REFUTATION OF GENERAL RELATIVITY: INCONSISTENCIES IN THE EINSTEINIAN THEORY OF PERIHELION PRECESSION

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1 Jounal of Foundations of Physics and Chemisty () REFUTATION OF GENERAL RELATIVITY: INCONSISTENCIES IN THE EINSTEINIAN THEORY OF PERIHELION PRECESSION M. W. Evans and H. Eckadt Alpha Institute fo Advanced Studies ( Peihelion pecessions ae calculated fo nealy cicula obits fo vaious foce laws used in astonomy. It is shown that the claims of Einsteinian geneal elativity (EGR) ae self inconsistent and ae not veified by expeimental data. Thee ae also philosophical self inconsistencies within the famewok of EGR, which means that it is not a satisfactoy theoy of natual philosophy. A simple suggestion is given fo a philosophically self consistent theoy within the famewok of ECE theoy. Keywods: ECE theoy, calculation of peihelion pecessions fo nealy cicula obits, efutations of Einsteinian geneal elativity.. INTRODUCTION Duing the couse of development of the well known ECE unified field theoy {-} many efutations emeged of Einsteinian geneal elativity (EGR). These efutations ae simple to undestand and thee is no logical answe to them, so new systems of cosmology have been foged in ecent papes of this seies. In Section a method is developed to calculate peihelion pecessions fo nealy cicula obits in the classical appoximation to ECE theoy and it is applied to potential and foce laws used in astonomy and cosmology. The Newtonian appoximation is adequate fo most pecessions but the EGR theoy claims that a change in philosophy is needed to descibe an anomalous pecession. The latte is

2 M. W. Evans and H. Eckadt defined as the diffeence between the expeimental pecession and a Newtonian calculation. This claim is not only inconsistent but also untenable in view of numeous citicisms and efutations of EGR {-} made ove nealy a centuy. The flaws in EGR become appaent with some calculations of peihelion pecessions given in Section. It is also well known that EGR fails entiely in systems such as whilpool galaxies, so it cannot be a pecise theoy inside the sola system. A theoy must descibe all data self consistently. In Section a simple suggestion is made fo the development of a philosophically self consistent elativity based on ECE theoy.. PERIHELION PRECESSIONS FOR NEARLY CIRCULAR ORBITS IN THE NEWTONIAN APPROXIMATION Conside the Lagangian in plane pola coodinates {, } fo obital motion in a plane: whee the educed mass is: ( ) ( ) () = µ θ + V mm µ= m+ M If m << M then the educed mass is nealy the same as m. The Eule Lagange equations ae: = d () θ dt θ and = d (4) dt and the total angula momentum is conseved: L = m = θ = constant (5) θ Fom Eq. (4): F θ ( ) = : = f ( ) (6) m Fom Eq. (5): L ħ = = = m θ constant (7) Conside small deviations {} fom nealy cicula obits such as the obit of a planet in the sola system. Then fom Eqs. (6) and (7): x ħ = f ( av + x) ; x: = av (8) ( + x) av ()

3 Fo small x, a Maclauin expansion gives: and fo a nealy cicula obit: so Eq. (6) becomes: Fom Eqs. (9) and (): Refutation of Geneal Relativity ħ x f x f ( av )+ ( av ) x (9) av av av () () f ( av ) f av x + ( ) x = () av av This is a hamonic oscillato equation with oscillation peiod: In the appoximation (): f f T = π av ħ θ av av f av = ( ) The angle by which θ inceases between a maximum and minimum of is the apsidal angle {}, and the time needed fo this is T/. The apsidal angle fo elliptical obits fo example is π. In geneal the apsidal angle is: ψ av / / () (4) = θ T (5) so: / av f ψ= π + (6) f av In ode fo the obit to be closed the apsidal angle has to be a ational function of Ψ. Fom Eqns. (5) and (6) π ψ = fo f ( )= c n / (7) + n and fo the invese squae law: then ( ) n = (8) ψ = π (9)

4 4 M. W. Evans and H. Eckadt The EGR theoy uses a metic based philosophy in which foce is not defined initially {-} but self inconsistently aives at a foce law using the classical Eule Lagange equations of this section. It uses a hugely elaboate method to calculate the peihelion pecession fom this foce law {}. Recently this method has been shown to be iddled with eos {-}, and also to poduce unphysical singulaities. The EGR method is well known and accepted to be neglect spacetime tosion and to be based on the wong field equation and incoect second Bianchi identity. It poduces a foce law of the type: so the apsidal angle fom this foce law is, fom Eq. (6): If: f ( k )= 4 () ψ= π + / ( kav ) + / ( kav ) / () k av () then and the apsidal angle advances by: In EGR: / ψ π + π + kav kav () ψ = π (4) k av LMG k = MG, =, (5) mc whee M is the mass of the attacting object (such as the sun), G is Newton s constant and the total angula momentum in the non elativistic appoximation. In the Newtonian appoximation {} L =α m MG (6) whee α is the half ight latitude of an elliptical obit. So: ψ= π α MG (7) c av In one complete evolution π the peihelion advances by: 6πMG α θ= ψ= (8) c av

5 Refutation of Geneal Relativity 5 Now use: α min =, + α max =, and so fo an appoximately cicula obit: (9) () and Fo an elliptical obit: and so: av min max α, () θ = 6πMG αc ( ) α= () a () α a (4) This gives: The esult given in efeence {} is: Fo the Eath: θ = 6πMG ac 6πMG θ = ac ( ) (5) (6) = (7) and so Eq. (5) is adequate. In pevious wok {-}: = x (8) θ π( ) so x = (9) k av Howeve, it is known fom pevious wok that the Einstein foce law () does not poduce the tue pecessing ellipse: α (4) cos xθ = + ( )

6 6 M. W. Evans and H. Eckadt The foce law needed fo Eq. (4) is the sum of tems invese and invese cubed in : In the appoximation: the apsidal angle fom foce law (4) is: which is appoximately: ψ= π and the peihelion advances by: L Lx F = ( x ) m m α (4), α (4) ψ= π + F / F / (4) x α π + ( ) ( α) ( + α) x (44) x x ( ) πα x θ = x In pevious wok it was shown that the peihelion advances by: so: and: which has the popeties: (45) = x (46) θ π( ) x x = x ( ) α x = ± + 8 / α 4 α The equivalent esult fom the Einstein theoy is: α (47) (48) as x (49) GM π( x)= 6π (5) ac so: GM x( Einstein)= (5) ac Taking the positive oot in Eq. (48) then, using: α (5)

7 Refutation of Geneal Relativity 7 and: Finally use: to find: x( tuepecessing ellipse) α (5) L a, α mmg x L mmga which is completely diffeent fom the Einstein esult: L x( Einstein)= αmca Theefoe EGR neve gives the tue pecessing ellipse, QED. Theefoe it is impossible to accept the vaious claims that EGR is a pecise theoy. In addition it is possible to show as follows that the EGR esult is given by a small petubation of a Newtonian potential. Conside the geneal expansion {}: = a n= a n n ( ) n ( ) (54) (55) (56) p cosθ p cos θ, a (57) in tems of Legende polynomials. In a plane: π θ= θ = (58) so: and: Theefoe: cosθ= cosθ = (59) p( )=, p( )=, p( )=, p( )=, p4 ( )= (6) 8 9 = + a 4 a a +..., a, (6) a geneal esult of mathematics which can be applied to calculate the aveage potential: φ= MG a (6)

8 8 M. W. Evans and H. Eckadt in the Newtonian theoy. Hee M is a mass at the oigin and m a mass sepaated fom M by a, a modulus o magnitude with mean distance vecto and fluctuations a. The potential is: MG φ= + a + a (6) 4 64 and the foce pe unit mass is: F d MG a a = = + 4 φ m d (64) 4 64 If: a (65) the foce is: mmg F a + 4 which is the type of potential encounteed in EGR in which: mmg L F = + ( mc ) Eqs. (66) and (67) ae the same if: Using: a L = mc (66) (67) (68) L =α m MG (69) and defining the obsolete Schwazschild adius as: then Eq. (68) becomes: MG = (7) c a = α (7) A small petubation (7) of a Newtonian obit will poduce the same pecession as EGR. If the obit is nealy cicula the apsidal angle is: ψ= π + df F d / and fo an EGR foce of type (67) poduces the peihelion pecession (7) 6πMGα = = = θ π α a π c (7)

9 Refutation of Geneal Relativity 9 This esult means that a peihelion pecession of any kind can always be thought of as a petubation of a Newtonian esult, without the need fo EGR. In Section it will be shown that a peihelion pecession of any kind can always be attibuted to a Catan spin connection, without the need fo EGR. In note 4(6) accompanying this pape on an example of this type of petubation theoy is given by consideing the sun of mass M, the planet Mecuy of mass m, and the planet Venus of mass m. The total potential is: whee: mmg mmg V = a Expanding Eq. (74) in tems of Legende polynomials: (74) a (75) mmg mm G V( )= + 4 a (76) a a 4 64 The Newtonian foce is: V mmg mm G F( )= = (77) a a a 6 and in the appoximations of note 4(6) poduces the apsidal angle: 45 ψ π m + + M a 4 a The pecession of the peihelion of Mecuy due to the influence of Venus is theefoe: m 45 Dθ π + M a 4 a whee and a ae the distances espectively of Mecuy and Venus fom the sun. Eq. (79) is the same as that deived by Fitzpatick {}, a check on the method and appoximations used. Using Eq. (79) and the data fo the sola system given by Fitzpatick himself in ef {}, a table of peihelion advances can be dawn up fo the vaious planets. These esults ae given in Table. The esults in Table ae completely diffeent fom those claimed by Fitzpatick, even though the same equation and same data ae being used. This does not give much confidence in the methods used in calculating these peihelion advances. It is claimed in standad physics that these Newtonian peihelion advances can be calculated with geat accuacy, and lead to a shotfall o anomaly when compaed with the obseved peihelion advance. This assumed anomaly is calculated with EGR and it is claimed that EGR poduces the anomaly with geat accuacy. Howeve, a slight petubation in (78) (79)

10 M. W. Evans and H. Eckadt Table Planetay Pecessions fom Eq. (79) in Ac Seconds a Centuy. Planet m/m T R (au) Δθ(T=) Δθ(T=.4) Mecuy Venus Eath Mas Jupite Satun Uanus Neptune negligible negligible the distance of a planet fom the sun would change the anomaly. The same philosophy should be used thoughout, in standad physics the Newtonian philosophy is used fo the planetay petubations and also fo the main equinoctial pecession, and EGR is used fo the anomaly. This is a nonsense. If EGR wee applied self consistently, Eq. (74) would change to: mmg mgl mmg mgl V = mc a mc a fo the equinoctial pecession and evey planetay petubation. Howeve Eq. (8) is not applied coectly in standad physics. The only EGR equation used is, fo a given planet: mmg mgl V = mc and all othe calculations ae Newtonian. This is not self consistent. Fo the sake of agument howeve, assume that the last tem in Eq. (8) is small. Then the foce is: mmg mgl mm G 9 F = + 4 mc a a a 6 and as in note 4(7) the total pecession fo a given planet is: π m 6πMG θ = + M a c Fo Mecuy the second tem fom EGR is the anomaly, and is the well known 4 acseconds a centuy. In standad physics this is simply added to the pecession due to the effect of the gavity of Venus on Mecuy. So an EGR tem is added to a Newtonian tem, wheeas both tems should be eithe EGR o Newtonian. Also the shape of the sun poduces a pecession, and small changes in the shape of the sun may completely change the expeimental anomaly, so EGR would be compaed with the wong expeimental data. (8) (8) (8) (8)

11 Refutation of Geneal Relativity In addition to these citicisms the standad physics assumes that the total foce due to N planets is: mmg mmg mmg F =... (84) a a This foce is assumed to account fo the petubations of planets m, m,... on m. The foce between m and the sun M is counted only once, and the EGR effect is also counted only once, to give: π m = + + m 6πMG θ.. + (85) M a M a c so the EGR coection is applied to many Newtonian coections. It is by no means clea that these fundamental assumptions ae logical. The easons ae given in detail in note 4(7). It is easily seen that the poblems with EGR multiply if the EGR coection is applied to some standad calculations in astonomy {}. Fo example the nutation of the eath is claimed in the standad physics to be due to a Newtonian potential of the type: mmg MG( T T ) V = + sin θ (86) whee m is the mass of the eath, M the mass of the sun, the mean distance between the eath and sun, and θ=. 44 (87) The elevant moments of inetia of the eath in this calculation ae: 7 T = 8 4. kg m (88) 7 T = 8 8. kg m (89) because the eath is a symmetic top. The foce fom Eq. (86) is: V F = k = (9) 4 whee: MG k = MG, = ( T T ) cos θ (9) so fom Eq. (4) the peihelion pecession due to nutation is: ( ) 6π T T θ = cos θ (9) m Evey centuy the eath s obit moves backwads by. 4 ac seconds because it is a symmetic top and nutates in its obit. Note caefully that this esult is entiely standad.

12 M. W. Evans and H. Eckadt Howeve, it is Newtonian, and if EGR wee valid the potential (86) should be ( ) mmg LMG mg T T V = + sin θ (9) mc If it is assumed that the elativistic coection to the quadupole tem is small, then the foce is: mmg LMG MG F = ( T ) 4 T sin θ (94) mc so the peihelion pecession becomes: ( ) θ= π α MG T T 6 cos θ (95) c m The eath s obit is nealy cicula so and the peihelion pecession of the eath due to EGR is: α (96) θ = 86. ac secod pe centuy (97) This is about fou odes of magnitude geate than the Newtonian effect due to the quadupole tem and in the opposite diection. This is obviously not obseved in astonomy, the pecession due to nutation being known accuately. What happens in the standad model is that the EGR effect is consideed to be a completely diffeent phenomenon and is neve associated with nutation. This is illogical and self inconsistent. EGR cannot be eseved fo one phenomenon isolated fom all othes, and as soon as it is used with all phenomena it poduces absud esults. Anothe clea efutation of EGR can be agued by consideing the eath as an oblate spheoid o symmetic top with a Newtonian potential enegy: V ( mmg )= + (98) whee = S R R (99) Hee R is the eath s equatoial adius (about 4, miles) and R the diffeence between the equatoial and pola adii (about miles). The foce fom Eq. (98) is: mmg mmg F( )= () 4 Assume that thee is a satellite in equatoial obit, then fom Eqs (4) and () its peihelion pecesses by: 6π θ = R R () 5

13 evey obit of the satellite, i.e. by Refutation of Geneal Relativity 6π 7 θ = 5. () adians pe obit. Howeve this is a Newtonian calculation and if EGR wee valid it should be coected to a potential enegy: mmg LMG V( )= + mc () so the peihelion pocession would be coected to: Fo the eath: so the effect on a satellite due to EGR is: MG θ= 6π + c (4) MG = 44. m (5) θ =. 7 4 / (6) ac seconds pe obit. An obit such as that of Gavity Pobe B was 65 km above the eath s suface. This is a distance of 4, miles plus 65 km above the cente of the eath, i.e m above the cente of the eath. This gives an EGR effect of θ = 4. (7) ac seconds pe obit. In one thousand obits this would incease to.4 ac seconds. This is a vey lage effect and should have been obseved by Gavity Pobe B with its well known high esolution. Howeve the only thing obseved by Gavity Pobe B was a Thomas pecession o geodetic dift of 6.6 ac seconds pe yea (many obits of Gavity Pobe B). The pecession due to the shape of the eath is.56 ac seconds pe obit at 65 km above the suface. It is assumed that this was taken into account by Gavity Pobe B. Consideing all the eos being uncoveed hee this seems like a big assumption. Poceeding in this way note 4() collects the esults of some EGR coections when applied self consistently to well known textbook {, } Newtonian calculations in astonomy. Fo example the moon is a satellite of the eath in a nealy cicula obit and the fact that the eath is a symmetic top causes a peihelion pecession of 4.7 adians pe obit of the moon. It is assumed that this is a well obseved effect in astonomy. Howeve this is again a Newtonian calculation and the EGR coection in this case is almost as lage,. adians pe obit of the moon (7 days). The lagest contibution to the pecession of planets aound the sun is the equinoctial pecession, which fo the eath is 5,9. ac seconds pe centuy, which is.9697 pe centuy. So π adians ae coveed in: 6 T = = 5, 77 yeas (8). 9697

14 4 M. W. Evans and H. Eckadt In UFT9 the equinoctial pecession was consideed to be due to the gavitomagnetic field. It can also be calculated staightfowadly by assuming that the sun is a symmetic top descibed by the Newtonian potential (98), which gives a peihelion pecession of the eath aound the sun of: The distance fom the eath to the sun is: and the equatoial adius of the sun is: 6π θ = (9) = 49. m () so R= m () R= 5. 8 m () would give a pecession of 5,9. ac seconds pe centuy. Compaed with this the EGR pecession of the eath is only.85 ac seconds pe centuy, and the combined planetay petubations ae of the ode of a tenth of the equinoctial pecession. Howeve all these calculations ae Newtonian with the exception of.85 ac seconds pe centuy. If EGR wee valid all the calculations should be EGR calculations. It is also clea that a change in R of the sun by less than one pat in a thousand would poduce the so called anomaly on which the test of EGR is based. In standad physics the EGR effect is consideed in isolation, i.e. is not associated with peihelion pecession. Similaly EGR is consideed in isolation of the effect of a petubing planet on the eath s obit. Citicisms such as these ae detailed in note 4(). The EGR potential: mmg MGL V( )= mc () should be used in all the Newtonian calculations, and if this is done, the EGR coection would occu many times. In standad physics howeve it is applied only once, and then only to a claimed expeimental anomaly.. SELF CONSISTENT ECE THEORY OF PRECESSION Conside the Catan tosion {-} a a a a b a b T = q q + ω q ω q (4) µ ~ µ ~ ~ µ µ b ~ ~ b µ whee q a is the Catan tetad and μ ωa the spin connection. By antisymmety: μb ( ) a a a b T = q + ω q (5) µ ~ µ ~ µ b ~ Fo the sake of simplicity conside the element:

15 Refutation of Geneal Relativity 5 and the indices: ( ) b T = q + ω q (6) ~ ~ b ~ b =,~ = (7) so: Define the gavitational potential as: It follows that the foce is: T = q + ω q (8) ( ) φ= φ ( ) q (9) Assume that the gavitational potential is Newtonian: φ= mmg If the spin connection is defined by: the foce is: F = φ Ω φ () Ω= L mc mmg MGL F = 4 mc () () () which is the foce law of EGR. It is known that this EGR foce law is incoect, and is being used hee only to show that it can be poduced by a paticula choice of spin connection in ECE theoy. Fo an appoximately cicula obit: Ω= whee the obsolete Schwazschild adius is defined by: The foce is theefoe: which gives the peihelion pecession: (4) MG = (5) c mmg F = + Ω (6) 6πMG θ = Ω = c (7)

16 6 M. W. Evans and H. Eckadt All obseved pecessions ae poduced within one self consistent philosophy in which the potential is Newtonian, but in which the foce contains the spin connection. This appoach will be developed in futue wok. ACKNOWLEDGMENTS The Bitish Govenment is thanked fo a Civil List Pension and the AIAS and othes fo many inteesting discussions. Dave Buleigh is thanked fo posting, and Alex Hill and Robet Cheshie fo tanslation and boadcasting. REFERENCES [] M. W. Evans, Ed., Definitive Refutations of the Einsteinian Geneal Relativity (Cambidge Intenational Science Publishing, CISP, ), Special Issue Six of efeence (). [] M. W. Evans, Ed., J. Found. Phys. Chem., (CISP six issues a yea). [] M. W. Evans, S. J. Cothes, H. Eckadt and K. Pendegast, Citicisms of the Einstein Field Equation (CISP, pepint on [4] M. W. Evans, H. Eckadt and D. W. Lindstom, Geneally Covaiant Unified Field Theoy (Abamis 5 to ) in seven volumes. [5] M. W. Evans, H. Eckadt and D. W. Lindstom, papes in the Sebian Academy of Sciences (-) and othe jounals. [6] L. Felke, The Evans Equations of Unified Field Theoy (Abamis 7, pepints in English and Spanish on [7] M. W. Evans and L. B. Cowell, Classical and Quantum Electodynamics and the B() Field (Wold Scientific ). [8] M. W. Evans and S. Kielich, Moden Nonlinea Optics (Wiley, 99, 99, 997, ) in two editions and six volumes. [9] M. W. Evans and J.-P. Vigie, The Enigmatic Photon, (Kluwe, Dodecht, 994 to ) in ten volumes hadback and softback. [] M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum Field Theoy (Wold Scientific 994). [] J. B. Maion and S. T. Thonton, Classical Dynamics (Hacout Bace, New Yok, 988, d. Ed.). [] R. Fitzpatick, An Intoduction to Celestial Mechanics (Cambidge Univesity Pess, ).