1 Deivation of the Gavitational Red Shift fom the Theoem of Obits by Myon W. Evans, Alpha Institute fo Advanced Study, Civil List Scientist. emyone@aol.com and www.aias.us Abstact The expeimentally obsevable gavitational ed shift is deived by otating the line element deived fom the Theoem of Obits. The latte is a simple special case of the Fobenius Theoem fo a spheically symmetic space-time. All known obits ae descibed by the geomety of the Theoem of Obits, and the gavitational ed shift is shown to be the pecession o phase shift caused by otating the line element of the Theoem of Obits. Keywods: ECE Theoy, gavitational ed shift, Theoem of Obits, line element otation. 1.1 Intoduction Recently in the ECE seies of papes [1 10] it has been shown that all known obits can be descibed diectly by the spheical symmety of space-time with tosion and cuvatue without having to use any field equation a pioi. The Theoem of Obits pape 111 has been deived fom the well known [11] Fobenius Theoem applied to a spheically symmetic space-time. Fom the Theoem of Obits the line element is deived, giving the obital equation. Theefoe the field of foce which becomes the Newtonian field of foce in the appopiate limit is deived diectly fom spheical space-time symmety. This pocedue is summaized in Section 1., and in Section 1.3 the well known gavitational ed shift is given a new meaning by deiving it fom otation of the line element of Section 1.. It is found that the gavitational ed
374 1 Deivation of the Gavitational Red Shift fom the Theoem of Obits shift is a pecession o phase shift - essentially a popety puely of spheical space-time and not of any field equation. In the standad model the gavitational ed shift is thought to be a wavelength change and incoectly deived fom a space-time that has no tosion. 1. Line Element and Obital Equation fom Theoem of Obits The Theoem of Obits is a simple example of the Fobenius Theoem [11] which defines the most geneal line element. The Theoem of Obits is: n = m = d = + µ 1.1 whee n and m ae functions of, the adial coodinate in spheical pola coodinates. The constant of integation is in geneal non-zeo, and goes to zeo in a Minkowski space-time. If the Fobenius Theoem is applied [11] to a spheically symmetic space-time the line element is: ds = nc dt + md + dω. 1. Fom the Theoem of Obits it is found that: n =1+ µ, 1.3 m = 1+ µ 1, 1.4 so that the line element becomes: ds = 1+ µ c dt + 1+ µ 1 d + dω 1.5 in spheical pola co-odinates. The obital equation is obtained by consideing the special case of obits in a plane, so the line element 1.5 educes to: ds = 1+ µ c dt + 1+ µ 1 d + dφ. 1.6
1. Line Element and Obital Equation fom Theoem of Obits 375 Define [1 11] the constant of motion: ds ɛ = = c dλ dλ = 1+ µ dt c dλ + 1+ µ 1 d dλ dφ + 1.7 dλ whee is the infinitesimal element of pope time. Now make the choice: to find: c = 1+ µ dt c + 1+ µ λ = τ 1.8 1 d dφ +. 1.9 To convet to S.I. units multiply thoughout by 1 m,wheemistobedetemined: 1 m dφ 1 1+ m µ dt c + 1 1+ m µ 1 d = 1 mc. 1.10 Multiply though by 1+ µ : to find that: 1 m dφ 1+ µ 1 m 1+ µ c = 1 mc 1+ µ dt + 1 m d. 1.11 This is the obital equation: The total enegy in S.I. units is: 1 d m + V = E. 1.1 E = 1 mc 1+ µ dt. 1.13
376 1 Deivation of the Gavitational Red Shift fom the Theoem of Obits The potential enegy in S.I. units is: V = 1 m 1+ µ c + L 1.14 whee: L = dφ 1.15 is a constant of motion having the units of angula momentum pe unit mass. The facto 1 is intoduced [11] to wite the equation in standad dynamical fom. The potential enegy is theefoe: V = 1 mc + 1 µ mc + 1 ml + 1 ml µ 3 1.16 and is made up of fou tems which ae identified below. Fo all obits excluding binay pulsas and the Cassini/Pionee anomaly it is found by expeimental obsevation that: Theefoe the potential enegy becomes: V = 1 mc m mg µ = mg c. 1.17 + 1 ml L mmg c 3. 1.18 Theefoe it becomes possible to identify the fou tems as follows. 1 A constant tem popotional to est enegy, 1 mc. The Newtonian potential of attaction, mm G/. 3 The centipetal epulsion, ml /. 4 The elativistic coection to the Newtonian attaction, L mmg/c 3. Theefoe the facto m is the mass of an object attacted by an object of mass M. The Theoem of Obits 1.1 is the geometical contol ove the way m and M inteact. The intoduction of m, M and G intoduces physics into pue geomety. The Newtonian limit is defined by: 1.19
1. Line Element and Obital Equation fom Theoem of Obits 377 when the familia Newtonian tems 1. and 1.3 dominate. The Newtonain foce of attaction is: F = V = mmg 1.0 which is the invese squae law of Newton. Fom Eqs 1.18 and 1.0 the total foce between m and M is: F = mmg + ml 3 3L mmg c 4. 1.1 This foce law descibes the vast majoity of known obits with geat accuacy. It descibes peihelion advance, deflection of light by gavity, fame dagging, Shapio time delay and all the phenomena incoectly attibuted in the standad model to the now obsolete [1 10] Einstein field equation. As agued, these phenomena ae due puely to the spheical symmety of space-time. The masses m and M ae intoduced following expeimental obsevation. The Newtonian foce is Eq. 1.0. Newton did not ealize the existence of the centipetal foce, and of couse did not ealize the existence of the elativistic coection. In binay pulsas pape 108 the obits decease by a few millimetes pe evolution. This effect is descibed by the addition of a vey small petubation as follows: mg µ = c + a which geneates an additional attaction potential: 1. and an additional foce of attaction: V = 1 c ma + L c 4 c F = ma 3 + L c 5 1.3 1.4 which causes the two objects of a binay pulsa to spial in towads each othe fom a elativistic obit whose peihelion advance is a few degees pe evolution. This is a vey small effect, but epoducible and epeatable. The same type of phenomenon is found in the sola system in the well known Pionee/Cassini anomalies. Both spacecaft see a tiny additional foce of
378 1 Deivation of the Gavitational Red Shift fom the Theoem of Obits attaction not pesent in Eq. 1.1. The complete foce law fo binay pulsas and the Pionee Cassini obits is theefoe: F = mmg + m 3 L ac 3LmMG c 4 the Newtonian foce in this case being only one of five tems. aml c 5, 1.5 1.3 The Gavitational Red Shift Conside the line element 1.5 in cylindical pola co-odinates: ds = 1+ µ c dt 1+ µ 1 d dφ dz. 1.6 Now otate it see pape 110 at an angula velocity ω as follows: The otated line element is theefoe: ds = φ = φ + ωt. 1.7 1+ µ c dt 1+ µ 1 d dφ dz 1.8 whee: and It is found that: dφ = dφ + ωdt 1.9 dφ = dφ +ωdφdt + ω dt. 1.30 ds µ v = 1+ c c dt Ωdφdt 1+ µ 1.31 1 d dφ dz whee the obital linea velocity of otation is defined by v = ω. 1.3
1.3 The Gavitational Red Shift 379 Identify the elativistic angula velocity compae pape 110 on www.aias.us as: Ω=ω 1+ 1 µ v c 1.33 and the infinitesimal of pope time by: = 1+ 1 µ v c dt. 1.34 The change of phase, o pecession, upon otating by π adians is: α =Ω ωdt =π 1+ µ 1 v c 1. 1.35 The limit 1.36 defines the Thomas pecession pape 110: αthomas = π 1 v c 1 1 1.37 and the limit: v 0 1.38 defines the gavitational ed shift: αgav = π 1+ µ 1 1. 1.39 Fo almost all obits, as agued in Section 1., it is found by expeimental obsevation that: µ = mg c 1.40
380 1 Deivation of the Gavitational Red Shift fom the Theoem of Obits so the gavitational ed shift is: αgav = π 1 mg 1 1 c πmg c. 1.41 as obseved expeimentally as is well known. The Thomas pecession is well obseved expeimentally in atomic and molecula specta in spin obit coupling. It is concluded that both the gavitational ed shift and the Thomas pecession ae due puely to the spheical symmety of space-time. The standad model s Einstein field equation is known to be geometically incoect because of its neglect of tosion, so the standad explanation of the gavitational ed shift cannot be coect. Similaly, the standad model s cosmological ed shift which is diffeent fom the gavitational ed shift is an atifact based on the Robeston Walke metic, which in pape 93 on www.aias.us was shown to violate basic geomety the Hodge dual of the Bianchi identity. Acknowledgments The Bitish Govenment is thanked fo the awad of a Civil List Pension and amoial ensigns fo pe-eminent contibutions to Bitain and the Commonwealth in science and the staffs of AIAS and many othe colleagues woldwide thanked fo inteesting discussions.
Refeences [1] M. W. Evans, Geneally Covaiant Unified Field Theoy Abamis 005 onwads, volumes one to fou, volumes five and six in pep. papes 71 onwads on www.aias.us. [] L. Felke, The Evans Equations of Unified Field Theoy Abamis, 007. [3] K. Pendegast, Cystal Sphees pepint on www.aias.us. [4] M. W. Evans, Omnia Opea section of www.aias.us fom 199 onwads. [5] M. W. Evans ed., Adv. Chem. Phys, vol. 119 001; M. W. Evans and S. Kielich eds., Adv. Chem. Phys., vol. 85 199, 1993, 1997. [6] M. W. Evans and L. B. Cowell, Classical and Quantum Electodynamics and the B1.3 Field Wold Scientific, 001. [7] M. W. Evans and J.-P. Vigie, The Enigmatic Photon Kluwe, 1994 to 00, hadback and softback, in five volumes. [8] M. W. Evans, fifteen papes in Found. Phys. Lett., 003 to 005. [9] M. W. Evans. Acta Phys. Polon., 38B, 11 007; Physica B, 403, 517 008. [10] M. W. Evans and H. Eckadt, Physica B, 400, 175 007. [11] S. P. Caoll, Space-time and Geomety: an Intoduction to Geneal Relativity Addison Wesley, New Yok, 004, and downloadable lectue notes, chapte 7.