Separating Equivalent Space-Time Theories

Transcription

1 Apeirn, Vl. 6, N. 3, July Separating Equivalent Space-Time Theries Stephan J. G. Gift Department f Electrical and Cmputer Engineering Faculty f Engineering The University f the West Indies St. Augustine, Trinidad, West Indies Stephan.Gift@sta.uwi.edu A cmplete set f Equivalent Space-Time Theries based n a grup f equivalent transfrmatins has been studied by Selleri. This equivalent set f space-time theries includes Special Relativity Thery invlving the Lrentz transfrmatins and a semi-classical Abslute Space Thery based n the generalized Galilean r inertial transfrmatins. In rder t separate these theries, the light speed predictins f the cmplete set are cmpared with the relative light speed fr light emanating frm I a satellite f Jupiter determined by direct calculatin in the space-time framewrk f these theries. Keywrds: Equivalent space-time theries, special relativity thery, abslute space thery, Lrentz transfrmatins, Inertial transfrmatins 009 C. Ry Keys Inc.

2 . Intrductin Apeirn, Vl. 6, N. 3, July The accepted thery f space and time is the Special Relativity Thery fr which the equivalence f all inertial frames and light speed invariance are fundatin pstulates [-4]. Despite this acceptance, there has been cntinuing interest in a semi-classical ether thery invlving a preferred r abslute frame where light speed varies with mvement relative t the preferred frame. Thus fr example Gagnn et. al. [5] studied this semi-classical Abslute Space Thery in which light prpagates istrpically in a preferred reference frame. These authrs referred t this abslute space mdel as the Generalized Galilean Transfrmatin since it invlves the classical Galilean transfrmatins adjusted t take int accunt the real effects f the Fitzgerald-Larmr-Lrentz (FLL) cntractins first experimentally cnfirmed by Ives [6-8]. In these FLL cntractins, a rd f length l in a preferred frame when mving with speed v relative t that preferred frame, is shrtened t a length l given by ( v c ) l l (.) and a system f frequency f when statinary in the preferred frame, has a reduced frequency f given by ( v c ) f f (.) bth changes resulting in ( x vt ), y y, z z, t t x γ γ (.3) Here x, y, z, t are the crdinates f space and time in the preferred reference frame, x, y, z, t are the crdinates in a reference frame mving at speed v relative t the preferred frame, c is the speed 009 C. Ry Keys Inc.

3 Apeirn, Vl. 6, N. 3, July f light in the preferred reference frame and γ is the FLL cntractin factr given by ( v ) γ c (.4) This Abslute Space Thery (AST) has been extensively investigated by ther researchers [9-] wh have highlighted the clse agreement between such a thery and Special Relativity Thery (SRT) fr nearly all predicted effects. Selleri in particular [9, 3] has shwn that the SRT and AST are members f a cmplete set f theries that differ nly accrding t the clck-synchrnizatin cnventin emplyed. This set is based n equivalent transfrmatins that incrprate the experimental facts f the cnstancy f the twway speed f light and clck retardatin. Selleri [3] has shwn that the entire set f theries makes the same predictins fr several phenmena and used acceleratin in an attempt t argue that the AST gives the best descriptin f the physical wrld. In rder t mre cnvincingly separate the theries, we nte that the theries in the set make different light speed predictins. We therefre cmpare these different light speed predictins fr light frm I a satellite f Jupiter detected n Earth with the light speed determined by direct calculatin in the space-time framewrk f these theries.. Equivalent Theries f Space and Time [3] Cnsider an inertial system S with space and time crdinates x, y, z, t in which the speed f light is c, and anther inertial system S having space and time crdinates x, y, z, t which is mving at speed v relative t S alng the x-axis. The tw systems are 009 C. Ry Keys Inc.

4 Apeirn, Vl. 6, N. 3, July cincident att t 0. Selleri [3] has shwn that the cmplete set f transfrmatin laws frm S t S is f the frm, ( ) x f x vt (.a) y g y (.b) z g z (.c) t e x + e4t (.d) where the factrs f, g, e, e4 can depend n the velcity v f S measured in S. The cnstancy f the tw-way velcity f light [4, 5] and experimentally established clck retardatin [6] reduce the generality f transfrmatins (.) by requiring f (.a) g (.b) e 4 + ev (.c) where v / c. As a result, transfrmatins (.) becme t x vt0 x (.3a) t + e y y (.3b) z z (.3c) ( x vt ) (.3d) 009 C. Ry Keys Inc.

5 Apeirn, Vl. 6, N. 3, July where e is nw the nly unknwn factr. The ne-way velcity f light ( S) relative t S in the directin f v that satisfies these cnditins is given by [3] c ( S) (.4) + + ce The transfrmatins (.3) represent the cmplete set f equivalent theries. Different theries in the set are btained by selecting different values f e which cnstitute different clcksynchrnizatin cnventins. The Lrentz transfrmatin and SRT result as a particular case when e / c (.5) and the Generalized Galilean r Inertial transfrmatin and AST result frm e 0 (.6) In rder t determine the remaining unknwn factr e, we directly evaluate light speed relative t S and cmpare the result with that predicted in (.4) by each thery f the set. 3. Light Speed Calculatin using Jupiter s Occulting Satellite I Fllwing Selleri [3], let Jupiter s satellite I be in a state f mtin alng the x axis f an inertial frame S. The Earth mves with cnstant speed v relative t S and cnstitutes anther inertial frame S. At the timet measured in S, I emits a light signal (ccultatin) when it is at the pint x I as determined in S. At this instant let the 009 C. Ry Keys Inc.

6 Apeirn, Vl. 6, N. 3, July psitin f the Earth as measured in mtin in S yields E ( T ) vt S be x ( ) E T 009 C. Ry Keys Inc. Its equatin f x (3.) Therefre the distance d in S between I and the Earth at the instant the light is emitted is given by d xe ( T ) xi vt xi x be the Earth s psitin in S crrespnding t x ( ) Let ( ) E T (3.) E T S and let x I be I s psitin in S crrespnding t x I. Then using the transfrmatin (.a), x T f x T vt (3.3) where E ( ) ( E ( ) ) x f ( x vt ) (3.4) I I f (.a) Therefre, nting frm (3.3) and (3.4) that x E ( T ) and x I are fixed crdinates in S that are independent f the time as measured in S, the distance d in S between I and the Earth at the instant the light is emitted is given by d x E ( T ) x I f( xe ( T ) xi ) ( xe ( T ) xi )(3.5) Frm (3.), equatin (3.5) becmes d d (3.6) in

7 Apeirn, Vl. 6, N. 3, July which can be written as d ( vt xi ) (3.7) In the right hand side f (3.7), all f v, c, T, xi are measured in S. Therefre as Selleri [3] has bserved fr many phenmena including length f rds mving with respect t S, the distance d in (3.7) bserved n the Earth is exactly the same fr all equivalent theries independently f the factr e. The time t r in S when the signal reaches the Earth is given by [3] ct xi tr (3.8) Frm (3.8), the elapsed time Δtmeasured in S is given by vt xi Δ t tr T (3.9) Using (3.) and (.d), the time T in S (i.e. n Earth) at which I emits the signal is given by T e x E ( T ) + e4t ( ev + e4 ) T (3.0) But Therefre e v + e4 (.c) T T (3.) The time t r indicated by the clck n Earth when the signal is received is given by 009 C. Ry Keys Inc.

8 Apeirn, Vl. 6, N. 3, July (3.) tr e xre + e4tr where xre is the Earth s psitin recrded in S at the time f receptin f the signal. Based n the equatin f mtin f the Earth, this psitin is given by This gives x re vt r (3.3) ( e v + e ) tr tr t r 4 (3.4) Using (3.8), the elapsed time t Δ measured in S is given by Δt t T Substituting fr Δt using (3.9) gives r ( t T ) Δt (3.5) 009 C. Ry Keys Inc. r ( vt x ) I Δt (3.6) In the right hand side f (3.6), all f v, c, T, xi are measured in S. Hence as ccurs fr many phenmena, the elapsed time Δt in (3.6) measured n the Earth is exactly the same fr all equivalent theries independently f the factr e. Using (3.6) fr Δt and (3.7) fr d, we can nw calculate the CAL f the received light relative t the Earth as speed ( ) determined frm Earth. This is given by d cr ( CAL) Δt (3.7) Cnsistency demands that the relative light speed ( CAL) calculated in (3.7) be equal t the relative light speed ( S) predicted by the set f theries in (.4). This requires that

9 Apeirn, Vl. 6, N. 3, July c c + + ce + It fllws frm (3.8) that (3.8) + + ce + (3.9) which fr v c yields e 0 (3.0) The cnditin e 0 crrespnds t the AST which frm (.4) predicts relative light speed ( AST ) as c ( AST ) (3.) + the same light speed value as that calculated in (3.7). In all ther theries f the set, e 0 giving predicted relative light speed ( SET ) as In particular fr SRT, c (3.) relative light speed ( ) ( SET ) + + e c e / c and hence SRT s predicted SRT is ( SRT ) c (3.3) Interestingly, the ne-way light speed invariance suggested by (3.3) has nt been cnfirmed despite numerus experimental tests [7]. The cnditin e 0 means that the AST is the nly thery f the set f equivalent theries that predicts a relative light speed AST CAL, a ( ) which is equal t the directly calculated value ( ) 009 C. Ry Keys Inc.

10 Apeirn, Vl. 6, N. 3, July value that has been experimentally cnfirmed [8]. All the ther theries including SRT which crrespnd t e 0 are therefre incnsistent and hence seem unable t represent the physical wrld since fr them cr ( SET ) cr ( CAL). If SRT is t remain the accepted thery f space and time, this incnsistency must be remved. 4. Cnclusin In this paper, light speed predictins by Einstein s Special Relativity Thery, the Abslute Space Thery and all ther members f the cmplete set f equivalent theries develped by Selleri [3] were cmpared with the directly calculated light speed value fr Jupiter s cculting satellite I. Of these the SRT predicts relative light speed ( SRT ) c in accrdance with its fundatin light speed invariance pstulate while the AST predicts relative light speed ( AST ) cnsistent with a preferred frame. The AST was the nly thery whse predictin was in agreement with the directly calculated relative light speed value ( CAL) crrespnding t a clck synchrnizatin parameter e 0. On this basis, the Abslute Space Thery with its preferred frame appears t be the best descriptin f physical space and time. References [] Einstein, A. On the Electrdynamics f Mving Bdies, in The Principle f Relativity by H.A. Lrentz, A. Einstein, H. Minkwski and H. Weyl, Dver Publicatins, New Yrk, 95. [] French, A.P., Special Relativity, Nelsn, Lndn, C. Ry Keys Inc.

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