An Elucidation of the Symmetry of Length Contraction Predicted by the Special Theory of Relativity

Transcription

1 pplied Phsis Researh; Vol 9, No 3; 07 ISSN E-ISSN Published b Canadian Center of Siene and Eduation n Eluidation of the Smmetr of ength Contration Predited b the Speial Theor of Relativit Chudaiji uddhist Temple, Isesaki, Japan Koshun Suto Correspondene: Koshun Suto, Chudaiji uddhist Temple, Isesaki, Japan Tel: koshun_suto9@mbrniftom Reeived: pril 7, 07 epted: pril 8, 07 Online Published: Ma 3, 07 doi:05539/aprv9n3p3 UR: bstrat In this paper, onsider a rod (inertial frame ) and rod (inertial frame ) moving at onstant veloit relative to eah other ssume that the lengths of two rods are equal when the are stationar ording to the STR, when length in the diretion of motion of rod, moving at onstant veloit, is measured from inertial frame, the rod ontrats in the diretion of motion lso, the time whih elapses on lok in inertial frame is delaed ompared to the time whih elapses on lok in inertial frame If, onversel, inertial frame is measured from inertial frame, rod ontrats in the diretion of motion, and the time whih elapses on lok is delaed However, aording to lassial ommon sense, if rod ontrats when measured from inertial frame, then rod measured from rod must be longer than rod Thus, this paper disusses the smmetr of rod ontration, and eluidates this problem It is found, based on the disussion in this paper, that the ontration of a rod inludes true phsial ontration, and relativisti ontration obtained due to measurement using the method indiated b Einstein However, in the STR, an two inertial frames are equivalent, and therefore is not possible to aept points suh as the fat that reasons for ontration are different This paper onludes that STR is not a theor whih desribes the objetive state of realit Kewords: Speial Theor of Relativit, Classial Stationar Sstem, Classial Moving Sstem, Relativisti Stationar Sstem, ength Contration, Veloit Vetor Introdution In the era of lassial phsis, as eemplified b Newtonian mehanis, it was thought that phsial laws eist independentl of the eistene of human beings The role of phsis was to disover phsial laws, and desribe them in the language of mathematis Now, onsider a rod (inertial frame ) and rod (inertial frame ) moving at onstant veloit relative to eah other ssume that the lengths of two rods are equal when the are stationar ording to the STR, when length in the diretion of motion of rod, moving at onstant veloit, is measured from inertial frame, the rod ontrats in the diretion of motion lso, the time whih elapses on lok in inertial frame is delaed ompared to the time whih elapses on lok in inertial frame If, onversel, inertial frame is measured from inertial frame, rod ontrats in the diretion of motion, and the time whih elapses on lok is delaed ording to Einstein s priniple of relativit, the two inertial frames are equivalent, and thus the same results are obtained no matter whih inertial frame measurement is arried out from The essene of STR is the smmetr of the theor However, aording to lassial ommon sense, if rod ontrats when measured from inertial frame, then rod measured from rod must be longer than rod The author has alread disussed the smmetr of time dela in another paper (Suto, 06-07): The smmetr of length ontration is derived quantitativel from the formula for orentz transformations However, the STR does not eplain the reason wh the rod ontrats Thus, this paper disusses the smmetr of rod ontration, and eluidates this problem 3

2 aprsenetorg pplied Phsis Researh Vol 9, No 3; 07 Setions to 5 are preparator stages for setion 6 Setion disusses the priniple of onstan of light speed adopted b Einstein Setion 3 eplains the method of disriminating between a lassial stationar sstem and lassial moving sstem Setions 4 and 5 disuss the ontration whih an be predited from a lassial perspetive Here, the term lassial is used when disussing objetive realit whih eists regardless of the observer Setion 6 emplos the method of lok snhronization proposed b Einstein It also eluidates the smmetr of rod ontration b appling the priniple of relativit The Priniple of Constan of ight Speed E Introdued b Einstein When Einstein developed the STR, he assumed the priniple of relativit and the priniple of onstan of light speed The latter inludes the following two priniples n ra of light moves in the stationar sstem of oordinates with the determined veloit, whether the ra be emitted b a stationar or b a moving bod (Einstein, 93): et a ra of light start at the time t diretion of, and arrive again at at the time t In agreement with eperiene we further assume the quantit from towards, let it at the time t be refleted at in the =, t t to be a universal onstant the veloit of light in empt spae (Einstein, 93): In this paper, we distinguish between the former priniple as the priniple of onstan of light speed I and the latter priniple as the priniple of onstan of light speed II The priniple of onstan of light speed I asserts that the light speed in vauum does not depend on the speed of the light soure The priniple of onstan of light speed II asserts that the light speed alulated from the round-trip travel time is onstant et there be a given stationar rigid rod of length as measured b a ruler whih is stationar, and assume that the rod is plaed along the positive diretion of the stationar sstem s -ais ssume that loks and of the same tpe are set up at points and on the rear and front end of this rod Here lok will be abbreviated as C, and lok as C Suppose a ra of light is emitted in the diretion of from at time t of C, reahes and is refleted at at time t of C, and then returns to at time t of C Einstein determined that if the following relationships hold between these times, then the two loks represent the same time b definition (Einstein, 93): t t = t t () ( ) t + t t = () If the relationship in Equation () does not hold for the times of C and C, then it is neessar to adjust the time of C so that the relationship in () holds (tuall, either lok an be adjusted) Net, assume that the stationar rod has been aelerated, and has attained the onstant veloit v (see Figure ) Figure rod is moving at onstant veloit v relative to stationar sstem Clok and are set up at and at eah end of this rod, and the times of eah of these loks are snhronized while the sstem is stationar 3

3 aprsenetorg pplied Phsis Researh Vol 9, No 3; 07 Then the time C must be adjusted again so that the relationship in Equation () holds between the times C and C Due to this operation, the light speed on the outward and return paths measured in the moving sstem of the rod is measured as on both paths Considered lassiall, an inertial frame in whih light propagates isotropiall is a stationar sstem, and an inertial frame in whih light propagates anisotropiall is a moving sstem However, if lok time is adjusted aording to the requirements of Einstein, light propagates isotropiall at the same speed in all inertial frames (Relativisti isotropi propagation) lso, all inertial frames beome stationar sstems in the sense of the theor of relativit In this paper, the priniple introdued b Einstein is alled the "priniple of onstan of light speed E" (where "E" stands for Einstein) That is, Priniple of onstan of light speed E: In all inertial frames, light speed of the outward path and return path is onstant () This priniple is not a universal priniple, but a personal priniple introdued b Einstein To maintain this priniple, the observer in a stationar sstem must adjust the time on a lok eah time the veloit of a moving sstem hanges If the observer neglets this task, the priniple of onstan of light speed E is no longer a priniple 3 Classial ength Contration Derived b ppling Priniple of Constan of ight Speed I and II et us imagine that times t, t, t of this moving sstem orresponds to times t, t, t of the stationar sstem Now when the time required for the light signal emitted from point at the rear of the rod to travel from point to point is measured with the lok in moving sstem, it is ( t t) lso, if this time is measured with the lok in the stationar sstem, it is epressed as ( t ) ording to the STR, the rod seen from stationar sstem ontrats b / times in the diretion of motion lso, the observer in stationar sstem applies the "priniple of onstan of light speed I" to the propagation of light emitted from moving sstem, and thus ( t ) is given b the following equation t = (s), = ( v / ) / (3) ( v) lso, the time ( t ) required for the light signal to return from point to point is given b the following equation t ' = (s) (4) ( + v) However, the denominator on the right side of Equations (3) and (4) does not signif that the light speed hanges ording to the STR, the relationship of ( t t ) and ( t ) is: t t ( ) = t t Here, if the right side of Equation (3) is substituted for ( t ) in Equation (5), (5) ( + v) t t = (s) (6) Similarl, if the time ( t' t) whih passes on the lok in moving sstem while the light signal returns from point to point is, ( v) t' t = (s) (7) If we set t = 0 to simplif the equation, t' beomes the time whih passes in moving sstem while the light signal makes a round trip between and Thus, the observer in moving sstem determines that the time of C when the light has arrived at is t' / This time an be found from Equations (6) and (7) That is, t' = t t + t t ( ) ( ) (8a) = (s) (8b) 33

4 aprsenetorg pplied Phsis Researh Vol 9, No 3; 07 However, sine ( + v)/ > /, the time on C must be later than the time on C to resolve this disrepan If this adjustment time is taken to be Δ t, Δ t = ( t t) t' (9a) v = (s) (9b) If the time of C is delaed b v / (s), then a state is ahieved where the times of C and C an be said to be simultaneous in moving sstem t the time Δ t = v/ (s), it an be determined that the oordinate sstem where the rod was initiall stationar was the oordinate sstem where light propagates isotropiall In this paper, this oordinate sstem is defined as the lassial stationar sstem S l The l subsript of Sl is taken to mean a lassial inertial frame Two loks whose times have been snhronized in S l math in an absolute sense On the other hand, at the time Δt v/ (s), it an be determined that the oordinate sstem where the rod was initiall stationar was the oordinate sstem where light propagates anisotropiall (Suto, 00): In this oordinate sstem, the priniple of onstan of light speed II holds, but the priniple of onstan of light speed E does not hold In this paper, this oordinate sstem is defined as the lassial moving sstem S l The ause of anisotropi propagation of light in S l is the veloit vetor attahed to this oordinate sstem (Suto, 05): The author has previousl presented a thought eperiment for disriminating between S l and S l However, Einstein believed it was impossible to disriminate these inertial frames through eperiment lso, Einstein proposed that the time on two loks in an inertial frame be adjusted so that the relationship in Equation () holds s a result, the speed of light beame for both the outward path and return path, even in S l lso, all inertial frames beame equivalent in the sense of the theor of relativit (a stationar sstem S re in the sense of the theor of relativit) The following summarizes the above: Classial phsis Speial theor of relativit Classial stationar sstem Sl Classial moving sstem S l Relativisti stationar sstem S re Now, how should we imagine S l? In the latter half of the 9th entur, it was thought that a medium was needed for light to propagate as a wave The phsiists at the time alled this medium the aether However, Einstein eliminated the aether from the STR, and thus disussion of the eistene of this hpothetial substane graduall disappeared However, if the priniple of onstan of light speed I holds, then there needs to be a medium for transmitting light as a wave Thus, this paper looks at the pairs of virtual partiles and antipartiles whih onstitute the vauum The ountless relative veloities between the S l oordinate sstem and the ountless virtual partile pairs in the viinit are indiated as vetors, and then omposed If the size of the vetor beomes zero at this time, then the oordinate sstem is S l where light propagates isotropiall On the other hand, if the omposed vetor has magnitude, then the oordinate sstem is S l where light propagates anisotropiall 4 ength Contration and Time Dela Eplainable using Classial Considerations Consider a laborator whose interior floor is a square The Mihelson interferometer is plaed in this laborator (see Figure ) t the enter of the room, there is a glass plate (beam splitter) P with a semi-transparent metal oating on its front fae The angle between this glass plate and the -ais is 45 ight emitted from the light soure S strikes this glass at an angle, and the light is split in two One beam passes through the plate, strikes a mirror M, is refleted, and retraes its path to the splitting point P On the seond light path, the beam is refleted b the glass plate P, arrives at mirror M, is refleted there, and returns to the splitting point P (Onl the essential parts of the eperimental instrument are shown here Equipment not needed for the disussion in this paper has been omitted) This laborator is moving at onstant veloit v along the -ais of S l The light path length PM measured indoors is taken to be and the path length PM is taken to be (However, in measurements in the laborator, and are equal) In addition, the light path length when is measured from S l is taken to be, and the light path length when is measured from S l is taken to be (However, and are equal) 34

5 aprsenetorg pplied Phsis Researh Vol 9, No 3; 07 Figure This figure shows the view from above of a laborator moving at onstant veloit with respet to S l Here, the time required for light to make a round trip over PM is measured from S l If this round trip time is taken to be t, then the observer in S l applies the priniple of onstan of light speed I to this light propagation, and thus: t = + = = v + v v ( v / ) Net, the time for light to make a round trip over PM is measured If this round trip time is measured in taken to be t, then: (0) S l and t = v ( / ) / () The method of deriving Equation () is eplained in man tetbooks so here it is omitted (Fenman, 963; Frenh, 968): Inidentall, the predited effet ould not be deteted from the Mihelson-Morle eperiment This means that t and t are equal In the end, the following relationship an be derived from Equations (0) and () = () Here, and are equal, so Equation () an be written as follows = (3) When measured from S l, the laborator ontrats b / times in the diretion of motion This ontration is phsial ontration due to the fat that some fore has ated on the laborator, and this an be regarded as true ontration (ontration I) Inidentall, an observer in the oordinate sstem S l of the laborator applies the priniple of onstan of light speed II to this light propagation, and thus the round trip times of light t and t are predited as follows: t = (4) t = (5) In the end, t elapses in S l while t elapses in S In addition, l = and thus Equation () an be written as follows: t = (6) 35

6 aprsenetorg pplied Phsis Researh Vol 9, No 3; 07 Net, if this is ompared with Equations (5) and (6): t = t (7) When observed from S, the time whih elapses in l S l is delaed ompared to the time whih elapses in S l tuall, this predition has been verified b eperiments where the life of elementar partiles is etended In the end, spae ontration and time dela in S l an be predited if the priniples of onstan of light speed I and II are assumed 5 Rod Contration whih an and annot be Classiall Eplained In this setion, the lengths of rod (inertial frame ) and (inertial frame ) moving at onstant veloit relative to eah other are measured using two tpes of methods ) Two methods for an observer in inertial frame ( S l ) to measure the length of rod moving at onstant veloit Measurement In this ase, observer is at the rear end and observer is at the front end of rod of length plaed on the -ais of S l lso, at an arbitrar time, a light signal is emitted from a point light soure S plaed in the enter of rod That light signal propagates isotropiall from S, and arrives at both ends of the rod with absolute simultaneit t this time, observers and read off the position of both ends of rod from the oordinates (This -ais is parallel to the -ais) Sine rod ontrats b / times in the diretion of motion, the length of rod read off from the -ais beomes if we refer to Equation (5) From this, ength of rod : ength of rod, Here, if the length of rod measured from : :, > (8) S l is taken to be, then Equation (8) an be written as follows; = (9) Contration in this ase is a result of the fat that some phsial fore ated on rod, and this an be alled true ontration (ontration I) Measurement First we onsider rod moving at onstant veloit v along the -ais of S l (ength when the rod is at rest is ) When the front end of the rod passes in front of observer in S l, observer starts the stopwath, and measures the time t until the rear end of the rod passes ording to the STR, the rod ontrats b / times in the diretion of motion at this time That is, = vt = (0) The results obtained from measurement and verif the ontration in Equation (3) ) Two methods for an observer in inertial frame ( S l ) to measure the length of rod Measurement 3a In this ase, ontrar to measurement, observers at both ends of rod ompare the length of rod and with absolute simultaneit The loks are used at both ends of rod have been snhronized when the rod was at rest in S l If Equation (9) is taken into aount, the length of rod read off from the oordinates b observers at both ends of rod is shorter than rod That is, ength of rod : ength of rod, : : () Considered lassiall, if rod ontrats, then rod is longer than rod Measurement 4a This ase is the inverse of measurement method Observer on rod measures the length of rod of length plaed on the -ais of S l If observer measures the time required to pass both ends of rod, and this is taken to be t, then lassiall t is, t = () v However, the time whih passes in the oordinate sstem of rod is delaed ompared to the time whih passes in S l Therefore, the time t whih passes in S l beomes / times Equation () That is, 36

7 aprsenetorg pplied Phsis Researh Vol 9, No 3; 07 t = (3) v Inidentall, it is impossible for rod to ontrat beause rod began to move at onstant veloit Thus, the observer of rod determines that time elapsing in his own oordinate sstem is delaed, and he does not regard rod as having ontrated In lassial measurement, ontration of rod annot be observed 6 Contration of Rod Interpreted b orrowing Einstein s Measurement Method The measurement in this setion emplos the following operation and priniple used when developing the STR ) Times on the loks at both ends of rod moving at onstant veloit are snhronized so that the relationship in Equation () holds ) The priniple of relativit is applied to the oordinate sstem of rod Measurement 3b Net, the moving observer uses the same method as measurement method, and reads off the position of both ends of rod from the oordinate in S l Observer is at the rear end and observer is at the front end of the moving rod t an arbitrar time, a light signal is emitted from S in the enter of rod n observer in S l applies the priniple of onstan of light speed I to this light propagation When the light signal emitted from S has arrived at both ends of the rod, observers and read off the oordinates in S l Then the two observers of rod ompare the length of the oordinate the themselves read off, and the length of the stationar rod Now, the observer in Sl measures time until the light signals emitted from S arrives at the observers and at both ends of the rod If these times are taken to be t and t, then sine the distane from S to the rod end is rod /, t = ( + v) t = (5) ( v) Inidentall, the observer in Sl determines the following values for the distane traveled b the light signal until it reahes both ends of the rod Travel distane in the negative diretion of the -ais = t = ( + v) Travel distane + in the positive diretion of the -ais + = t = (7) ( v) The observers at both ends of rod obtain the following values as the length of the rod read off from the -ais of the stationar sstem, based on Equations (6) and (7) = + + =, < (8) Contrar to Equation (), the length of rod in this ase is longer than rod That is, ength of rod : ength of rod, : : (9) Inidentall, if the priniple of relativit is applied to the oordinate sstem of rod, the length of must math Equation (9) Thus, the observers on rod make the following judgment based on Equation (8) : : (30) When the length of rod is measured from the oordinate sstem of rod, rod is ontrated b / times in the diretion of motion (ontration II) (4) (6) 37

8 aprsenetorg pplied Phsis Researh Vol 9, No 3; 07 Measurement 4b n observer of rod who has applied the priniple of relativit believes that his own oordinate sstem is a stationar sstem Therefore, Equation (3) is eplained using the ontration of rod rather than a dela in the time of S l (ontration III) If it is assumed that the priniple of relativit holds in S l, then the results of measurement 3b and 4b math the values of measurement and 7 Disussion In setion 4, an observer in S l applied the priniple of onstan of light speed I, and an observer in S l applied the priniple of onstan of light speed II, to propagation of light emitted from a light soure in S l t this time, the length of the laborator measured b the observer in S l ontrated in the diretion of motion This ontration is phsial ontration whih ourred as a result of some fore having ated on the moving laborator, and is true ontration (ontration I) Net, in measurement 3a, loks snhronized in S l were used as the two loks in S l used for measurement Therefore, the times of the two loks mathed absolutel The observer in S l determined the rod to be shorter than rod (Equation ()) Thus, the method of lok snhronization proposed b Einstein was used in this paper s a result, in measurement 3b, rod was determined to be longer than rod (Equation (9)) However, even this is unaeptable, and therefore in this ase the priniple of relativit was applied to this oordinate sstem ording to the priniple of relativit, an two inertial frames are equivalent, and thus measured values must math In measurement 3b, the ratio of the lengths of rod and rod was interpreted as :/ (ontration II) However, atuall it is not the ase that rod has phsiall ontrated This is relativisti ontration whih ours when measurement is done using the method indiated b Einstein The dela in time whih elapses in the oordinate sstem of rod was observed in measurement 4a, but it was not determined that rod ontrated based on Equation (3) However, in measurement 4b the priniple of relativit was applied to inertial frame, and therefore, observer believed his own oordinate sstem to be a stationar sstem With regard to the fat that Equation (3) was obtained, it was determined to be the result of rod having ontrated This ontration is tentative ontration (ontration III) observed beause the passage of time in inertial frame was delaed With this, the values for measurements to 4 all math Now, how is this problem handled in the STR? The STR assumes the priniple of relativit If ontration of rod is observed in measurement and, then b definition ontration of rod is also observed in measurement 3b and 4b If phsis is a siene whih pursues the nature of realit as it is, then ontration II and ontration III annot be aepted In the end, the STR should be regarded not as something whih desribes phsial law eisting in the natural world, but as a mathematial epression of the universe as imagined b Einstein The veloit vetors present in the natural world are missing from the STR 8 Conlusion Through the disussion in this paper, it was determined that there are the following three tpes of ontration of a rod moving at onstant veloit Contration I (phsial ontration): This is the ontration obtained from measurements and, and it is true ontration due to fat that some phsial fore has ated on rod whih is moving at onstant veloit It was possible to eplain this ontration with the lassial disussion in setion 5 Contration II (relativisti ontration): Reasons wh ontration of rod was observed in measurement 3b: ) True ontration of rod whih is moving at onstant veloit (ontration I) ) Times of the loks at both ends of rod were adjusted to ahieve simultaneit in the sense of the theor of relativit 3) The priniple of relativit was applied to inertial frame It was possible to predit Equation (9) from ) and ) In addition, b appling the priniple of relativit, it was possible to interpret Equation (9) like Equation (30), and eplain the ontration of rod Contration III (relativisti ontration): The priniple of relativit was applied to inertial frame in measurement 4b Therefore, with regard to the fat that Equation (3) was obtained, it was determined to be the result of rod having ontrated In measurement and 3b, there was ontration I and II so results mathing the preditions of the STR were obtained lso, in measurement and 4b, ontration I and III are the reason wh smmetr of length ontration was observed 38

9 aprsenetorg pplied Phsis Researh Vol 9, No 3; 07 In the end, the fat that smmetr of rod ontration, whih is lassiall impossible, ould be eplained in this paper is due to the following three reasons ) The inertial frame was assumed to be a lassial stationar sstem ) n operational definition of simultaneit was used in measurement 3b 3) In measurement 3b and measurement 4b, the priniple of relativit was applied to inertial frame s a result, relativisti ontration II and III ourred, and the measurement results in measurement 3b and measurement 4b mathed the results of measurement and measurement The STR is an astonishing theor in whih rod, undergoing no hange in itself, is fored to ontrat This paper onludes that there should be serious disussion of whether or not the STR an reall be alled a phsial theor knowledgments I would like to epress m thanks to the staff at CN Translation Servies for their translation assistane lso, I wish to epress m gratitude to Mr H Shimada for drawing figures Referenes Einstein, (93) The Priniple of Relativit (pp 40-4) New York: Dover Publiation, In Fenman, R P (963) The Fenman etures on Phsis (pp 5-4) ddison-wesle Publishing Compan Frenh, P (968) Speial Relativit (p 55) New York & ondon: WWNORTON&COMPNY Suto, K (00) Violation of the speial theor of relativit as proven b snhronization of loks Phsis Essas, 3(3), Suto, K (05) Demonstration of the eistene of a veloit vetor missing from the speial theor of relativit Phsis Essas, 8(3), Suto, K (06) Thought Eperiment Revealing a Contradition in the Speial Theor of Relativit pplied Phsis Researh, 8(6), Suto, K (06) Eluidation of Time Smmetr Predited b the Speial Theor of Relativit IOSR Journal of pplied Phsis, 8(6) IV, Suto, K (07) New Problem with the Twin Parado pplied Phsis Researh, 9(), /aprv9np Coprights Copright for this artile is retained b the author(s), with first publiation rights granted to the journal This is an open-aess artile distributed under the terms and onditions of the Creative Commons ttribution liense ( 39