Derivation of Transformation and One-Way Speed of Light in Kinematics of Special Theory of Ether

Transcription

1 Amerian Journal of Modern Physis 07; 66: doi: 0.648j.ajmp ISSN: Print; ISSN: Online Deriation of Transformation and One-Way Speed of Light in Kinematis of Speial Theory of Ether Karol Szostek, Roman Szostek Department of Thermodynamis and Fluid Mehanis, Rzeszow Uniersity of Tehnology, Rzeszow, Poland Department of Quantitatie Methods, Rzeszow Uniersity of Tehnology, Rzeszow, Poland address: K. Szostek, R. Szostek To ite this artile: Karol Szostek, Roman Szostek. Deriation of Transformation and One-Way Speed of Light in Kinematis of Speial Theory of Ether. Amerian Journal of Modern Physis. Vol. 6, No. 6, 07, pp doi: 0.648j.ajmp Reeied: September 8, 07; Aepted: Otober 0, 07; Published: Otober 3, 07 Abstrat: At present, it is belieed that the Speial Theory of Relatiity STR is the only theory eplaining the Mihelson- Morley and Kennedy-Thorndike eperiments. This artile proed that another theory in aordane with these eperiments is possible. In this artile, we derie the new theory of kinematis of bodies from the uniersal frame of referene UFR, ether, whih we alled the Speial Theory of Ether STE. The artile eplains why Mihelson-Morley and Kennedy-Thorndike eperiments ould not detet the uniersal frame of referene. In artile, a different transformation of time and position than the Lorentz transformation is deried on the basis of the geometri analysis of the Mihelson-Morley and Kennedy-Thorndike eperiments. The formula for summation of speeds for absolute speed has been deried. Based on the deried transformation, we derie the formula for the eloity of light in auum measured in any inertial referene system. The entire artile ontains only original researh onduted by its authors. Keywords: Kinematis of Bodies, Uniersal Frame of Referene, Coordinate and Time Transformation, One-Way Speed of Light. Introdution The artile an eplanation the results of the Mihelson- Morley [] and Kennedy-Thorndike [] eperiments, assuming the eistene of the uniersal frame of referene UFR, in whih the eloity of light is onstant, is presented. In inertial frames of referene moing in the UFR, the oneway eloity of light may be different. The transformations from the inertial system to the UFR and from the UFR to the inertial system was deried by the geometri method. The eloity of light in one diretion has neer been aurately measured. In all aurate laboratory eperiments, as in the Mihelson-Morley eperiment, only the aerage eloity of light, traelling on a losed trajetory, was measured. In these eperiments, light always omes bak to the soure point. Therefore, the assumption about the onstant eloity of light instantaneous eloity adopted in the Speial Theory of Relatiity is not eperimentally justified. The deriation presented in this artile is based on the assumption resulting from these eperiments, that is for eery obserer, the aerage eloity of light traelling the way to and bak is onstant. The transformation «UFR- inertial system» 7-8 deried in this artile by the geometri method was already deried in artiles [3] and [4] by other method. In artile [3] the author obtained this transformation from the Lorentz transformation thanks to the synhronization of loks in inertial frames by the eternal method. The transformation obtained in the work [3] is the Lorentz transformation differently written down after a hange in the manner of time measurement in the inertial frame of referene, this is why the properties of the Speial Theory of Relatiity were attributed to this transformation. The transformation 7-8 has a different physial meaning than the Lorentz transformation, beause aording to the theory outlined in this artile, it is possible to determine the speed with respet to a uniersal frame of referene by loal measurement. So the uniersal referene system is real, and this is not a freely hosen inertial system.

2 Amerian Journal of Modern Physis 07; 66: The Assumptions In the presented analysis of the Mihelson-Morley and Kennedy-Thorndike eperiments, the following assumptions are adopted: I. There is a uniersal frame of referene UFR with respet to whih the eloity of light in auum is the same in eery diretion. II. The aerage eloity of light on its way to and bak is for eery obserer independent of the diretion of light propagation. This results from the Mihelson-Morley eperiment. III. The aerage eloity of light on its way to and bak does not depend on the eloity of the obserer in relation to the UFR. This results from the Kennedy- Thorndike eperiment. IV. In the diretion perpendiular to the diretion of the eloity of the body, moing in relation to the UFR, there is no ontration or elongation of its length. V. The transformation «UFR-inertial system» is linear. The transformation deriation presented in this artile differs from the deriation by the geometri method of the Lorentz transformation whih is the basis for the STR. In STR in the deriation of the Lorentz transformation, it is assumed that the reerse transformation has the same form as the original transformation. Suh an assumption stems from the belief that all inertial frames are equialent. In the deriation presented in this artile, we do not assume what form the reerse transformation has. Assumptions onerning the eloity of light adopted in this artile are also weaker than those adopted in the STR. In the STR, it is assumed that the eloity of light is absolutely onstant, despite the fat that it has not been proen by any eperiment. In this artile, the assumption resulting from eperiments is adopted, i.e. the aerage eloity of light on the way to the mirror and bak is onstant assumption II and III. In the presented onsiderations, the eloity of light by assumption is onstant only in one highlighted frame of referene - the UFR assumption I. Assumptions IV and V are idential to those on whih the STR is based. In works [7] and [8], idential transformations were deried as in this artile, but with the adopted additional assumption. For this, it was neessary to ondut the full analysis of the Mihelson-Morley eperiment in whih also the seond stream of light, parallel to eloity, is taken into aount. In that ase, only one stream of light was analyzed. 3. Time and Way of the Light Flow in the UFR Let us onsider inertial system U', whih moes in relation to system U related with the UFR at eloity Figure. In system U', there is a mirror at distane D' from the beginning of the system. Light in the system U moes at onstant eloity. From system U', from point '0 in time t0, a stream of light was sent in the diretion of the mirror. Haing reahed the mirror, the refleted light moes in the system U in the opposite diretion at eloity with the negatie alue. We assume the following symbols for the obserer from the system U: t is the time of the light flow to the mirror, t is the time of the light return to the starting point. L and L are ways whih were traelled by light in the system U in one diretion and in another. When light moes in the diretion of the mirror, then the mirror runs away from it at eloity. When light omes bak to point '0 after the refletion from the mirror, then this point runs towards it at eloity. For an obserer from system U, distane D' parallel to eloity etor is seen as D. We obtain t L D t, L D t L D t L D t, t Figure. The Time and the Flow Path of Light to the Mirror and Bak. a the way of light seen from the inertial system U', b the way of light seen from the UFR. Dependenies should be soled due to t and t. We then obtain time and way of flow in the UFR t D D, t L t D, L t D 4. The Geometrial Deriation of the Transformation We analyze the results of the Mihelson-Morley eperiment, as shown in Figure. The inertial system U' moe sat a relatie eloity to the inertial system U, assoiated with the UFR, parallel to the ais. Aes and ' lie on one straight line. At the moment when origins of systems oerlap, loks in both systems are synhronized. Cloks in system U related to the UFR are synhronized by the internal method [3]. Cloks in system U' are synhronized by the eternal method in suh a manner that if the lok of system U indiates time t0, then the lok of system U' net to it is also reset, that is t'0. In the system U', an eperiment measuring the eloity of 3 4

3 4 Karol Szostek and Roman Szostek: Deriation of Transformation and One-Way Speed of Light in Kinematis of Speial Theory of Ether light in auum perpendiular and parallel to the diretion of moement of the system U' in relation to the UFR was onduted. In eah of these diretions, light traels to the mirror and bak. Figure presents in part a the flow path of light seen by the obserer from the system U', while in part b the path seen by the obserer from the system U. In system U light has always onstant eloity assumption I. Considerations onern the flow of light in auum. In aordane with onlusions resulting from the Mihelson-Morley eperiment it has been assumed that the aerage eloity of light p on the way to the mirror and bak in system U' is the same in eery diretion, in partiular in the parallel diretion to the ais y' assumption II. It has also been assumed that the aerage eloity of light p on the way to the mirror and bak does not depend on the eloity of an obserer in relation to the UFR assumption III. ais ', the seond one on the ais y'. We assume that the distane D', whih is perpendiular to the eloity is the same for obserers from both systems assumption IV. Therefore, in Figure, there is the same length D' in part a and part b. The flow time of light in the system U, along the ais, in the diretion to the mirror is marked as t. The flow time bak is marked as t. The flow time of light in the system U', along the ais ', in the diretion to the mirror is marked as t'. The flow time bak to the soure is marked t'. Total time is marked respetiely as t and t' t t t and t' t' t'. The light stream, moing parallel to the ais y', from the point of iew of the system U moes along the arms of an isoseles triangle of side length L. Sine the eloity of light is onstant in the system U, therefore, the time of moement along both arms is the same and is equal to t. In the system U, the light stream parallel to the ais, in the diretion of the mirror oeromes distane L during time t. On the way bak, it traels distane L during time t. These distanes are different due to the moement of the mirror and the soure point of light in the UFR. In the eperiment, both light streams ome bak to the soure point at the same time, both in system U and system U'. It results from assumption II and from the mirrors setting at the same distane from the point of light emission. For an obserer of U' and U, the eloity of light an be written as D D L L L t t t t t t 6 From equation 6 light paths L and D' as a funtion of the eloity of light and the light flow times t, t' respetiely in the systems U and U' an be determined Figure. Paths of Two Streams of Light. a seen by an obserer from the system U', b seen by an obserer from the system U UFR. From assumption II and III it follows that the aerage eloity of light p in the inertial frame of referene is the same as the eloity of light in the system U. If we allow that the aerage eloity of light p in the system U' is a funtion of the eloity of light in the system U dependent on the eloity, we an write f 5 p From assumption III the aerage eloity of light is the same for different eloities of the Earth relatie to the UFR, so f f. Sine f 0, therefore f for eery eloity. It follows that p. The mirrors are assoiated with the system U' and plaed at distane D' from the origin. One mirror is loated on the t t L ; D 7 The eloity of the system U' relatie to the absolute frame of referene U, i.e. the UFR is marked by. Sine p is the path that the system U' traelled in time t, of the light flow, we hae p ; p t 8 t Using the geometry of Figure, the length L an be epressed as L D t D 9 p Haing squared equation 9 and taken 7 into aount, we obtain t t t 0

4 Amerian Journal of Modern Physis 07; 66: After arranging we obtain t t t t for 0 The aboe relation desribes only times t and t' that inole the full light flow to the mirror and bak. It should be noted that these are times measured in point '0. Howeer, if we assume that the length D' an be hosen so that time flow of light is any time, so the relationship is true for any time. Length D' assoiated with the system U' that is parallel to the ais, and is seen from the system U as D. If light flows in the absolute frame of referene U to the mirror, is hasing the mirror, whih is away from it at length D. After refletion, light returns to the soure point, whih runs against him. Using equations 4, we obtain the equations for light flow paths in both diretions along the ais ' in the system U L t D ; L t D 3 From equations 3 the sum and differene in length the L and L, whih light traelled in the system U, an be determined L L D D D, L L D D D 4 From the seond equation, the distane that the system U' traelled in half of the light flow time tan be determined, so we hae p t L L D 5 Sine it was assumed that in the system U the eloity of light is onstant, therefore both distanes, whih are traelled by light L and L L are the same L L L 6 After substituting 9 and the first equation 4 we obtain t D D 7 After reduing by two, raising to the square and taking 5 into aount we an write D D D 8 From equation 8 a dependene for the length ontration an be determined D D D D D 9 D D 0 Lengths D and D' whih are distanes between mirrors and the point of light emission our in the aboe dependene. Sine length D' an be seleted on a oluntary basis; therefore, dependene 0 is true for any alue of D'. Haing introdued to 8, we hae p t for 0 We assume that the transformation from the inertial system U' to the system U is linear assumption V. If linear fators dependent on ' are added to the transformation of time and position,, transformations with unknown oeffiients a, b an be obtained t t a t b Transformation should be alid for any time and position. In a partiular ase, it is alid at the moment of loks' synhronization, that is when tt'0 for the point with oordinates D' in system U'. In this onnetion, we introdue tt'0, 'D' and D into. Haing taken 0 into aount, we obtain 0 ad We obtain oeffiients a and b D bd a 0 b 3 4 Finally, the transformation from any inertial system U' to the system U, assoiated with the UFR takes the form t t t 5 6

5 44 Karol Szostek and Roman Szostek: Deriation of Transformation and One-Way Speed of Light in Kinematis of Speial Theory of Ether After transformations of the aboe equations, we obtain the inerse transformation, that is the transformation from the system U, assoiated with the UFR to the inertial system U' t t 7 t Due to the assumption IV also ours 8 y y and z z 9 The eloity is the eloity of the inertial system relatie to the uniersal referene system. We introdue differentials 3 into equations 30. We obtain That is V Vy Vz d dy dz The Transformation of Veloity Aes of the inertial system U' and the system U onneted with the ether were determined in suh a way that they were parallel to eah other Figure 3. The inertial system moes at the eloity in parallel to the ais and '. V dy Vy dz Vz d 34 Based on 3, we obtain the searhed transformation of eloity Figure 3. The Moement Seen from the Ether and the Inertial System. Differentials from the transformation 7-9 hae the form d d dy dy dz dz 30 A moing body is obsered from the ether U and the inertial system U'. In the ether, it moes at the eloity V, while in the inertial system, it moes at the eloity V'. Components of these eloities are presented in Figure 3. The eloity of the body in the system of the ether U an be written in the form V d dy dz, Vy, Vz 3 The eloity of the body in the inertial system U' an be written in the form d dy dz V, Vy, Vz 3 V V Vy Vy Vz Vz 35 Transformation 35 epresses the relatie eloity V' from the absolute eloities V and. From the first equation of this transformation an obtain a formula for summing the parallel eloities in the form V and formula for relatie eloity in form V 36 V V The Veloity of Light in Vauum for a Moing Obserer Generally, the light flow ours along paths presented in Figure 4. Aes of oordinate systems are set in suh a way that 0 38 z z

6 Amerian Journal of Modern Physis 07; 66: Figure 4. The Light Flow at Any Angle. In aordane with the Figure based on the Pythagorean theorem, we obtain y 39 y 40 The following also ours os 4 When V and V' ', then in aordane with 35 the following ours 4 y y The First Dependene for the Veloity of Light Haing introdued dependenies 4 and 43 into 39, we obtain y 44 4 y 45 ] [ y 46 Haing taken 40 into aount, we obtain ] [ On this basis, we obtain the first dependene for the eloity of light in the inertial system epressed from The Seond Dependene for the Veloity of Light Based on 4 we obtain 5 After introduing it into 5, we obtain On this basis we obtain the seond dependene for the eloity of light in the inertial system, epressed from ' The Third Dependene for the Veloity of Light Based on 56 we obtain From this equation based on 4 we obtain the third dependene for the eloity of light in the inertial system, epressed from ' Figure 5

7 46 Karol Szostek and Roman Szostek: Deriation of Transformation and One-Way Speed of Light in Kinematis of Speial Theory of Ether 6 os This formula is idential to formula 377 deried by the geometri method in the work [6]. In work [6] a formula for the eloity of light running in any diretion in a material medium motionless in relation to the obserer, more general than formula 6, is deried by means of the geometri method. It has the form of s is the aerage eloity of light traeling the path to and bak in this material medium s s 65 os s 7. Conlusion Figure 5. The Veloity of Light ' ' in the Inertial System for 0, 0.5, 0.5, 0.75,. We will now determine the aerage eloity of light whih in any inertial system traels the path with the length L', is refleted from the mirror and returns along the same path to the soure point Figure 6. If t' is the time needed for light to trael the path L' in one diretion, while t' is the time needed for light to trael the same path in the other diretion, then the aerage eloity of light along the path bak and forth is equal to L L sr t t L L 63 os os π sr os os 64 It follows that the aerage speed of light is onstant and equal to the one-way speed of light seen from the ether. This aerage eloity does not depend on the angle ' nor the eloity. For this reason, the rotation of the interferometer in Mihelson-Morley and Kennedy-Thorndike eperiments does not influene interferene fringes. Therefore, these eperiments ould not detet the ether. Figure 6. The Veloity of Light in the Mihelson-Morley Eperiment. It follows from the onduted analysis that the eplanation of the results of the Mihelson-Morley eperiment on the basis of the uniersal frame of referene is possible. Stating that the Mihelson-Morley eperiment proed that the eloity of light is absolutely onstant is untrue. Stating that the Mihelson-Morley eperiment proed that there is no uniersal frame of referene in whih light propagates and moes at a onstant eloity is also untrue. From the deried transformations 5-6 and 7-8 it follows that the measurement of the eloity of light in auum by means of the preiously applied methods will always gie the aerage alue equal to. This happens despite the fat that for the moing obserer the eloity of light has a different alue in different diretions. The aerage eloity of light is always onstant and independent of the eloity of the inertial frame of referene. Due to this property of the eloity of light, the Mihelson-Morley and Kennedy-Thorndike eperiments ould not detet the uniersal frame of referene. Admitting that the eloity of light may depend on the diretion of its emission does not differentiate any diretion in spae. The eloity of light whih is measured by the moing obserer is signifiant here. It is the eloity at whih the obserer moes in relation to the uniersal frame of referene that differentiates the harateristi diretion in spae, but only for this obserer. For the obserer motionless in relation to the uniersal frame of referene, the eloity of light is always onstant and does not depend on the diretion of its emission. If the obserer moes in relation to the uniersal frame of referene, then from his perspetie spae is not symmetrial. The ase of this obserer will be similar to the ase of the obserer moing on water and measuring the eloity of the wae on water. Despite that the wae propagates on water at the onstant eloity in eery diretion, from the perspetie of the obserer moing on water, the eloity of the wae will be different in different diretions. In works [6]-[0], a new physial theory of kinematis and dynamis of bodies based on the transformation determined here, alled by the authors the Speial Theory of Ether, was deried. In work [0] it has been shown that it is possible to weaken the assumption IV and derie a more general form of transformation 5-6 and 7-8.Thus many kinematis an be deried in aordane with the Mihelson-Morley and Kennedy-Thorndike eperiments. In the work [6] has been shown that within eah suh kinematis an derie infinitely many dynamis. In order to derie dynamis, it is neessary

8 Amerian Journal of Modern Physis 07; 66: to adopt the additional assumption, whih will allow for introdution into theory of the onepts of mass, kineti energy, and momentum. Based on this kinematis an naturally eplain the anisotropy of the mirowae bakground radiation, whih was disussed at work [5]. This allows determine the speed at whih the solar system is moing relatie to a uniersal referene system, that is kms. This has been shown in [8] and [0]. The Mihelson-Morley eperiment and Kennedy- Thorndike eperiment were onduted many times by different teams. Eah of the eperiments only onfirmed that the aerage eloity of light is onstant. Therefore, assumptions, on whih the presented deriation is based, are eperimentally justified. Referenes [] Mihelson Albert A., Morley Edward W., On the relatie motion of the earth and the luminiferous ether. Am. J. Si. 34, , 887. [] Kennedy Roy J., Thorndike Edward M., Eperimental Establishment of the Relatiity of Time. Physial Reiew, 4 3, , 93. [3] Mansouri Reza, Sel Roman U., A Test Theory of Speial Relatiity: I. Simultaneity and Clok Synhronization. General Relatiity and Graitation, Vol. 8, No. 7, 977, [4] Tangherlini Frank R., The Veloity of Light in Uniformly Moing Frame. A Dissertation. Stanford Uniersity, 958 reprint in The Abraham Zelmano Journal, Vol., 009, ISSN [5] Smoot George F., Nobel Leture: Cosmi mirowae bakground radiation anisotropies: Their disoery and utilization in English. Reiews of Modern Physis, Volume 79, , 007. Smoot George F., Anizotropie kosmiznego mikrofalowego promieniowania tła: ih odkryie i wykorzystanie in Polish. Postępy Fizyki, Tom 59, Zeszyt, 5-79, 008. Смут Джордж Ф., Анизотропия реликтового излучения: открытие и научное значение in Russian. Успехи Физических Наук, Том 77,, 94-37, 007. [6] Szostek Karol, Szostek Roman, Speial Theory of Ether in English, Publishing house Amelia, Rzeszow, Poland, 05, ISBN Szostek Karol, Szostek Roman, Szzególna Teoria Eteru in Polish, Wydawnitwo Amelia, Rzeszów, Polska, 05, ISBN [7] Szostek Karol, Szostek Roman, The Geometri Deriation of the Transformation of Time and Position Coordinates in STE, IOSR Journal of Applied Physis IOSR-JAP, Volume 8, Issue 4, Version III, -30, 06, ISSN , doi: [8] Szostek Karol, Szostek Roman, Выделенная в космологии система отсчета и возможная модификация преобразований Лоренца in Russian: The preferential referene system in osmology and the possible modifiation of Lorentz transformation. Ученые Записки Физического Факультета МГУ Memoirs of the Faulty of Physis Lomonoso Mosow State Uniersity,, 07, 70, ISSN [9] Szostek Karol, Szostek Roman, The eplanation of the Mihelson-Morley eperiment results by means of the theory of ether in English. ixra 06, Szostek Karol, Szostek Roman, Wyjaśnienie wyników eksperymentu Mihelsona- Morleya przy pomoy teorii z eterem in Polish. ixra 07, [0] Szostek Karol, Szostek Roman, The Deriation of the General Form of Kinematis with the Uniersal Referene System in English. ixra 07, Szostek Karol, Szostek Roman, Wyprowadzenie ogólnej postai kinematyki z uniwersalnym układem odniesienia in Polish. ixra 07,