On the Logical Inconsistency of the Special Theory of Relativity. Stephen J. Crothers. 22 nd February, 2017

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1 To ite this paper: Amerian Journal of Modern Physis. Vol. 6 No pp doi: 0.648/j.ajmp On the Logial Inonsisteny of the Speial Theory of Relatiity Stephen J. Crothers thenarmis@yahoo.om nd February 07 Abstrat: Einstein s Speial Theory of Relatiity requires systems of lok-synhronised stationary obserers and the Lorentz Transformation. Without both the Theory of Relatiity fails. A system of lok-synhronised stationary obserers is proen inonsistent with the Lorentz Transformation beause it is Galilean. The Speial Theory of Relatiity insists that Galilean systems must transform not by the Galilean Transformation but by the non-galilean Lorentz Transformation. The Theory of Relatiity is therefore inalid due to an intrinsi logial ontradition. Keywords: Speial Relatiity Simultaneity Stationary Obserers Clok-Synhronised Obserers Lorentz Transformation Time. Introdution It has reently been proen by Engelhardt [] that Einstein s method of synhronising loks in his Speial Theory of Relatiity is inonsistent with the Lorentz Transformation. This inonsisteny is in fat due to an inherent logial ontradition in Speial Relatiity beause Einstein inorretly assumed that systems of loksynhronised stationary obserers are onsistent with the Lorentz Transformation. Now howeer as we know how to judge whether two or more loks show the same time simultaneously and run in the same way we an ery well imagine as many loks as we like in a gien CS. The loks are all at rest relatie to the CS. They are good loks and are synhronised whih means that they show the same time simultaneously. [ 3] We hae so far defined only an A time and a B time. We hae not defined a ommon time for A and B for the latter annot be defined at all unless we establish by definition that the time required by light to trael from A to B equals the time it requires to trael from B to A. Let a ray of light start at the diretion of A and arrie again at A at the A time t' A. In aordane with definition the two loks synhronize if t B - t A = t' A - t B. [3 3] We enisage a rigid sphere of radius R at rest relatiely to the moing system k and with its entre at the origin of oordinates of k. A rigid body whih measured in a state of rest has the form of a sphere therefore has in a state of motion - iewed from the stationary system - the form of an ellipsoid of reolution with the aes R R R. Thus whereas the Y and Z dimensions of the sphere (and therefore of eery rigid body of no matter what form) do not appear modified by the motion the X dimension appears shortened in the ratio : ( - / ) i.e. the greater the alue of the greater the shortening. [3 4] Therefore ( ) τ = t = t t whene it follows that the time marked by the lok (iewed in the stationary system) is slow by - ( - / ) seonds per seond [3 4]. Thus with the help of ertain imaginary physial eperiments we hae settled what is to be understood by synhronous stationary loks loated at different plaes and hae eidently obtained a definition of simultaneous or synhronous and of time. The time of an eent is that whih is gien simultaneously with the eent by a stationary lok loated at the plae of the eent this lok being synhronous and indeed synhronous for all time determinations with a speified stationary lok. [3 ] In I mathematially onstrut a system of stationary obserers and apply the Lorentz Transformation to proe that no obserer an be lok-synhronised. In 5 I mathematially onstrut a system of lok-synhronised obserers and apply the Lorentz Transformation to proe that not all obserers are stationary. That systems of lok-synhronised stationary obserers are logially inonsistent with the Lorentz Transformation entirely suberts the Theory of Relatiity.. Systems of Stationary Obserers A system of lok-synhronised stationary obserers is an essential feature of Speial Relatiity. Einstein [3 3] holds that the Lorentz Transformation assoiates oordinates y z t of his stationary system K with the oordinates ξ η ζ τ of

2 Stephen J. Crothers: On the Logial Inonsisteny of the Speial Theory of Relatiity his moing system k. A system of lok-synhronised stationary obserers and the Lorentz Transformation are the bases for Einstein s time dilation and length ontration. It is regarded in general by physiists [4.] that a stationary system of obserers k whih are lok-synhronised when at rest are not synhronised when they all moe together with respet to a lok-synhronised stationary system K as illustrated in figure. obserer and 0. All the obserers are stationary. All obserers hae an Einstein lok reading the time t. The time t assoiated with obserer loated at must now be quantified by means of the Lorentz Transformation. The equation for ξ in () annot be used for this purpose beause it leads to a ontradition onerning the presene of arbitrarily many obserers. Therefore only the equation for τ in () an be used. Therefore set t = t t = (3) whene t ( ) = t +. (4) Then using the relation for ξ of the Lorentz Transformation () Figure. All the synhronised loks in the stationary system K read the same time at all positions in the K system. All the loks in the moing system k do not read the same time aording to the K system despite being synhronised with respet to the k system. Only at = ξ = 0 do the loks depited read the same time in both systems where t = τ = 0. In figure loks to the left of the entral lok in the moing system k are ahead of the entral lok and those to the right of it lag it aording to the stationary system K where all the loks therein always read the same time t. After a time t > 0 in K the moing obserers loks adane to the right and the hands on the moing loks adane but aording to any obserer in K they do not read the same time. As time t inreases all the hands of the stationary loks adane by the same amount and all obserers loks in K still read the ommon time t they are synhronised. Howeer for any and t in the stationary system K there is always an obserer loated at someplae with a lok that reads t thereby ontraditing Einstein s assumption of synhronisation of loks in his stationary system K. Reall the Lorentz Transformation equations for the time τ at position ξ in the moing system k aording to the stationary system K: τ = β η = y ς = z ( t ) ( t ) ξ = β β =. To ensure a system of stationary obserers K by mathematial onstrution set () = () where R denotes the obserer the loation of that ξ = β ( t ) = β + t β = β + t leading to some sample alues in table. TABLE : Sample alues of obserer loation and assoiated time aording to the stationary system K. t ξ τ t ½ ( ) t 3 3 t + (5) β t β t ( + ) t β β t β t t β ( ) t ( ) t + t β t β t β t β t β ( 3 ) t β t

3 Amerian Journal of Modern Physis 07; X(X): XX-XX 3 The Stationary Lorentz Transformation is τ = β ( t ) = ξ = β ( t ) = β + t β ( ) t = t + η = y ς = z β = R. From table for any time t 0 of the stationary system K there are any number of loations at whih there is a stationary obserer whose lok reading is t t thereby ontraditing Einstein s assumption of lok synhronisation for all obserers in the stationary system K. Moreoer eery stationary obserer in K obseres the same time τ on the loks in k. This is onsistent with two different interpretations both of whih are opposite to those of Einstein: (a) Aording to eery obserer in the stationary system of obserers K all the loks in the system k are synhronised but not stationary beause system k is moing and all the loks in the system K are stationary but not synhronised. (b) Eery stationary obserer at position in the system K obseres the same lok in k finding that it indiates the same time τ for them all but also find that the obsered lok is at a different loation ξ in k. Either way Einstein s system of lok-synhronised stationary obserers is inonsistent with the Lorentz Transformation. Suppose an eent ours in the stationary system of obserers K at at assoiated time t. Then aording to all other obserers in K the eent ours at t t. No eents in K an be simultaneous aording to any of its obserers. Aording to all the obserers of system K all the obserers in the moing system see the eent in K simultaneously. This is reealed mathematially by the Inerse Stationary Lorentz Transformation: (6) ( ξ ) t = β τ + ξ = ξ = β ( ξ + τ ) = β + ξ + τ β ( ) ξ τ = τ y = η z = ς β = R. None of the stationary obserers in k an be loksynhronised and all see the same time t at all obserer loations in the moing system K for any eent at ξ at time τ in k whih annot be simultaneous for any obserer in k. For a system of stationary obserers relatiity of simultaneity is opposite to that adaned by Einstein from his Speial Theory of Relatiity. His relatiity of simultaneity obtains effetiely from his unwitting restrition of obserers to just one obserer = in table in his stationary system K and letting it speak for all obserers in K due to his false assumption that a system of lok-synhronised stationary obserers is onsistent with the Lorentz Transformation. Setting = in (6) the Lorentz Transformation used by Einstein is reoered. A system S that ontains only one obserer annot by its singular harater synhronise its lok with anything or judge simultaneous eents in S. Physiists sine Einstein hae only eer inoked the ase of obserer = in table ; in other words they hae only eer onsidered this one obserer in the system K on the inorret assumption that any desired system of obserers an be a system of lok-synhronised stationary obserers onsistent with the Lorentz Transformation. From this inorret assumption they onlude that the onditions and effets are the same for all obserers in the system K; hene a ommon time dilation and length ontration relatie to any and all obserers of a moing system. Table howeer reeals that this is not the ase. 3. Length Contration for a System of Stationary Obserers (7) At any instant of time in the stationary system K desribed by equations () (7) let a rigid rod in the moing system k hae a length ξ = l 0 when at rest relatie to the moing system k. From (6) Therefore ( ρ ) ξ = l0 = ξ ξρ = =. β β 3

4 4 Stephen J. Crothers: On the Logial Inonsisteny of the Speial Theory of Relatiity = β l = 0 l 0 Hene all the obserers of the stationary system K obsere not length ontration but length etension. 4. Time Dilation for a System of Stationary Obserers By means of the Inerse Stationary Lorentz Transformation (7). (8) t τ = τ = = t. (9) β Thus although no obserer in the stationary system K is lok-synhronised eery obserer of the stationary system K obseres the same time interal in K and the same time-dilated interal in k but at the epense of length ontration and of lok-synhronised stationary obserers. This is irreonilable with Einstein s theory. 5. Systems of Clok-Synhronised Obserers The loations of the obserers assoiated with the ommon time t = t for all alues of must now be quantified by means of the Lorentz Transformation. The equation for ξ in () annot be used for this purpose beause it leads to a ontradition onerning the presene of arbitrarily many obserers. Therefore only the equation for τ in () an be used. Therefore set τ τ t = + = + β β τ = τ (0) where labels an obserer and is the loation of that obserer in the lok-synhronised system K. Thus all read the same time t. Soling (0) for gies From (0) ( ) τ Putting () into () gies = +. () β τ = β t. () ( ) t = +. (3) Although all obserers in K are lok-synhronised by (0) to a ommon time t only is not a funtion of the time t. Thus only is stationary. All other obserers in K annot be stationary. The Clok-Synhronised Lorentz Transformation is ( t ) τ = β = τ ξ β ( t ) ( ) t = + η = y ς = z = β = (4) The Inerse Clok-Synhronised Lorentz Transformation is ( ) t = β τ + ξ = t β ( ξ τ ) ( ) τ ξ = + ξ y = η z = ς = + β =. (5) Although all obserers in k are lok-synhronised by (5) to a ommon time τ only ξ is not a funtion of the time τ. Thus only ξ is stationary. All other obserers in k annot be stationary. From (4) whene From (5) whene d dt Combining (6) and (7) ( ) = < < (6) ( ) dξ = < dτ < +. (7) < < +. (8) 4

5 Amerian Journal of Modern Physis 07; X(X): XX-XX 5 All obserers in the lok-synhronised system K obsere that none of the loks in system k are synhronised. Conersely all obserers in the lok-synhronised system k obsere that none of the loks in system K are synhronised. Setting = in (4) reoers the Lorentz Transformation used by Einstein. It pertains to only one obserer in system K and one obserer in system k. By assuming systems of loksynhronised stationary obserers onsistent the Lorentz Transformation Einstein inorretly permitted the obserer = to speak for all obserers. 6. Length Contration for a System of Clok-Synhronised Obserers From (4) at any instant of time t Therefore ( ) ξ = ξ ξ = β ρ ρ ( ) = β ρ = β ξ = = ξ. β This is Einstein s length ontration equation. Cloksynhronised obserers and ρ of the system K see the same length ontration at the epense of being stationary obserers. 7. Time Dilation for a System of Clok-Synhronised Obserers Sine all obserers are lok-synhronised with respet to their own systems all obserers in the K system obsere the ommon lok time-interal t. Obserer of system K wathes the lok of ξ in the moing system k and obseres the lok time-interal τ of ξ in system k. Then from (4) τ = β t. (9) Thus after a time interal t in K any obserer in the lok-synhronised system K reads the time-interal τ at ξ in the k system. Eah obserer obseres a different time and a different time-interal on the orresponding lok held by obserer ξ in system k. For eample the obserer = loated at in system K obseres not time dilation at ξ but time epansion at ξ : τ = β t. The obserer = /β loated at /β in system K obseres no hange in the time-interal of the lok of ξ /β in system k: τ /β = t. The obserer = /β loated at β obseres the time interal τ β at ξ to β be t τ = = t (0) β β whih is Einstein s time dilation equation. Note that in all ases τ = τ in aordane with (4) sine τ = β t. Conersely all obserers ξ in the k system read a ommon time τ and ommon time interal τ finding that the lok at in K reads from (5) the time interal 8. The Systems of Obserers t = β τ. () To failitate omparison of obserers write (6) with apital letters τ = β X ( t X ) ξ = β ( X t ) = β + X t β ( ) X t = t + η = y ς = z = X β = R. () Now set =. The stationary obserer = is loated at X. This obserer is not lok synhronised with any other obserers in its stationary system K. Now set = in (4). The obserer although stationary (the only stationary obserer in its system) is lok synhronised with all other obserers in its lok-synhronised system K. Moreoer the range on in () is not the same as in (4) sine in the latter the alues of are onstrained by the speed and the speed of light. Thus obserers loated at X and are not equialent obserers. Nonetheless owing to his false assumption Einstein makes them not only the same but also the speaker for all other obserers in his stationary system K effetiely making eah of his two systems K and k systems that ontain only one obserer in eah. To reaffirm that only the obserer = is ontained in Einstein s theory set the position in the stationary system (6) equal to the position of the lok-synhronised system (4) ( ) t = +. Thus the only obserer is = for eah system. Similarly set the time for the stationary system (6) equal to the time for the lok-synhronised system (t = t in the latter sine its loks are all synhronised) ( ) t = t + = t = t. 5

6 6 Stephen J. Crothers: On the Logial Inonsisteny of the Speial Theory of Relatiity Thus one again only the obserer = is present. 9. Conlusions By his lok synhronisation method Einstein attempted to ensure that time at all plaes within a gien stationary system of obserers is the same despite subsequently inoking the Lorentz Transformation. Yet lok-synhronised stationary systems of obserers are inonsistent with the Lorentz Transformation. Speial Relatiity is thereby inalid due to an insurmountable logial ontradition. Systems of loksynhronised stationary obserers are Galilean. Einstein [3 ] defined time by means of his loks. Howeer time is no more defined by a lok than pressure is defined by a pressure gauge speed by a speedometer heat by a thermometer or graity by a spring. Measuring instruments are inented to measure something other than themseles. Einstein's loks measure only themseles. By defining time by his loks Einstein detahed time from physial reality. Nonetheless all tetbook writers on the subjet reiterate Einstein s false assumptions for eample [4-8]. They all suffer neessarily from the same logial inonsisteny as Einstein s 905 paper. Referenes [] W. Engelhardt Physis Essays [] S. J. Crothers Einstein s Anomalous Clok Synhronisation [3] A. Einstein On the eletrodynamis of moing bodies Ann. Phys [4] D. J. Griffiths Introdution to Eletrodynamis 4 th Ed. PearsonEduation In. 03 [5] A. Einstein The Meaning of Relatiity Prineton Uniersity Press 988 [6] A. Einstein Relatiity the Speial and the General Theory Methuen London 954 [7] E. F. Taylor J. A. Wheeler Spaetime Physis nd ed. W. H. Freeman and Company New York 99 [8] R. d Inerno Introduing Einstein s Relatiity Oford Uniersity Press 99 [9] B. W. Carroll D. A. Ostlie Modern Astrophysis Addison Wesley Longman 996 [0] R. A. Mould Basi Relatiity Springer 994 [] R. C. Tolman Relatiity Thermodynamis and Cosmology Doer Publiations In.New York 987 [] S. H. Radin R. T. Folk Physis for Sientists and Engineers Prentie Hall In. 98 [3] D. MMahon Relatiity Demystified MGraw-Hill 006 [4] B. Shutz A First Course in General Relatiity Cambridge Uniersity Press 009 [5] A. Shadowitz Speial Relatiity Doer Publiations In. 988 [6] W. Rindler Introdution to Speial Relatiity Clarendon Press 98 [7] W. Pauli The Theory of Relatiity Doer Publiations In. New York 98 [8] R. E. Turner Relatiity Physis Routledge & Kegan Paul 984 6