Stellar Aberration, Relative Motion, and the Lorentz Factor

Transcription

1 ong Beah 010 PROCEEDINGS of the NP 1 Stellar berration, Relatie Motion, and the orentz Fator Joseph. Rybzyk 139 Stetson Drie, Chalfont, P jarybzyk@erizon.net Presented are the results of an in-depth inestigatie analysis onduted to determine the relationship between stellar aberration and the priniple of relatie motion. Introdued are two new priniples of physis that were disoered during the proess. The first of these priniples inoles the orentz fator and its inherent relationship to the priniple of stellar aberration. The seond inoles the newly defined priniple of distane perspetie that integrates the priniple of inerse stellar aberration with the priniple of relatie motion. 1. Introdution fter dispelling a possible misoneption inoling the standard priniple of stellar aberration [1] a systemati analysis is onduted to determine the relationship of the standard effet to that of the inerse effet introdued in a preious work [] by this inestigator. In the proess a onnetion is made between the standard effet and the orentz transformation fator [3] that seems to imply that relatiisti effets are of a loal nature and do not apply oer great distanes. The final disoery inoles a new priniple alled distane perspetie that defines the mathematial relationships between the priniples of inerse stellar aberration and relatie motion.. Displaement of Objets in Spae as a Funtion of Distane Gien the obserational eidene that the angle of stellar aberration is the same for all objets in the same loality of spae it diretly follows that the spatial displaement of those objets is a diret funtion of their distane to the obserer. This behaior an be onfusing depending on the perspetie from whih it is iewed. For example, in referring to Figure 1 Image Image Image Soure Fig. 1. Image Displaement for Soures at Different Distanes where the distane traeled at speed and angle of obseration is the same for eah of the obserers as they obsere light propagating in a perpendiular diretion to their path of motion, the distane by whih the soure s image is displaed to the left is learly a funtion of the distane between the soure and obserer. To the asual obserer this might seem to indiate that stars are seen in a sattered arrangement from their true positions in spae as a funtion of their distane to the obserer. That is not the ase, howeer, as shown in Figure where the relationships ight between the stars and obserer are gien from a different perspetie. Image Soure Image Image ight Fig.. Image Displaement for Soures at Different Distanes s an be seen in Figure where speed and the angle of obseration are the same as in Figure 1, the farther the soure is from the obserer, the greater its image displaement must be in order to be seen at the same loation in spae as loser soures loated in the same perpendiular diretion from the point of obseration. From this seond perspetie it is quite lear that the stars, regardless of their distane to the obserer, will be obsered in the same general arrangement relatie to eah other as in their true position but simply at a shifted loation in spae. It should also be understood that the amount of shift for a low orbital speed, similar to that of Earth s orbit, is ery small, too small, in fat, to illustrate faithfully. Subsequently, unless otherwise stated, suh shifts are greatly exaggerated in this work for the sole purpose of illustratie pratiality. 3. Stellar berration and the orentz Fator In preparation for the proess of extending the priniple of stellar aberration to its inerse form inoling its relationship to the priniple of relatie motion one final aspet of its behaior in the standard form must be onsidered. It inoles the newly disoered diret relationship of stellar aberration to the orentz transformation fator that to the inestigator s knowledge went undeteted until this present researh effort. Referring to Figure 3, the reised form of stellar aberration as gien in the inestigator s earlier work is illustrated whereby light speed is assigned to the path the light traels in the moing frame of the Earth, and light speed is assigned to the path light follows in the frame of the Sun. This is a reersal of the two paths as originally defined by James Bradley in order to bring the standard theory of stellar Soure Soure

2 Rybzyk: Stellar berration, Relatie Motion, and the orentz Fator Vol. 6, No. aberration into onformane with the eidene that supports relatiisti theory. With the understanding that light path from the soure is perpendiular to the orbital plane of the Earth, where is Earth s orbital speed around the Sun, we hae the Pythagorean triangle shown in the illustration. ight path through telesope Fig. 3. Stellar berration - Reised s preiously shown [], if is the angle of stellar aberration with regard to standard theory, then the standard formula for stellar aberration is gien by asin (1) where is the speed of the obserer and is the speed of light. Using = 9783 m/s for Earth s orbital speed, and = m/s for the speed of light we obtain = degrees, or = ar seonds that is in full agreement with the eidene. lternatiely, howeer, we ould hae soled for the angle of stellar aberration using the formula aos () where is the speed of light in the frame of the Sun (i.e., the stationary frame relatie to whih speed is referened) Sine is gien by (3) we hae by way of substitution into equation () where the resulting term aos pparent diretion to soure (Earth Frame) is diretion of light in Sun Frame (5) is the Millennium Relatiity [4] form of the orentz transformation fator. This fator is simply another form of the more familiar ersion 1 (6) used in Speial Relatiity [5]. In the interest of ompleteness, the gamma form of the fator is gien by ase (7) (4) or ase 1 ase 1 giing ase (10) for the final ersion using the γ gamma symbol in plae of the gamma fator. Of ourse, all of these ersions are alid only for the ase where the light propagating at speed is reeied in a diretion that is perpendiular to the path of obserer motion defined by speed. 4. Inerse Stellar berration and the ight Clok If the motion defined by speed is assigned to the soure instead of the obserer and if the soure s path of motion is loal to the loation of the obserer and the reeied light traels a path that is perpendiular to the soure s path of trael, we then hae the simplest ase of inerse stellar aberration as illustrated in Figure 4. In suh ase the ratio of distanes D to D, is equal to the ratio of speeds to as gien by D (11) D were D and D are the respetie distanes traeled at speeds and in the frame of the obserer. The reason for this distintion will beome learer in Setions (5) and (6) when the equality just defined is broken. Referring now to Figure 4, the illustrated Pythagorean triangle is simply an upside-down ersion of the abbreiated form of the speial relatiity light lok used in the millennium theory of relatiity. (By abbreiated it is simply meant that the seond half of the light lok that gies the refleted path of the light is omitted on the basis of redundany.) Image Soure Diretion of light in obserer frame Diretion of light in soure frame Fig. 4. Inerse Stellar berration (oal) Sine the triangle of Figure 4 has the same trigonometri relationships as the triangle of Figure 3 (disounting the reersal of the relationship between the soure and obserer) all of the formulas deried from Figure 3 also apply to Figure 4. Of speial interest are equations (4) and (8) that establish beyond doubt the relationship of the orentz fator with the inerse nature of stellar aberration. The importane of this disoery will beome learer in Setions (5) and (6) when these fators take on a modified form as determined by the eidene. 5. The Priniple of Distane Perspetie The first step in deeloping the mathematial relationships required to orretly integrate the priniple of inerse stellar ab- (8) (9)

3 ong Beah 010 PROCEEDINGS of the NP 3 erration with the priniple of relatie motion is that inoling the apparently preiously oerlooked priniple of distane perspetie. lthough the rationale for this priniple is straightforward and easily understood, the mathematial deriation an be ery onfusing. Beginning with the rationale used, it is easily understood that all objets appear to grow smaller as they moe off into the distane. If that is the ase then all of the dimensions assoiated with suh objets appear to shrink in diret omparison to idential objets in the iinity of the obserer. Stated another way, all far off distanes appear shrunken in inerse proportion to their distane from the obserer. To define this priniple mathematially we need to simply, in the geometrial sense, plae an objet or dimension at a defined distane from an idential objet oriented in an idential manner at the loation of the obserer. Or, otherwise stated, if the loal distane traeled at speed is proportional to the loal distane traeled at the speed of light, then the remote distane traeled at speed is the same as the loal distane traeled at speed, but the distane traeled at the speed of light is the inreased distane between the light soure and the obserer. What we are doing then, is holding the distane traeled at speed onstant while extending the distane traeled by light at speed to the distane between the soure and the point of obseration. Then, as with the loal relationships between distanes,, and, the distane traeled by light at speed remains a diret trigonometri funtion of the distanes traeled at speeds and. The resulting triangular relationships will be gien in the next Setion. 6. The Priniples of Relatie Motion and Inerse Stellar berration To integrate the priniple of relatie motion with the priniple of inerse stellar aberration we need to simply inoke the priniple of distane perspetie disussed in the preious Setion. For the simpler ase where the distane defined by is perpendiular to the distane traeled at speed by the soure we hae the Pythagorean triangle relationships illustrated in Figure 5. Image Soure T Diretion of light in obserer frame Fig. 5. Inerse Stellar berration Diretion of light in soure frame T In this ase the basi formula for the angle of inerse stellar aberration is gien by the modified ersion of equation (1) in the form of asin (1) T where is the speed of the soure, is the speed of light in the frame of the obserer and T is the time interal during whih the light propagated from the soure to the obserer, at speed in the frame of the soure, and at speed in the frame of the obserer. Time interal T is normally gien in seonds and is alid in the range 1 T (13) as determined in the stationary frame of the obserer. It is understood here that is the ariable speed of light in the frame of the obserer and is subsequently alid for use with stationary frame interal T. This is a perfetly alid alternatie to the method where light has a onstant speed in the moing frame of the soure and subsequently requires time in the soure s frame to ary relatie to the time in the stationary frame as defined by the time transformation formula t T (14) using the millennium relatiity ersion of the orentz transformation fator. Similar to the treatment used for standard stellar aberration, an alternatie formula for the angle of inerse stellar aberration is aailable in the form of a modified ersion of equation () gien as aos T T (15) where T is the stationary frame time during whih the light traeled from the soure to the obserer. lthough it might seem that this is atually the same equation as equation () sine the ariable T anels out of the fration, suh is not atually the ase. The reason is simple. Variable has a different definition in this latter ersion of the equation from that defined by equation (3) used in the original equation (). That is, in the present ase is defined by T ( T) (16) ( T) (17) T T (18) T T (19) that when substituted into equation (15) (after the anelation of the T ariables) gies T aos (0) for the omplete ersion of the alternate formula for inerse stellar aberration. Of great interest is the fat that the fator inluded in the brakets of equation (0) is a modified form of the millennium relatiity transformation fator arried at earlier in equation (4). Simplifiation of equation (0) in the manner of

4 4 Rybzyk: Stellar berration, Relatie Motion, and the orentz Fator Vol. 6, No. then gies T aos aos 1 T aos 1 T aos 1 T (1) () (3) (4) for the final ersion based on a modified form of the orentz transformation fator. The importane of this finding will be disussed later in Setion The Complete Formulas for Inerse Stellar berration Now that the priniples hae been established inoling the true nature of stellar aberration we an derie the general formulas for inerse stellar aberration at all angles of obseration from 0 to 180. We begin by referring to Figure 6 where the trigonometri relationships of an approahing and a reeding soure are illustrated. pproah Reession Image Soure Image Soure R s R s T T T T Fig. 6. Inerse Stellar berration In referring to Figure 6 the following definitions apply: Constant Speed of ight in the Stationary Frame of the Speed of Soure in the Stationary Frame of the Variable Speed of ight in the Stationary Frame of the T Stationary Frame Time Interal during whih Distanes Were Traeled R ngle of Reession (Image What is atually seen) ngle of pproah (Image What is atually seen) ngle of Reession (Soure The atual physial objet) s ngle of pproah (Soure The atual physial objet) ngle of Separation between Image and Soure Inerse Stellar berration Next we apply the aw of Cosines to the propagation triangles of Figure 6 to obtain ( T) ( T) Tos (5) where the onstant and ariables hae the definitions just gien. From equation (5) we then derie T ( T) Tos (6) for light propagation distane T. This is the atual distane between the soure and obserer at the instant of obseration. For our next step we apply the aw of Sines to the same propagation triangles to obtain sin sin (7) T that through the proess of substitution with equation (6) gies sin sin (8) ( T) Tos that in turn yields sin for the sine of. This then gies asin sin ( T) Tos sin ( T) Tos (9) (30) where is the angle of inerse stellar aberration. In applying the aboe formula in relation to the propagation triangles of Figure 6 it is understood that the relationship between supplementary angles and R is gien by 180 R and R 180 (31) where is the angle approah and R is the angle of reession with respet to the point of obseration. 8. Confirming the Priniple of Relatie Motion In omparing the trigonometri relationships of Figure 6 that equation (30) is based on to the trigonometri relationships of Figure 7 that follows, it is easily seen that, exept for the ariable T used in Figure 6, the two relationships are that of relatie motion. That is, if speed of the soures of Figure 6 is attributed instead to the Earth in Figure 7 the relationships are unhanged. pparent diretion to soure in diretion of approah (Image) is light path in Sun frame in both ases shown Soure s R R s ight path in Earth Frame (Earth Frame) Fig. 7. Stellar berration - Reised This is equialent to assigning the alue of unity to time interal T in Figure 6 and subsequently means that the priniple of distane perspetie is a fator in relatie motion as already established in the preious Setions. If we next, reerse the diretion of motion of the Earth in Figure 7 as shown in Figure 8 we an pparent diretion to soure in diretion of reession (Image) ight path in Earth Frame Soure

5 ong Beah 010 PROCEEDINGS of the NP 5 make a diret omparison with the trigonometri relationships gien preiously in Figure 3 that equation (1) is based on. Then, setting light speed in Figure 8 to the alue gien by equation (3) we hae the Pythagorean triangle of Figure 3 that equation (4) is based on. In that ase equation (1) also applies to the trigonometri relationships of Figure 8. Soure s ight path in Earth Frame pparent diretion to soure in diretion of reession (Image) R s ight path in Earth Frame Fig. 8. Stellar berration - Reised There is, howeer, a problem with equation (1) that has neer been properly addressed before. Equation (1) is based on an angle that an only be arried at indiretly sine it is not diretly disernable through isible means. This shortoming will be disussed in greater detail in the next Setion. 9. Reinterpreting the Standard Formula for Stellar berration It appears that most explanations of stellar aberration are based on formulas suh as equation (1) in the present work that are in turn based on a Pythagorean triangle. With that being the ase, the angle gien by the formula is alid only for a right triangle suh as that shown in Figure 3 and repeated below in Figure 9 with angles and added. ight path through telesope Fig. 9. Stellar berration - Reised It is understood then in referring to Figure 9 that is a 90 angle. With that being the ase, if the angle at whih the soure is iewed an be independently established to be = ar seonds in adane of the diretion of the Earth s orbital motion, as disussed in Setion 3, then the physial soure must atually be loated diretly aboe the obserer in a diretion perpendiular to the Earth s motion at that instant. It s really that simple. The problem, howeer, is that formulas suh as equation (1) are only alid for the right triangle relationships just disussed. For all other triangular relationships a more general formula suh as equation (30) is required. nd now that we understand the priniple of distane perspetie it is lear that equation (30) redues to Soure (Earth Frame) pparent diretion to soure (Earth Frame) pparent diretion to soure in diretion of approah (Image) R is light path in Sun frame in both ases shown is diretion of light in Sun Frame asin sin os (3) for use in determining stellar aberration angle at any angle. This, of ourse, follows from the fat that time interal T disappears at the loation of the obserer beause it takes on the alue of unity. It is important to remember here that we only see the image of the soure and not the soure itself. Or, stated otherwise, angle and its supplementary angle R, are the only angles that are diretly assoiated with the obsered image along its path of motion relatie to the obserer. (Note: Equation (3) erified in preious work [6].) 10. Summarizing the Priniple of Distane Perspetie There is no doubt that the onept of distane perspetie an be ery diffiult to ompletely omprehend. That is most likely the reason why it was not disoered earlier. Yet, with a little thought it is readily seen that the priniple of distane perspetie is absolutely alid and an no longer be ignored in the appliation of the ombined sienes of physis, mathematis, and osmology. With that objetie in mind, we an summarize the priniple of distane perspetie as follows: ording to the priniples of Newtonian mehanis we obsere an objet to moe a ertain distane during a ertain interal of time. When onsidering this proposition it is quite obious that suh distane to time relationship does not hange simply beause the objet is farther away in one ase as opposed to another. More speifially, it is quite obious that the distane an objet traels at a gien speed during a speifi time interal T is independent of the distane to the obserer. Thus, we an determine the distane an objet traels at speed during time interal T loally and plae that same distane as far away as neessary to ealuate its geometrial relationship relatie to its distane from the point of obseration. Or simply stated, the distane traeled at speed remains onstant while the distane to the obserer inreases to whateer distane neessary to define the atual distane of obseration. Referring to Figure 10 we an isualize what was just disussed in a more graphial sense. In iewing the illustration it should be quite apparent that the distane traeled by the moing objet at speed is dependent only on the time interal during whih the motion takes plae and is not affeted by the distane to the obserer. Hene for an obserer at twie the distane D illustrated, the distane traeled at speed during the same interal T is unhanged. Subsequent = Speed of Objet D = Distane to Objet Fig. 10. Distane Perspetie

6 6 Rybzyk: Stellar berration, Relatie Motion, and the orentz Fator Vol. 6, No. ly, a triangle onstruted from the two distanes illustrated beomes narrower in inerse proportion to the distane to the obserer as shown preiously in this work. It should also be obious that the angle at whih the motion of the objet is obsered also has no affet on the distane traeled. gain, only the distane between the objet and the obserer matters and the farther away the objet, the narrower the resulting triangle. When the priniple of distane perspetie beomes part of our normal thought proess the following harateristis beome eident: 1. We an hold distane D onstant and inrease or derease distane beause speed inreased or dereased.. We an hold distane onstant and inrease or derease distane D beause the distane to the obserer inreased or dereased. 3. We an inrease or derease both distane and distane D beause both speed and the distane to the obserer inreased or dereased. 4. We an inrease or derease distane while doing the opposite to distane D beause speed inreased or dereased while the distane to the obserer did the opposite. nd so on. 11. Summarizing Stellar berration, Inerse Stellar berration, and orentz, Effets Figure 11 was added as a final step to aid in the assimilation proess inoling all of the priniples introdued in this latest researh effort by this inestigator. Soure (at great distane) ight Speed (in MF) ight Speed (in SF) Speed (in MF) Fig. 11. oal (Stellar berration) (in SF) Fig. 11B. oal (orentz Effet) Soure (apparent loation) ight Distane (in SF) T t ight Distane (in MF) T Soure Distane (in SF) Soure Speed (in SF) ight Speed (in MF) ight Speed (in SF) (in SF) Fig. 11C. oal (Inerse Stellar berration) Soure Distane (in SF) ight Distane (in MF) T T ight Distane (in SF) (in SF) Fig. 11D. Remote (Inerse Stellar berration) Fig. 11. Stellar berration ight Clok Inerse Stellar berration Reiewing eah of the four inluded illustrations should be helpful with regard to isualizing how the different effets relate to eah other. nd whereas, similar to the ase with distane perspetie, the manner in whih all of these effets are integrated together an be ery diffiult to omprehend, the need to do so is of ultimate importane if we wish to adane our understanding of the priniples that appear to goern the unierse. This is beause from an almost paradoxial point of iew what has been shown in this work is so fundamental as to be obious in terms of alidity. et us now disuss one of the more obious reperussions of this new point of iew. 1. Fundamental Change in Our View of the Unierse Based on our preious understanding of the unierse, that did not inlude the priniple of distane perspetie, we erroneously onluded that the objets we see at great distanes in spae are no longer at the loations in whih we see them. That is, due to the expansion of the unierse they hae ontinued to moe away from us during the time it took the light they emitted to reah us. Thus, we are seeing them as they appeared in the past when the light was emitted and at the loations they were in at that time. From what was shown here, howeer, due to the priniple of distane perspetie, the farther away an objet is the less effet its motion has on where it is seen by the distant obserer. Subsequently, when we iew objets far out in spae, een those at the edge of the isible unierse that are traeling away from us at near the speed of light, we are seeing them where they are and not where they were so long ago. In summary, light does take on the speed of the soure as has been shown throughout this work. It is simply beause of the priniple of inerse stellar aberration that we see the light soure at a preious loation and obsere the speed of its light at the onstant alue of. Howeer, as has also been shown here, the effets of stellar aberration and inerse stellar aberration are loal effets due to the priniple of distane perspetie. Subsequently, we obsere far off objets right where they are at the instant the light from them is reeied. While it is true that we see suh objets as they appeared in the past when the light from them was emitted, we see them where they are at the time of obseration. This onlusion an be arried at mathematially as will be shown next. For the purpose of emphasis, let us take the worse ase example of an objet moing at light speed in a transerse diretion

7 ong Beah 010 PROCEEDINGS of the NP 7 to the point of obseration. Using equation (4) it is found that the angle of inerse stellar aberration goes to zero een if taken to 17 deimal plaes when the distane between the soure and obserer inreases to 4.55 light-years. More speifially, when T is inreased to a alue of T = seonds in equation (4), that equation will produe a result of zero up to 17 deimal plaes. Sine the number of seonds in a year is gien by 7 ys seonds (33) where y s is the number of seonds in a year, we an diide the alue gien preiously for interal T by y s to obtain T 10 y 4.55 years y s (34) where y is the time interal in years. In referring to Figure 11-D that equations (15) through (4) represent it is seen that if the angle = 0 then =. With that being the ase the distane D between the soure and obserer is gien by DT T where T 4.55 years (35) that equates to traeling at light speed for 4.55 years, or a distane of 4.55 light-years. Sine the nearest star to our Sun is Proxima Centauri at a distane of 4.43 light-years, this means that the 0 result for applies to irtually the entire unierse with the exeption of the loal iinity of our solar system and its immediate loation within the Milky Way galaxy. Of ourse, similar would be true at any other loation in the unierse. That is, the standard laws of physis are alid only in the immediate iinity of that loation and dissipate entirely within a distane of approximately 4.55 light-years. By way of referene, Proxima Centauri is the ompanion star of the binary system lpha Centauri B that ontains the next nearest stars to our Sun at a distane of 4.37 light-years. s a final point of interest it should be noted that when T = T as gien in equation (35), the fration in equation (15) redues to 1 thereby giing a result of = 0 as already disussed. 13. Conlusion It has long been apparent to this inestigator that experimental and obserational eidene founded upon loal onditions an be misleading and een inalid when applied to great distanes. This does not neessarily mean that the eidene is inalid but rather that it an be inorretly applied to great distanes beause of misoneptions deried from its loal appliation. striking example of suh misoneptions is the speed of light. t 300 million meters per seond it is the fastest thing known to mankind, yet it takes thousands, millions, and een seeral billions of years to reah us from relatiely nearby stars. Due to loally indued biases we attribute this extraordinarily long trael time to the great distanes inoled. Yet, we ould hae just as alidly onluded that light only seems fast beause it is obsered loally. There are endless examples of biases and misoneptions suh as this founded on the priniple that our knowledge is primarily deried from loal experienes. In fat suh biases form the ery basis of our mathematial and physial sienes sine suh sientifi disiplines were deeloped primarily to adane our understanding of loal experienes and permit benefiial returns. If we wish now to further our understanding of the unierse we must break free from these biases and ommene with the deelopment of a new branh of physis and mathematis that deals with the opposite extremes assoiated with the great distanes inoled. This just ompleted researh by this inestigator is but a first step in that diretion. Referenes [ 1 ] James Bradley, stronomer, disoered the priniple of stellar aberration in 177 while searhing for eidene of stellar parallax, a ompletely different phenomenon. [ ] Joseph. Rybzyk, The Complete Nature of Stellar berration, (010), [ 3 ] H.. orentz, Duth physiist, disoered a new way to transform distane and time measurements between two moing obserers so that Maxwell s equations would gie the same results for both, (1895). [ 4 ] Joseph. Rybzyk, series of works, inluding the present work, beginning with the Millennium Theory of Relatiity, (001), in whih the onept of spherial referene frames was introdued. ll of the works are aailable at [ 5 ] lbert Einstein, On the Eletrodynamis of Moing Bodies, now known as the Speial Theory of Relatiity, nnalen der Physik, (1905). [ 6 ] Joseph. Rybzyk, The Complete Nature of Stellar berration, (010), Referene Setion 5, equation (8).