Communiating Speial Relatiity Theory s Mathematial Inonsistenies Steen B Bryant Primitie Logi, In, 704 Sansome Street, San Franiso, California 94111 Stee.Bryant@RelatiityChallenge.Com Einstein s Speial Relatiity Theory is belieed to be mathematially onsistent. Here we find subtle and diffiult to detet mathematial mistakes in eah of Einstein s deriations. In his 191 paper, the mistake is traed to the misuse of set algebra instead of statement algebra. Speifially, he uses the = relation on the Real set instead of the = relation on the Binary set, inorretly establishing the equialene of equations. Beause the two = relations operate on different sets, they annot be used interhangeably. In his 1905 paper, he begins the deriation of the transformations with the equation ξ = τ. Mathematially, his final transformation equations fail the internal alidity hek beause the stand-alone equation he deries for τ does not always equal ξ τ =. This mistake is traed to the mistreatment of the time funtion, whih is the result of the partial differential equation, as an equation rather than as a funtion. Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies Einstein s Speial Relatiity Theory (SRT) is well-onfirmed experimentally and is generally aepted as internally onsistent. 1 It is generally aepted that any attempt to modify or disproe [SRT] has to rely upon either the quantitatie preditions of different experimental results or the disoery of an internal logial inonsisteny. Quantitatie results based on experimental hallenges 3,4 an only separate theories into two lasses, those that are onsistent with the results and those that are not. It is unlikely that an experiment will inalidate SRT without the establishment of a new theory that is onsistent with the existing results while being mathematially distinguishable from SRT in its quantitatie preditions. Logial onsisteny hallenges to SRTT5,6 are largely based on the analysis of the impliations and theoretial preditions assoiated with the SRT equations (e.g., time dilation, moing loks and twins). These hallenges, while initially onining, hae not disproed SRT sine the paradoxes and impliations hae already been explained by the sientifi ommunity. 7 Logial onsisteny hallenges an also be based on the mathematial onsisteny of the deriations. Mathematial hallenges to SRT are infrequent beause of the amount of srutiny plaed on the theory 8 and the general aeptane that SRT is mathematially orret. 9 Howeer, a mathematial hallenge stands the greatest hane of suess beause of its preision. Mathematial arguments are unambiguous and always stik to the preise mathematial definition, regardless of any olloquial usage. 10 The sope of Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 3 this paper is to ommuniate mathematial inonsistenies in eah of Einstein s SRT equation deriations. Suess Criteria Suess riteria must be established for a mathematial hallenge to proe suessful. Therefore, the following riteria are suggested. 1. The steps performed mathematially outweigh any supporting written text desribing those steps. In other words, the hallenge must be mathematial in nature and annot be based on non-mathematial meanings assoiated with the equations or terms.. The steps must iolate existing mathematial rules or fail to adhere to internal alidity heks. With the suess riteria defined, we now show the inonsistenies in Einstein s deriations. Einstein s 1905 Deriation Consider the equation a = * b, where is a onstant and a and b are ariables. Importantly, and mathematially, if a is known and was deried as the equation a = * b, b an always be found by diiding a by the onstant (e.g., a b = ). This proides a means to hek for internal onsisteny. Speifially, if we diide a by the Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 4 onstant to independently arrie at b, and find that the result is different than if we simply used an equation to arried at b, we hae found a mathematial error sine b annot simultaneously hae two different alues. Consider the steps Einstein performs in reating the transformation equations. In 3 of his 1905 manusript, he states that 11 ξ = τ. Eq. 1 x He begins with the equation, ξ = τ, and sine he earlier found that τ = α ( t ), x he replaes τ with the expression α ( t ), produing x ξ = τ = α ( t ). Eq. Einstein then states that x t = and replaes t with the expression x, resulting in x x ξ = τ = α( ), Eq. 3 whih Einstein simplifies to x ξ = τ = α Eq. 4 or x ξ = τ = α. Eq. 5 Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 5 Einstein s deriation of the ξ equation is presented in Figure 1. Soure: Annalen der Physik 17, 891 (1905) FIG 1. Einstein begins with the equation x transformation equations, resulting in ξ = τ = α. ξ = τ as the foundation for deriing his x Yet, in reating the stand-alone τ equation, he only simplifies α ( t ) by replaing x with x t to produe x t τ = α Eq. 6 Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 6 Notably, in produing the τ equation, he does not first replae t with the expression x. Einstein ompletes his ξ deriation by replaing x with x t, and then multiplies all of the equations by to produe the normalized equations ξ = τ = x t Eq. 7 and x t τ =. Eq. 8 Einstein proides a means to test the alidity of the equations by using the equation ξ = τ. We hae found that x t ξ = τ = and τ = x t. While ξ should always equal τ, we find that generally 1 x t x t. Thus, Einstein s 1905 deriation is not mathematially onsistent using the rules of modern algebra. Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 7 The root ause of the problem in Einstein s 1905 deriation is the mistreatment of x τ = α ( t ) as an equation rather than as a funtion. 13 In fat, the author has shown that Einstein s time transformation, τ = x t, is inorret 14 and should be alulated as * τ = x t. This error might hae been disoered sooner had Einstein x written the linear funtion as τ ( x, y, z, t) = α ( t ), onfirming τ as a funtion. Readers familiar with Speial Relatiity may want to assoiate the Greek ariables ξ and τ with a wae-front and not with fixed point transformations. This paper s finding, while ontrary to the aepted understanding, implies that the transformation equations are speifi instanes of the wae-front equations. This important exploration is beyond the sope of this paper, but is explained in Reexamining Speial Relatiity 15 and in Understanding and Correting Einstein s 1905 Time Transformation. 16 x t x * This equation is more properly written as τ = α or τ = α. 1 1 Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 8 Einstein s 191 Deriation Consider the equations a = b = d. Eq. 9 Now onsider these equations rewritten as a b = 0 d = 0. Eq. 10 It is easy to mathematially show that Equations 10 are mathematially equialent to Equations 9. In other words, any solution that satisfies Equations 10 will also satisfy Equations 9, and isa-ersa. What happens if we assoiate Equations 10 with one another, to produe a b = d? Eq. 11 While at first appearane, Equations 10 an be ombined using the transitie relation, it is mathematially inorret for us to state that Equation 11 is an expression of the equialene of the two statements omprising Equations 10. The equialene of Equations 10 is mathematially expressed using statement algebra, as a b = 0 d = 0. Eq. 1 Tehnially, Equations 10 annot be assoiated with one another using the transitie relation. The transitie relation definition is x, y, z A, if x = y and y = z then x = z. Sine 0 is a onstant in Equations 10, it annot take on all alues in the set and therefore does not satisfy the onditions for the transitie relation. Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 9 The use of the = relation in Equation 11, instead of the operator, to assoiate Equations 10 with one another requires us to explore set algebra, speifially equialene relations. Sine statements, as expressed by Equations 9 and 10, are either True or False, they are members of the Binary set. This allows us to use the equialene relation = for the Binary set. In this ase, the "=" relation assoiates a b = 0 with d = 0 suh that ( a b = 0) = ( d = 0). Eq. 13 Mathematially, "=" is the operator and it takes two Binary operands, resulting in the finite relation set {(True, True), (False, False)}. There is an infinite solution set for the ariables a,b,, and d that make this relation possible. The information that a b must equal 0 and that Binary set. b must equal 0 is ontained within the = relation set on the Equation 11 also uses the = relation to reate a statement. Howeer, the "=" operator takes Real operands, not Binary operands. (Compare " a b " with a Real result ersus " a b = 0 " with a Binary result). Thus, the resulting infinite relation set is {( x, x) x Real}. There is an infinite solution set for the ariables a, b,,and d that make this relation possible. The information that a b must equal 0 and that b must equal 0 is neer established. Beause the set defined by the "=" relation in Equation 11 is infinite and operates on the Real set, it annot be used as a replaement for the set defined Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 10 by the "=" relation used in Equation 13 that is finite and operates on the Binary set. Thus, Equation 11 does not expresses the equialene of a b = 0 with d = 0. Finally, we must determine whether this analysis hanges if Equation 11 is written as a b = l( d) Eq. 14 where l is a generalized multiplier. The introdution of the generalized multiplier does not hange the result sine this statement still requires the = relation to operate on the Real set instead of the Binary set. We alidate this finding by heking if Equation 1 is equialent to Equations 14 suh that [ a b = 0 d = 0] [ a b = l( d)]. Eq. 15 All solutions for the ariables a,b,,and d in Equation 1 are solutions to equations 14, where is l a generalized multiplier that is able to take on all Real alues. Howeer, the In order to better illuminate this mathematial finding, readers with a programming bakground that inludes C should onsider that the following two oerloaded operator== methods would not be onfused with one another. Assume a Binary (a.k.a. Boolean) type and the oerloaded AddToRelationSet methods exist. Binary operator==( Binary op1, Binary op ) { if (op1==op) AddToRelationSet( op1, op ); return op1==op; } Binary operator==( Real op1, Real op ) { if (op1==op) AddToRelationSet( op1, op ); return op1==op; } Both methods may at first appear equialent to one another beause eah tests two operands, adds them to a set if they aren t already members, and returns a Binary result. Howeer, the two methods are not equialent. The ompiler uses the operand type of the inoking statement to orretly identify whih one of the operator== methods would be alled. The set reated from the operator== taking Binary arguments would ontain, at most, two entries, while the set reated from the operator== taking Real arguments would not hae an upper boundary on the number of entries. It is a programming error to use one of these sets as a replaement for the other. Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 11 reerse is not true. All solutions for the ariables a,b,, d and l in Equations 14 are not solutions to Equations 1. Consider the ase where l is 0, whih is a required ondition if the equations are equialent. As presented in Equation 15, equialene requires that Equation 1 imply Equations 14, and isa-ersa. Sine Equations 14 does not always imply Equations 1 (e.g., l = 0, a = 10, b = 10, = 10, and d = 11), the statements are not equialent. Thus, Equation 15 is False, and the use of Equation 14 to express the equialene of Equations 1 is a mathematial mistake. In his 191 manusript, Einstein begins his deriation by stating that 17 x x y y z z = t = t. Eq. 16 These equations are in the form a = b and = d. He then rewrites them to produe 18 x x y y z z t = 0 t = 0 Eq. 17 whih are in the form a b = 0 and d = 0. Einstein ombines these equations to produe 19 λ ( x y z t ) = x y z t, Eq. 18 whih he states expressed the equialene of the two statements omprising Equations 17. 0 Equation 18 is a mathematially inorret statement for the equialene of Sine we hae already shown that Equation 1 is equialent to Equation 10, then in a similar manner, Equation 14 is not equialent to Equation 10. Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 1 Equations 17 based on our preious analysis. The equialene of Equations 17 an be expressed using statement algebra as Eq. 19 0 0 = = t z y x t z y x or using set algebra as. Eq. 0 0) ( 0) ( = = = t z y x t z y x Importantly, Equation 18 an be produed by using the transitie relation on Eq. 1 g t z y x g t z y x = = ) ( λ whih are not the equations for spheres, exept of ourse in the speial ase where g is 0 and is 1. Equations 1 hae a larger solution set than Equations 17. λ Einstein ontinues his deriation to find a solution for Equations 18 using his Minkowski-based matrix. 1 While his matrix omputation may be orret, Einstein s 191 deriation is mathematially inonsistent beause he has already iolated the mathematial equialene rules. Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 13 Einstein s Relatiity Book Deriation Einstein begins the deriation in Appendix 1 of his Relatiity book by stating that x = t x = t. Eq. As with his 191 deriation, these equations are in the form a = b and = d. He then finds 3 x t = 0 x t = 0 Eq. 3 whih are in the form a b = 0 and d = 0. Equations and 3 are equialent. He then mathematially onludes that 4 λ ( x t) = x t Eq. 4 expresses the equialene of the two statements omprising Equation 3. Again, using the preious analysis, we hae shown that this is a mathematially inorret statement of the Equialene of Equation 3. The equialene of Equations 3 an be expressed using statement algebra as x t = 0 x t = 0 Eq. 5 or using set algebra as ( x t = 0) = ( x t = 0). Eq. 6 Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 14 Equation 4 an be produed by using the transitie relation on λ( x t) = g x t = g, Eq. 7 whih has a larger solution set than Equations 3. Equation 7 is equialent to Equations 3 only in the speial ase where g is 0 and deriation is inalidated beause he iolated the equialene rules. λ is 1. As in his 191 manusript, his entire Conlusion The goal of this paper was to highlight and ommuniate the mathematial inonsistenies in Einstein s deriations. We hae analyzed Einstein s deriations to reeal mathematial errors in eah. Eah hallenge satisfies the suess riteria sine they are not based on the olloquial meaning of any ariable or equation and an be mathematially erified. While Einstein s equations proide preditions that are suffiiently lose to the experimental results, 5 the theory itself is hallenged on mathematial onsisteny grounds. No amount of experimental eidene an make an internally inonsistent theory orret, no matter how lose the theory may be in its preditie harateristis. A omplete analysis of the mathematial orretion, the meaning of the orreted equations, and the relationship of the orreted equations to existing experimental results are beyond the sope of this paper and are explained in Reexamining Speial Relatiity. Importantly, the author s alternatie theory, as presented in Reexamining Speial Relatiity, remains Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 15 onsistent with the experimental results while being mathematially distinguishable from SRT. 1 Personal e-mail orrespondene with Dr. G. Lombardi, 003. Ibid. 3 A. Mihelson and E. Morley, Amerian Journal of Sienes 34, 333 (1887). 4 H. Ies and G. Stilwell, J. Opt. So. Am. 8, 15 (1938) 5 P. Daies, About Time - Einstein's Unfinished Reolution (Touhstone, New York, 1995), Chap., p.55,59-69. 6 H. Dingle, Siene at the rossroads (Martin, Brian and O'Keefe Ltd, London, 197) 7 Daies, About Time - Einstein's Unfinished Reolution (Touhstone, New York, 1995), Chap., p.55,59-69. 8 G. Lombardi, 003, op. it. (see referene 1). 9 H. Dingle, Nature 17, 19-0 (1968), (Obtained in the publi domain at <http://www.heretial.om/siene/dingle.html> ). 10 E. Bloh, Proofs and Fundamentals: A first ourse in Abstrat Mathematis (Birkhäuser, Boston, 003), Chap. 1-11 A. Einstein, Annalen der Physik 17, 891 (1905), (German ersion obtained in publi domain at <http://www.wiley-h.de/berlin/journals/adp/890_91.pdf> and the English translation obtained in publi domain from <http://www.fourmilab.h/etexts/einstein/sperel/www/> ). 1 S. Bryant, Reexamining Speial Relatiity: Reealing and orreting speial relatiity s mathematial inonsisteny (Currently unpublished) 13 Ibid. Copyright 005 Steen Bryant.
Communiating SRT s Mathematial Inonsistenies 16 14 Ibid. 15 Ibid. 16 S. Bryant, Understanding and Correting Einstein's 1905 Time Transformation (Submitted to Galilean Eletrodynamis, April 005) 17 A. Einstein, Einstein's 191 Manusript on the Speial Theory of Relatiity (George Braziller, In, New York, 1996,003), p.4-8. 18 Ibid. 19 Ibid. 0 Ibid. 1 Ibid. A. Einstein, Relatiity - The Speial and the General Theory (Three Riers Press, 1961). 3 Ibid. 4 Ibid. 5 S. Bryant, 005, op. it. (see referene 1). Copyright 005 Steen Bryant.