like that used by Podkletnov in his expeiment: a hoizontally oiented dis suspended in ai due to an altenating high-fequent magneti field geneated by an eletomagnet loated beneath the dis. Suh a dis undegoes an osillatoy boune along the vetial axis with a fequeny whih is the same as that of the altenating magneti field. We apply meti 3 to this ase, i.e. the meti of the spae nea suh a dis. Fist, beause the initial deviation of suh a dis fom the est point is vey small z 0 z, we have GM GM z+z 0 os Ω u 1 z 0 z z os Ω u GM z. 4 Seond, the elativisti squae is K = 1. Thid, unde the onditions of a eal expeiment like Podkletnov s, the tems Ω z 0 Ω z 0 Ωz, 0, v and v have suh small num- eial values that they an be omitted fom the equations. The meti 3 then takes the muh simplified fom, shown as expession 5 at the top of this page. In othe wods, the expession 5 epesents the meti of the spae of a dis whih undegoes an osillatoy boune othogonal to its own plane, in the onditions of a eal expeiment. This is the main meti whih will be used henefoth in ou study fo the Podkletnov effet. We alulate the main obsevable haateistis of suh a spae aoding to Appendix. In suh a spae the gavitational potential w and the omponents of the linea veloity of the spae otation v i ae w = GM + Ωz 0 sin Ω z u v, 6 v 1 = v os ϕ + sin ϕ v = v os ϕ sin ϕ v 3 = v Ωz 0 sin Ω u. 7 The omponents of the gavitational inetial foe F i ating in suh a spae ae F 1 = Ωz 0 sin Ω u v + os ϕ + sin ϕ v t F = Ωz 0 sin Ω u v ϕ + os ϕ sin ϕ v t F 3 = Ωz 0 sin Ω, 8 u v z GM z + v t + + Ω z 0 os Ω u whee the quantities v, v ϕ, v z, v t denote the espetive patial deivatives of v. D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 63

8 Volume 3 PROGRESS IN PHYSICS July, 007 In suh a spae the omponents of the tenso of the angula veloities of the spae otation A ik ae A 1 = 1 os ϕ + sin ϕ v ϕ os ϕ sin ϕ v A 3 = os ϕ sin ϕ v z 1 v ϕ. 9 A 13 = 1 os ϕ + sin ϕ v z 1 v Beause we omit all quantities popotional to v, the h.inv.-meti tenso h ik = g ik + 1 v i v k the obsevable 3-dimensional meti tenso beomes h ik = g ik. Its omponents fo the meti 5 ae h 11 = 1, h =, h 33 = 1 h 11 = 1, h = 1, h33 = h = det h ik =, ln h x 1 = 1 Fo the tenso of the spae defomation D ik we obtain D 33 = D 33 = 0, D = h ik D ik = Among the h.inv.-chistoffel symbols Δ i km within the famewok of ou appoximation, only two omponents ae non-zeo, Δ 1 =, Δ 1 = 1, 3 so, despite the fat that the obsevable uvatue tenso C ik whih possesses all the popeties of Rii s tenso R αβ on the 3-dimensional spatial setion see Appendix isn t zeo in the spae, but within the famewok of ou assumption it is meant to be zeo: C ik = 0. In othe wods, although the spae uvatue isn t zeo, it is so small that it is negligible in a eal expeiment suh as we ae onsideing. These ae the physial obsevable haateistis of a spae volume element loated in an Eath-bound laboatoy, whee the non-holonomi bakgound of the spae is petubed by a dis whih undegoes osillatoy bounes othogonal to its own plane. We have now obtained all the physial obsevable haateistis of spae equied by Einstein s equations. Einstein s equations desibe flows of enegy, momentum and matte. Using the deived equations, we will know in peisely those flows of enegy and momentum nea a dis whih undegoes an osillatoy boune othogonal to its own plane. So if thee is any additional enegy flow o momentum flow geneated by the dis, Einstein s equations show this. 3. Einstein s equations in the spae. Fist onlusion about the oigin of the Podkletnov effet Einstein s equations, in tems of the physial obsevable quantities given in Appendix, wee deived in the 1940 s by Zelmanov [15, 16, 17] as the pojetions of the geneal ovaiant 4-dimensional Einstein equations R αβ 1 g αβ R = κ T αβ + λg αβ 33 onto the time line and spatial setion of an obseve. We omit the λ-tem due to its negligible effet. In onsideing a eal situation like Podkletnov s expeiment, we assume the same appoximation as in the pevious Setion. We also take into aount those physial obsevable haateistis of the spae whih ae zeo aoding to ou alulation. Einstein s equations expessed in the tems of the physial obsevable quantities see Appendix fo the omplete equations then take the following simplified fom F i x i A ika ik + ln h x i F i = κ ρ + U A ij x j + ln h x j A ij = κj i Fi x k + F k A ij A j k x i Δm ikf m = = κ ρ U h ik + κu ik whee ρ = T00 g 00, J i = T i 0 g00 and U ik = T ik ae the obsevable pojetions of the enegy-momentum tenso T αβ of distibuted matte on the ight side of Einstein s equations the ight side detemines distibuted matte whih fill the spae, while the left side detemines the geometial popeties of the spae. By thei physial sense, ρ is the obsevable density of the enegy of the matte field, J i is the obsevable density of the field momentum, U ik is the obsevable stess-tenso of the field. In elation to Podkletnov s expeiment, T αβ is the sum of the enegy-momentum tenso of an eletomagneti field, geneated by an eletomagnet loated beneath the dis, and also that of the othe fields filling the spae. We theefoe attibute the enegy-momentum tenso T αβ and its obsevable omponents ρ, J i, U ik to the ommon field. Is thee additional enegy and momentum podued by the field of the bakgound spae non-holonomity in ode to ompensate fo a petubation theein, due to a dis undegoing osillatoy bounes othogonal to its own pane? This is easy to answe using Einstein s equations, owing to the fat that given the unpetubed field of the bakgound spae non-holonomity, the linea veloity of the spae otation v isn t a funtion of the spatial oodinates and time v f, ϕ, z, t. Afte F i, A ik, D ik, and Δ i kn speifi to the spae of a suspended, vetially osillating dis ae substituted into Einstein s equations 34, the left side of the equations should ontain additional tems dependent on the deivatives of v by the spatial oodinates, ϕ, z, and time t. The additional tems, appeaing in the left side, build 64 D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity

9 July, 007 PROGRESS IN PHYSICS Volume 3 os ϕ + sin ϕ v t + os ϕ sin ϕ v tϕ 1 + os ϕ sin ϕ v ϕ + os ϕ sin ϕ v v ϕ + os ϕ sin ϕ v ϕv z + GM z v tz 1 os ϕ sin ϕ v v z + os ϕ + sin ϕ v v z + Ωz 0 sin Ω u v + v ϕϕ + v zz + v = κ ρ + U os ϕ sin ϕ v ϕ v ϕ 1 [ v ϕz + os ϕ + sin ϕ os ϕ + sin ϕ v z + os ϕ + sin ϕ vϕϕ v ϕ v ϕ + v zz + v + 1 v z = κj 1 ] os ϕ sin ϕ v + v zz = κj os ϕ sin ϕ v ϕz 1 v + v ϕϕ 1 os ϕ sin ϕ v os ϕ sin ϕ v v ϕ + os ϕ + sin ϕ v t os ϕ + sin ϕ v v z + + v = κj os ϕ sin ϕ + Ωz 0 sin Ω u v = κ v ϕ + v z + ρ U + κu 11 [ 1 os ϕ sin ϕ vz + v v ϕ os ϕ sin ϕ v v z os ϕ + sin ϕ v ϕ v z + + os ϕ + sin ϕ v tϕ + os ϕ sin ϕ v t + [ 1 Ωz 0 sin Ω u v ϕ v ϕ ] = κu 1 v t + os ϕ + sin ϕ v tz os ϕ sin ϕ v v ϕ + 1 os ϕ sin ϕ v v z + + os ϕ + sin ϕ v ϕ os ϕ sin ϕ v ϕ v z + Ωz 0 sin Ω ] u v z = κu 13 1 os ϕ sin ϕ + os ϕ sin ϕ v tϕ v + vz os ϕ sin ϕ v v ϕ os ϕ sin ϕ v ϕ + os ϕ sin ϕ v ϕv z + Ωz 0 sin Ω vϕϕ u + v = κ [ 1 v tϕ + os ϕ sin ϕ v tz + os ϕ sin ϕ v os ϕ + sin ϕ v v ϕ os ϕ sin ϕ v v z os ϕ sin ϕ v ϕ v z + Ωz 0 sin Ω ] u v ϕz = κu 3 ρ U + κu 35 v + v z os ϕ + sin ϕ v v z + v ϕ os ϕ sin ϕ v ϕv z + Ωz 0 sin Ω u v zz = κ ρ U + κu 33 + v tz + GM z 3 + D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 65

10 Volume 3 PROGRESS IN PHYSICS July, 007 espetive additions to the enegy and momentum of the ating eletomagneti field on the ight side of the equations. Following this line, we ae looking fo the enegy and momentum podued by the field of the bakgound spae non-holonomity due to petubation theein. We substitute F i 8, A ik 9, D ik 31, and Δ i kn 3, speifi to the spae of suh an osillating dis, into the h.inv.-einstein equations 34, and obtain the Einstein equations as shown in fomula 35. These ae atually Einstein s equations fo the initial homogeneous non-holonomi spae petubed by suh a dis. As seen fom the left side of the Einstein equations 35, a new enegy-momentum field appeas nea the dis due to the appeaane of a non-unifomity of the field of the bakgound spae non-holonomity i.e. due to the funtions v of the oodinates and time: 1. The field beas additional enegy to the eletomagneti field enegy epesented in the spae see the sala Einstein equation;. The field has momentum flow J i. The momentum flow speads fom the oute spae towad the dis in the -dietion, twists aound the dis in the ϕ- dietion, then ises above the dis in the z-dietion see the vetoial Einstein equations whih desibe the momentum flow J 1, J, and J 3 towad, ϕ, and z-dietion espetively. This puely theoetial finding explains the Podkletnov effet. Aoding to Eugene Podkletnov, a membe of his expeimental team smoked a pipe a few metes away fom the yostat with the supeonduting dis, duing opeation. By a stoke of luk, Podkletnov notied that the tobao smoke was attated towads the yostat, then twisted aound it and ose above it. Podkletnov then applied a high peision balane, whih immediately showed a weight loss ove the yostat. Now it is lea that the tobaosmoke evealed the momentum flow podued by the bakgound spae non-holonomity field petubed nea the vetially osillating dis; 3. The field has distibuted stesses whih ae expessed by an addition to the eletomagneti field stesstenso see the Einstein tenso equations. In the simplest ase whee Podkletnov s expeiment is un in a ompletely holonomi spae v = 0 the Einstein equations 35 take the simplest fom GM z 3 = κρ J 1 = 0, J = 0, J 3 = 0 U 11 = 0, U 1 = 0, U 13 = 0, U = 0, U 3 = 0 GM z 3 = κu This is also tue in anothe ase, whee the spae is nonholonomi v 0 but v isn t funtion of the spatial oodinates and time v f, ϕ, z, t, that is the unpetubed homogeneous field of the bakgound spae non-holonomity. We see that in both ases thee is no additional enegy and momentum flow nea the dis; only the eletomagneti field flow is put into equilibium by the Eath s gavity, dieted vetially along the z-axis. So Einstein s equations show lealy that: The Podkletnov effet is due to the fat that the field of the bakgound spae non-holonomity, being petubed by a suspended, vetially osillating dis, podues enegy and momentum flow in ode to ompensate fo the petubation theein. 3.3 Complete geometization of matte Looking at the ight side of the Einstein equations 35, whih detemine distibuted matte, we see that ρ and U ae inluded only in the sala fist equation and also thee tenso equations with the indies 11,, 33 the 5th, 8th, and 10th equations. We an theefoe find a fomula fo U. Then, substituting the fomula bak into the Einstein equations fo ρ and U 11, U, U 33, we an expess the haateistis of distibuted matte though only the physial obsevable haateistis of the spae. This fat, oupled with the fat that the othe haateistis of distibuted matte J 1, J, J 3, U 1, U 13, U 13 ae expessed though only the physial obsevable haateistis of the spae by the nd, 3d, 4th, 6th, 7th, and 9th equations of the Einstein equations 35, means that onsideing a spae in whih the homogeneous non-holonomi bakgound is petubed by an osillating dis, we an obtain a omplete geometization of matte. Multiplying the 1st equation of 35 by the 3d, then summing with the 5th, 8th, and 10th equations, we eliminate ρ. Then, beause U = h ik U ik =U 11 + U +U 33, we obtain a fomula fo U expessed only via the physial obsevable haateistis of the spae. Substituting the obtained fomula fo κu into the 1st equation, we obtain a fomula fo ρ. Afte that it is easy to obtain ρ + U and ρ U. Using these in the thee Einstein tenso equations with the diagonal indies 11,, 33, we obtain fomulae fo U 11, U, U 33, all expessed only in tems of the physial obsevable haateistis of the spae. The esulting equations, oupled with those of the Einstein equations 35 whih detemine J 1, J, J 3, U 1, U 13, and U 13, build the system of the equations 37, whih ompletely detemine the popeties of distibuted matte the density of the enegy ρ, the density of the momentum flow J I, and the stess-tenso U ik only in tems of the physial obsevable haateistis of the spae. So: Matte whih fills the spae, whee a homogeneous non-holonomi bakgound is petubed by an osillating dis is ompletely geometized. 66 D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity

11 July, 007 PROGRESS IN PHYSICS Volume 3 1 os ϕ sin ϕ κu = v 1 + os ϕ sin ϕ + os ϕ sin ϕ v ϕ v z os ϕ + sin ϕ os ϕ sin ϕ v tϕ κρ = 3 v tz 4GM z 3 v ϕ + v z os ϕ sin ϕ v v z os ϕ + sin ϕ v t Ωz 0 sin Ω u v + v ϕϕ [ 1 os ϕ sin ϕ v os ϕ sin ϕ v ϕ + v z os ϕ sin ϕ v v ϕ κ ρ U 1 os ϕ sin ϕ = os ϕ sin ϕ v ϕ v z + v tz + GM z 3 + κj 1 = os ϕ sin ϕ os ϕ sin ϕ v ϕv z v 1 + os ϕ sin ϕ + os ϕ + sin ϕ Ωz 0 sin Ω u v + v ϕϕ v ϕ v ϕ κj = 1 [ v ϕz + os ϕ + sin ϕ os ϕ + sin ϕ os ϕ + sin ϕ v v z ] v ϕ + v z v v ϕ + v zz + v os ϕ sin ϕ v v ϕ v v z + os ϕ + sin ϕ v t + os ϕ sin ϕ v tϕ + + v zz + v vϕϕ + v zz + v + 1 v z ] v ϕ v ϕ os ϕ sin ϕ v + v zz κj 3 os ϕ + sin ϕ os ϕ sin ϕ v ϕz = v z + 1 v + v ϕϕ + v 1 os ϕ sin ϕ v κu 11 = v ϕ os + os ϕ sin ϕ + ϕ sin ϕ v v ϕ os ϕ + sin ϕ v z v v z + os ϕ sin ϕ v ϕ v z + os ϕ sin ϕ v tϕ v tz GM z 3 Ωz 0 sin Ω vϕϕ u + v zz + v κu 1 = 1 [ os ϕ sin ϕ vz + v v ϕ os ϕ sin ϕ v v z os ϕ + sin ϕ v ϕ v z + + os ϕ + sin ϕ v tϕ + os ϕ sin ϕ v t + Ωz 0 sin Ω u v ϕ v ] ϕ κu 13 = 1 [ v t + os ϕ + sin ϕ v tz os ϕ sin ϕ v v ϕ + os ϕ + sin ϕ v ϕ os ϕ sin ϕ v ϕ v z κu v os ϕ sin ϕ v v z + + Ωz 0 sin Ω ] u v z = os ϕ sin ϕ os ϕ sin ϕ v v ϕ os z ϕ sin ϕ v v ϕ os ϕ sin ϕ v ϕ v z os ϕ+ sin ϕ + v v z os ϕ+ sin ϕ v t v tz GM z 3 Ωz 0 sin Ω u v +v zz κu 3 = 1 [ v tϕ + os ϕ sin ϕ v tz + os ϕ sin ϕ v os ϕ + sin ϕ v v ϕ os ϕ sin ϕ v v z os ϕ sin ϕ v ϕ v z + Ωz 0 sin Ω ] u v ϕz os ϕ sin ϕ κu 33 = os ϕ sin ϕ v ϕ v z v v ϕ + v z os ϕ + sin ϕ v v z + os ϕ + sin ϕ v t os ϕ sin ϕ v tϕ os ϕ sin ϕ v v ϕ Ωz 0 sin Ω u v + v ϕϕ + v 37 D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 67

12 Volume 3 PROGRESS IN PHYSICS July, 007 Thee is just one question still to be answeed. What is the natue of the matte? Among the matte diffeent fom the gavitational field, only the isotopi eletomagneti field was peviously geometized that fo whih the meti is detemined by the Rainih ondition [3, 4, 5] R = 0, R αρ R ρβ = 1 4 δβ α R ρσ R ρσ = 0 38 and the Nodtvedt-Pagels ondition [6] η μεγσ R δγ;σ R ετ R δε;σ R γτ = The Rainih ondition and the Nodtvedt-Pagels ondition, being applied to the left side of Einstein s equations, ompletely detemine the popeties of the isotopi eletomagneti field on the ight side. In othe wods, the afoementioned onditions detemine both the geometi popeties of the spae and the popeties of a pevading isotopi eletomagneti field. An isotopi eletomagneti field is suh whee the field invaiants F αβ F αβ and F αβ F αβ, onstuted fom the eletomagneti field tenso F αβ and the field pseudo-tenso F αβ = 1 ηαβμν F μν dual, ae zeo F αβ F αβ = 0, F αβ F αβ = 0, 40 so the isotopi eletomagneti field has a stutue tunated to that of an eletomagneti field in geneal. In ou ase we have no limitation on the stutue of the eletomagneti field, so we use the enegy-momentum tenso of the eletomagneti field in the geneal fom [0] T αβ = 1 4π F ασ F σ β F μνf μν g αβ, 41 whene the obsevable density of the field enegy ρ = T00 g 00 and the tae U = h ik U ik of the obsevable stess-tenso of the field U ik = T ik ae onneted by the elation ρ = U. 4 In othe wods, if besides the gavitational field thee is be only an eletomagneti field, we should have ρ = U fo distibuted matte in the Einstein equations. Howeve, as seen in the nd equation of the system 37, ρ U 0 in the Einstein equations, fo the only eason that, in the ase we ae onsideing, the laboatoy spae is filled not only by the Eath s gavitational field and an altenating magneti field whih suppots the dis in ai, but also anothe field appeaed due to the fat that the osillating dis petubs the non-holonomi bakgound of the spae. The petubation field, as shown in the pevious Setion, beas enegy and momentum, so it an be taken as a field of distibuted matte. In othe wods, The fat that the spae non-holonomity field beas enegy and momentum was fist shown in the ealie publiation [7], whee the field of a efeene body was onsideed. We have obtained a omplete geometization of matte onsisting of an abitay eletomagneti field and a petubation field of the non-holonomi bakgound of the spae. 3.4 The onsevation law When onsideing the geodesi equations in a spae, the hon-holonomi bakgound of whih is petubed by a dis undegoing osillatoy bounes othogonal to its own plane, we need to know the spae distibution of the petubation, i.e. some elations between the funtions v t = v, v t = v, v ϕ = v, v ϕ z = v, whih ae espetive patial deivatives of z the value v of the linea veloity of the spae otation v i. The funtions v t, v, v ϕ, v z ae ontained in the left side geomety of the Einstein equations we have obtained. Theefoe, fom a fomal point of view, to find the funtions we should integate the Einstein equations. Howeve the Einstein equations ae epesented in a non-empty spae, so the ight side of the equations is not zeo, but oupied by the enegy-momentum tenso T αβ of distibuted matte whih fill the spae. Hene, to obtain the funtions v t, v, v ϕ, v z fom the Einstein equations, we should expess the ight side of the equations the enegy-momentum tenso of distibuted matte T αβ though the funtions as well. Besides, in ou ase, T αβ epesents not only the enegymomentum of the eletomagneti field but also the enegymomentum podued by the field of the bakgound spae non-holonomity ompensating the petubation theein. Yet we annot divide one enegy-momentum tenso by anothe. So we must onside the enegy-momentum tenso fo the ommon field, whih pesents a poblem, beause we have no fomulae fo the omponents of the enegy-momentum tenso of the ommon field. In othe wods, we ae enfoed to opeate with the omponents of T αβ as meely some quantities ρ, J i, and U ik. How to expess T αβ though the funtions v t, v, v ϕ, v z, aside fo by the Einstein equations? In anothe ase we would be led to a dead end. Howeve, ou ase of distibuted matte is ompletely geometized. In othe wods, the geometial stutue of the spae and the spae distibution of the enegy-momentum tenso T αβ ae the same things. We an theefoe find the funtions v t, v, v ϕ, v z fom the spae distibution of T αβ, via the equations of the onsevation law σ T ασ = The onsevation law in the h.inv.-fom, i.e. epesented as the pojetions of equation 43 onto the time line and spatial setion of an obseve, is [15] ρ t +Dρ + 1 D iju ij + J k t + D k i +A k i J i + i 1 F i J i 1 F ij i =0 i 1 F i U ik ρf k = D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity

13 July, 007 PROGRESS IN PHYSICS Volume 3 ρ t + J 1 + J ϕ + J 3 z + 1 J 1 = 0 J 1 t [ ] os ϕ + sin ϕ v ϕ os ϕ sin ϕ v J [ ] os ϕ + sin ϕ v z v J U U 1 ϕ + U 13 z + 1 U 11 U ρ [Ωz 0 sin Ω ] u v + os ϕ + sin ϕ v t = 0 J [ t os ϕ + sin ϕ v ϕ os ϕ sin ϕ v ] [ J 1 os ϕ sin ϕ v z v ] ϕ J U U ϕ + U 3 z + 3 [ U 1 ρ Ωz 0 sin Ω ] u vϕ os ϕ sin ϕ + v t = 0 J 3 t + [ ] os ϕ + sin ϕ v z v J 1 + [ ] os ϕ sin ϕ v z v ϕ J + + U U 3 ϕ + U 33 z + 1 U 13 ρ [Ωz 0 sin Ω u v z GM z + v t + Ω z 0 os Ω u ] = 0 46 whee ρ = T00 g 00, J i = T 0 i g00 and U ik = T ik ae the obsevable pojetions of the enegy-momentum tenso T αβ of distibuted matte. The h.inv.-onsevation equations, taking ou assumptions fo eal expeiment into aount, take the simplified fom ρ t + J i x i + ln x i J i = 0 J k t + A k i J i ik U + x i h + ln h. 45 x i U ik + + Δ k imu im ρf k = 0 Substituting into the equations the fomulae fo D, Di k, A k i, ln h, Δ k x i im, and F k, we obtain a system of the onsevation equations 46 wheein we should substitute ρ, J i, and U ik fom the Einstein equations 37 then, eduing simila tems, aive at some elations between the funtions v t, v, v ϕ, v z. The Einstein equations 37 substituted into 46 evidently esult in intatable equations. It seems that we will have no hane of solving the esulting equations without some simplifiation aoding to eal expeiment. We should theefoe take the simplifiation into aount fom the beginning. Fist, the sala equation of the onsevation law 44 unde the onditions of a eal expeiment takes the fom of 45, whih in anothe notation is ρ t + i J i = The nd equation of 37 detemines ρ: the quantity is ρ 1. Omitting the tem popotional to 1 as its effet is negligible in a eal expeiment, we obtain the sala equation of the onsevation law in the fom i J i = 0, 48 The h.inv.-diffeential opeatos ae ompletely detemined, aoding to [15, 16], in Appendix. i.e. the h.inv.-deivative of the ommon flow of the spatial momentum of distibuted matte is zeo to within the appoximation of a fist-ode expeiment. This finding has a vey impotant meaning: Given a spae, the non-holonomi bakgound of whih is petubed by an osillating dis, the ommon flow of the momentum of distibuted matte on the spatial setion of suh a spae is onseved in a fistode expeiment. Seond, thee ae thee states of the dis in Podkletnov s expeiment: 1 unifom otation; non-unifom otation aeleation/deeleation; 3 non-otating dis. To study the ase of a otating dis we should intodue, into the meti 5, additional tems whih take the otation into aount. We don t do this now, fo two easons: 1 the additional tems intodued into the meti 5 make the equations of the theoy too ompliated; the ase of a non-otating dis is that main ase whee, aoding Podkletnov s expeiments, the weight-loss effet appeas in the basi fom; aeleating/deeleating otation of the dis podues only additions to the basi weight-loss. So, to undestand the oigin of the weight-loss phenomenon it is most easonable to fist onside petubation of the bakgound field of the spae non-holonomity by a non-otating dis. Beause suh a dis lies hoizontally in the plane ϕ hoizontal plane, we should assume v z = 0, while the fat that thee v 0 and v ϕ 0 means feedom fo osillation in the plane ϕ aeleating o deeleating twists in the plane as a esult of vetial osillation of suh a dis othewise, fo no osillation in the plane ϕ, the onsevation equations would beome zeo. The fat that ϕ onst in the equations means the same. As a esult, the onsevation equations 46, with the afoementioned assumptions taken into aount, take the fom 49. The haateistis of distibuted matte suh as the momentum flow J i and the stess-tenso U ik, esulting fom D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 69

14 Volume 3 PROGRESS IN PHYSICS July, 007 J 1 t [ ] os ϕ + sin ϕ v ϕ os ϕ sin ϕ v J + v J 3 + U U 1 ϕ J [ t os ϕ + sin ϕ v ϕ os ϕ sin ϕ v ] J 1 + v ϕ J 3 + J 3 t v J 1 v ϕ J + U U 3 ϕ + 1 U 13 = 0 κj 1 os ϕ sin ϕ = κj = 1 [ os ϕ + sin ϕ v + v ϕϕ v ϕ v ϕ os ϕ + sin ϕ vϕϕ + v v ϕ v ] ϕ os ϕ sin ϕ v + v κj 3 = 1 os ϕ sin ϕ κu 11 = v + os ϕ sin ϕ v ϕ os ϕ sin ϕ GM z 3 Ωz 0 sin Ω vϕϕ u + v κu 1 = 1 κu 13 = 1 κu GM z 3 κu 3 = 1 κu 33 = [ v v ϕ + os ϕ + sin ϕ v tϕ + os ϕ sin ϕ v t + [ v t os ϕ sin ϕ v ] v ϕ + os ϕ + sin ϕ v ϕ = os ϕ sin ϕ v Ωz 0 sin Ω u v os ϕ + sin ϕ vϕ + os ϕ sin ϕ [ v tϕ + os ϕ sin ϕ v os ϕ + sin ϕ v v ϕ ] os ϕ sin ϕ v v ϕ + Ωz 0 sin Ω u v + v ϕϕ + v os ϕ sin ϕ v v ϕ U1 v v ϕ U 11 U = U ϕ + 3 U 1 os ϕ sin ϕ v tϕ Ωz 0 sin Ω u v ϕ v ϕ v v ϕ ] os ϕ + sin ϕ v t os ϕ + sin ϕ v t os ϕ sin ϕ v tϕ = the Einstein equations 37, wee olleted in omplete fom into the system 37. Unde the afoementioned assumptions they take the fom 50. We substitute the espetive omponents of J i and U ik 50 into the onsevation equations 49. Afte algeba, eduing simila tems, the fist two equations of 49 beome identially zeo, while the thid equation takes the fom: v = v ϕ, 51 The solution v = vϕ we have obtained fom the onsevation equations satisfies by the funtion v = B t e ϕ, 5 whee B t is a funtion of time t. Speifi fomula fo the funtion B t should be detemined by natue of the phenomenon o the onditions of the expeiment. The solution indiates a dependeny between the distibutions of v in the -dietion and ϕ-dietion in the spae, if the non-holonomi bakgound is petubed by a dis lying in the ϕ plane and osillating in the z-dietion. In othe wods, the onsevation equations in ommon with the Einstein equations we have obtained mean that: A dis, osillating othogonally to its own plane, petubs the field of the bakgound non-holonomity of the spae. Suh a motion of a dis plaes a limitation on the geometi stutue of the spae. The limitation is manifested as a speifi distibution of the linea veloity of the spae otation. This distibution means that suh a dis should also have small twists in its own plane due to the petubed nonholonomi bakgound. 70 D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity

15 July, 007 PROGRESS IN PHYSICS Volume 3 [ ] [ ] os ϕ + sin ϕ v ϕ os ϕ sin ϕ v ϕ os ϕ + sin ϕ vz v ż os ϕ + sin ϕ v t ϕ Ωz 0 sin Ω u v = 0 [ ϕ + os ϕ + sin ϕ v ϕ os ϕ sin ϕ v t + ṙ os ϕ sin ϕ v ϕ Ωz 0 sin Ω u vϕ = 0 ] [ ṙ os ϕ sin ϕ v z v ] ϕ ż z + [ ] [ ] os ϕ + sin ϕ v z v ṙ + os ϕ sin ϕ vz v ϕ ϕ + GM z v t Ωz 0 sin Ω u v z Ω z 0 os Ω u = The geodesi equations in the spae. Final onlusion about the foes diving the Podkletnov effet This is the final pat of ou mathematial theoy of the Podkletnov effet. Hee, using the Einstein equations and the equations of the onsevation law we have developed, we dedue an additional foe that podues the weightloss effet in Podkletnov s expeiment, i.e. the weight-loss ove a supeonduting dis whih is suppoted in ai by an altenating magneti field. As is well known, motion in a gavitational field of a fee test-patile of est-mass m 0 is desibed by the equations of geodesi lines the geodesi equations. The geodesi equations ae, fom a puely mathematial viewpoint, the equations of paallel tansfe of the fou-dimensional veto of the patile s momentum P α dx = m α 0 along the patile s ds 4-dimensional tajetoy dp α ds + Γα μνp μ dxν ds = 0, 53 whee Γ α μν ae Chistoffel s symbols of the nd kind, while ds is the 4-dimensional inteval along the tajetoy. The geodesi equations 53, being pojeted onto the time line and spatial setion of an obseve, and expessed though the physial obsevable haateistis of a eal laboatoy spae of a eal obseve, ae known as the h.inv.- geodesi equations. They wee dedued in 1944 by Zelmanov [15, 16]. The elated sala equation is the pojetion onto the time line of the obseve, while the 3-dimensional veto equation is the pojetion onto his spatial setion, and manifests the 3d Newtonian law fo the test-patile: dm dτ m F iv i + m D ikv i v k = 0 dmv i +m 54 D i dτ k+a i k v k mf i +mδ i nkv n v k = 0 whee m is the elativisti mass of the patile, v i is the 3- dimensional obsevable veloity of the patile, and τ is the physial obsevable o pope time [15, 16] This is that eal time whih is egisteed by the obseve in his eal m = dτ = g 00 dt + m 0 1 v /, vi = dxi dτ, 55 g 0i g 00 dx i = g 00 dt 1 v idx i. 56 With the simplifiations fo the eal expeiment we ae onsideing, the h.inv.-geodesi equations 54 take the fom dm dτ = 0 dmv i dτ + ma i k v k mf i + mδ i nkv n v k = 0 that is, in omponent notation, dm dτ = 0 d m dv1 dτ dτ d m dv dτ dτ d m dv3 dτ dτ +ma 1 +ma +ma 3 k v k mf 3 = 0 57 k v k mf 1 +mδ 1 v v = 0 58 k v k mf +mδ 1v 1 v = 0 whih ae atual h.inv.-equations of motion of a fee testbody in the spae, whose non-holonomi homogeneous bakgound is petubed by an osillating dis. The sala geodesi equation of 58 says m = onst, 59 so taking this fat into aount and intoduing the notation v 1 = d = ṙ, dτ v = dϕ = ϕ, dτ v3 = dz = ż, we obtain a system dτ of thee veto equations of motion of the test-body 60, wheein v t = v, v t = v, v ϕ = v, v ϕ z = v. z laboatoy spae. Intevals of the physial obsevable time dτ and the obsevable spatial oodinates dx i ae detemined, by the theoy of physial obsevable quantities honometi invaiants as the pojetions of the inteval of the 4-dimensional oodinates dx α onto the time line and spatial setion of an obseve, i.e.: b α dx α = dτ, h i α dxα = dx i [15, 16]. See Appendix fo the details of suh a pojetion. D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 71

16 Volume 3 PROGRESS IN PHYSICS July, 007 Beause the tems ontaining z 0 in equations 60 ae vey small, they an be onsideed to be small hamoni oetions. Suh equations an always be solved using the small paamete method of Poinaé. The Poinaé method is also known as the petubation method, beause we onside the ight side as a petubation of a hamoni osillation desibed by the left side. The Poinaé method is elated to exat solution methods, beause a solution podued with the method is a powe seies expanded by a small paamete see Lefshetz, Chapte XII, of [1]. Howeve ou task is muh simple. We ae looking fo an appoximate solution of the system of the veto equations of motion in ode to see the main foes ating in the basi Podkletnov expeiment. We theefoe simplify the equations as possible. Fist we take into aount that, in the ondition of Podkletnov s expeiment, the suspended test-body has feedom to move only in the z-dietion i.e. up o down in a vetial dietion, whih is the dietion of the ating foe of gavity. In othe wods, onening a fee test-body falling fom above the dis, we take ṙ = 0 and ϕ = 0 despite the foes and ϕ ating it in the -dietion and the ϕ- dietion ae non-zeo. Seond, otational osillation of the dis in the ϕ plane is vey small. We theefoe egad ϕ as a small quantity, so sin ϕ ϕ and os ϕ 1. Thid, by the onsevation equations, v ϕ = v. Taking all the assumptions into aount, the equations of motion 60 take the muh simplified fom + v ż 1 + ϕ v t Ωz 0 sin Ω u v = 0 ϕ + v ż 1 ϕ v t Ωz 0 sin Ω u v = 0 z + g v t Ω z 0 os Ω u = 0 61 whee g = GM is the aeleation podued by the Eath s z foe of gavity, emaining onstant fo the expeiment. Fo Podkletnov s expeiment, v t = onst, and this value depends on the speifi paametes of the vetially osillating dis, suh as its diamete, the fequeny and amplitude of its vibation. The hamoni tem in the thid equation is a small oetion whih an only shake a test-body in the z-dietion; this tem annot be a soue of a foe ating in just one dietion. Besides, the hamoni tem has a vey small numeial value, and so it an be negleted. In suh a ase, the thid equation of motion takes the simple fom z + g v t = 0, 6 whee the last tem is a oetion to the ating foe of gavity due to the petubed field of the bakgound spae non-holonomity. Integating the equation z = g + v t, we obtain z = g v t τ + C 1 τ + C, 63 whee the initial moment of time is τ 0 = 0, the onstants of integation ae C 1 = ż 0 and C = z 0. As a esult, if the test-body is at est at the initial moment of time ż 0 = 0, its vetial oodinate z at anothe moment of obsevable time is z = z 0 g τ + v t τ. 64 The esult we have obtained isn t tivial beause the additional foes obtained within the famewok of ou theoy oiginate in the field of the bakgound spae non-holonomity petubed by the dis. As seen fom the final equation of motion along the z-axis 6, suh an additional foe ats eveywhee against the foe of gavity. So it woks like negative gavity, a tuly anti-gavity foe. Within the famewok of Classial Mehanis we have no spae-time, hene thee ae no spae-time tems in the metis whih detemine the non-holonomity of spae. So suh an anti-gavity foe is absent in Classial Mehanis. Suh an anti-gavity foe vanishes in patiula ases of Geneal Relativity, whee the pseudo-riemannian spae is holonomi, and also in Speial Relativity, whee the pseudo- Riemannian spae is holonomi by definition in addition to the absene of uvatue, gavitation, and defomation. So the obtained anti-gavity foe appeas only in Geneal Relativity, whee the spae is non-holonomi. It should be noted that the anti-gavity foe F = mv t isn t elated to a family of foes of inetia. Inetial foes ae fititious foes unelated to a physial field; an inetial foe appeas only in mehanial ontat with that physial body whih podues it, and disappeas when the mehanial onnexion eases. On the ontay, the obtained anti-gavity foe oiginates fom a eal physial field a field of the spae non-holonomity, and is podued by the field in ode to ompensating fo the petubation theein. So the anti-gavity foe obtained within the famewok of ou theoy is a eal physial foe, in ontast to foes of inetia. Conening Podkletnov s expeiment, we should take into aount the fat that a balane suspended test-body isn t fee, due to the foe of eation of the pie of the balane whih ompletely ompensates fo the ommon foe of attation of the test-body towads the Eath the body s weight. As a esult suh a test-body moves along a nongeodesi wold-tajetoy, so the equations of motion of suh a patile have non-zeo ight side ontaining the foe of the eation of the pie. In the state of stati weight, the ommon aeleation of the test-body in the z-dietion is zeo z = 0, hene its weight Q is Q = mg mv t. 65 The quantity v t ontained in the additional anti-gavity foe F = mv t is detemined by the paametes of the small twists of the dis in the hoizontal plane, the fequeny of whih is the same as the fequeny Ω of vetial osillation of the dis, while the amplitude depends on paametes of the 7 D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity

17 July, 007 PROGRESS IN PHYSICS Volume 3 dis, suh as its adius and the amplitude z 0 of the osillation. A alulation fo suh an anti-gavity foe in the ondition of a eal expeiment is given in the next Setion. As we will see, ou theoy gives good oinidene with the weight-loss effet as measued in Podkletnov s expeiment. The geodesi equation we have obtained in the field of an osillating dis allows us to daw a final onlusion about the oigin of the foes whih dive the weight-loss effet in Podkletnov s expeiment: A foe podued by the field of the bakgound spae non-holonomity, ompensating fo a petubation theein, woks like negative gavity in the ondition of an Eath-bound expeiment. Being podued by a eal physial field that beas its own enegy and momentum, suh an anti-gavity foe is a eal physial foe, in ontast to fititious foes of inetia whih ae unelated to physial fields. In the onditions of Podkletnov s expeiment, a hoizontally plaed supeonduting dis, suspended in ai due to an altenating magneti field, undegoes osillatoy bounes in a vetial dietion othogonal to the plane of the dis with the same fequeny of the magneti field. Suh an osillation petubs the field of the bakgound spae non-holonomity, initially homogeneous. As a esult the bakgound nonholonomity field is petubed in thee spatial dietions, inluding the hoizontal plane the plane of the dis, esulting in small amplitude osillatoy twists about the vetial dietion. The osillatoy twists detemine the anti-gavity foe, podued by the petubed field of the bakgound spae non-holonomity, and at in the vetial dieting against the foe of gavity. Any test-body, plaed in the petubed nonholonomity field above suh a vetially osillating dis, should expeiene a loss in its weight, the numeial value of whih is detemined by the paametes of the dis and its osillatoy motion in the vetial dietion. If suh a dis otates with aeleation, this should be the soue of an addition petubation of the bakgound non-holonomity field and, hene, a substantial addition to the weight-loss effet should be obseved in expeiment. Unifom otation of the dis should give no effet. Heein we have been onened with only a theoy of a phenomenon disoveed by Podkletnov we efe to this as the Podkletnov effet, to fix the tem in sientifi teminology. Aoding to ou theoy, supeonduto tehnology aounts in Podkletnov s expeiment only fo levitation of the dis and diving it into small amplitude osillatoy motion in the vetial dietion. Howeve, it is evident that this isn t the only way to ahieve suh a state fo the dis. Futhemoe, we show that thee ae also both mehanial and nulea systems whih an simulate the Podkletnov effet and, hene, be the soues of ontinuous and explosive enegy fom the field of the bakgound spae nonholonomity. Suh a mehanial system, simulating the onditions of the Podkletnov effet, povides a possibile means of ontinuous podution of enegy fom the spae non-holonomity field. At the same time we annot ahieve high numeial values of the osillatoy motion in a mehanial system, so the ontinuous podution of enegy might be low althopugh it may still eah useful values. On the ontay, poesses of nulea deay and synthesis, due to the instant hange of the spin onfiguation among nuleons inside nulei, should have high numeial values of v t, and theefoe be an explosive soue of enegy fom the field of the bakgound spae non-holonomity. Both mehanial and nulea simulations of the Podkletnov effet an be ahieved in patie. 4 A new expeiment poposed on the basis of the theoy 4.1 A simple test of the theoy of the Podkletnov effet altenative to supeonduto tehnology Aoding ou theoy, the Podkletnov effet has a puely mehanial oigin, unelated to supeondutivity the field of the bakgound spae non-holonomity being petubed by a dis whih undegoes osillatoy bounes othogonal to its own plane, podues enegy and momentum flow in ode to ompensate fo the petubation theein. Owing to this, we popose a puely mehanial expeiment whih epodues the Podkletnov effet, equivalent to Podkletnov s oiginal supeonduto expeiment, whih would be a heap altenative to ostly supeonduto tehnology, and also be a simple mehanial test of the whole theoy of the effet. What is the aangement of suh a puely mehanial system, whih ould enable epodution of the Podkletnov effet? Seahing the sientifi liteatue, we found suh a system. This is the vibation balane [], invented and tested in the s by N. A. Kozyev, a famous astonome and expeimental physiist of the Pulkovo Astonomial Obsevatoy St. Petesbug, Russia. Below is a desiption of the balane, extated fom Kozyev s pape []: The vibation balane is an equal-shoulde balane, whee the pie of the ental pism is onneted to a vibation mahine. This vibation mahine podues vetial vibation of the pie. The aeleation of the vibation is smalle than the aeleation of the Eath s gavitation. Theefoe the pism doesn t lose ontat with the pie, only altenating pessue esults. Thus the distane between the ente of gavity and the one of the pism emains onstant while the weight and the balane don t hange thei own measuement peision. The vetial guiding ods, set up along the pie, exlude the possibility of hoizontal motion of the pie. One of two samples of the same mass is igidly suspended by the yoke of the balane, while the seond sample is suspended by an elasti mateial. Hee the foe equied to lift the yoke is just a small peentage of the foe equied to lift the igidly fixed sample. Theefoe, duing vibation of the balane, thee is stable kinemati of the yoke, whee the point O the point of had suspension D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity 73

18 Volume 3 PROGRESS IN PHYSICS July, 007 doesn t patiipate in vibation, while the point A the point of elasti suspension has maximal amplitude of osillation whih is double the amplitude of the ental pism C. Beause the additional foe, podued duing vibation, is just a few peent moe than the stati foe, the yoke emains fixed without inne osillation, i.e. without twist, in aodane with the equiement of stati weight. We tested diffeent aangements of balanes unde vibation. The tested balanes had diffeent sensitivities, while the elasti mateial was tied with ubbe, a sping, et. Hee is detailed a desiption of the vibation balane whih is uently in use. This is a tehnial balane of the seond lass of sensitivity, with a maximum payload of 1 kg. A 1 mm deviation of the measuement aow, fixed on the yoke, shows a weight of 10 mg. The ente of gavity of the yoke is loated 1 m below the pie of the ental pism. The length of the shouldes of the yoke is: OC = CA = l = = 16 m. The amplitude of vibation is a 0. mm. Thus the maximum speed of the ental pism is v = π a m/se, while T a = 10 is about 0% of the its maximum aeleation π T aeleation of the Eath s gavitation. We egulaly used samples of 700 g. One of the samples was suspended by a ubbe, the stain of whih fo 1 m oesponds 100 g weight. So, duing vibation, the additional foe on the yoke is less than 10 g and annot destoy the igidity of the yoke. The elasti ubbe suspension absobs vibation so that the sample atually ests. This balane, as well as all eently tested systems, showed eah time the inease of the weight of the elastially suspended sample. This additional foe Q is popotional to the weight of the sample Q, besides Q/Q = Hene, having Q = 700 g, Q = 1 mg and the foe momentum twisting the yoke is 300 dynes m. [... ] Fom fist view one an think that, duing suh a vibation, the pie makes twists aound the esting point O. In a eal situation the points of the pie ae aied into moe ompliated motion. The ental pism doesn t lose ontat with the pie; they ae onneted, and move only linealy. Theefoe the ental pat of the yoke, whee its main mass is onentated, has no entifugal aeleation. What is about the point O, this point in ommon with the igidly suspended sample is fixed in only the vetial dietion, but it an move feely in the hoizontal dietion. These hoizontal displaements of the point O ae vey small. Natually, they ae a l, i.e. 0.1 μm in ou ase. Despite that, the small displaements esult a vey speifi kinemati of the yoke. Duing vibation, eah point of the yoke daws an element of an ellipse, a small axis of whih lies along the yoke in the aveage position of it. The onavities of the elements in the yoke s setions O C and C A ae dieted opposite to eah othe; they podue oppositely dieted entifugal foes. Beause ˉv is geate in the setion C A, the entifugal foes don t ompensate eah othe ompletely: as a esult thee in the yoke a entifugal foe ats in the A-dietion the dietion at the point of the elastially suspended sample. This entifugal aeleation has maximum value at the point A. We have ˉv = 4π T a = 6 m /se. Fom hee we obtain the uvatue adius of the ellipse: ρ = 4l = 60 m. So the entifugal aeleation is ˉv = 0.1 ρ m/se. Suh a vibation balane is shown in the uppe pitue of Fig. 4. An analogous vibation balane is shown in the lowe pitue of Fig. 4: thee the vibation mahine is onneted Fig. 4: The vibation balane a mehanial test of the whole theoy of the Podkletnov effet a simple altenative to ostly supeonduto tehnology. not to the pie of the ental pism, but to he elasti suspension, while the pism s pie is suppoted by a sping; suh a system should podue the same effet. To undestand how the Podkletnov effet manifests with the vibation balane, we onside the opeation of the balane in detail see Fig. 5. The point O of the yoke undegoes osillatoy bounes in the -dietion with the amplitude d, given by d = l l os α = l l 1 sin α = = l l 1 1 a l l l a l while b is b=d tan α=d a l, 66 a l os α a 3 l a3 1 a l a. 67 l The point A undegoes osillatoy bounes in the z- dietion with the amplitude a, while its osillatoy motion in the -dietion has the amplitude = l l os α d = d. 68 The osillatoy bouning of the points O and A along the elements of an ellipse is an aeleating/deeleating otational motion aound the fous of the ellipse. In suh a ase, by definition of the spae non-holonomity as the nonothogonality of time lines to the spatial setion, manifest 74 D. Rabounski and L. Boissova. A Theoy of the Podkletnov Effet Based on Geneal Relativity