Espen Gaarder Haug Norwegian University of Life Sciences January 5, 2017

Transcription

1 Einstein ersus FitzGerald, Lorentz, and Larmor Length Contration Einstein s Length Contration is Also Consistent with Anisotropi One-Way Speed of Light Espen Gaarder Haug Norwegian Uniersity of Life Sienes espenhaug@ma.om. January 5, 07 Abstrat This paper disusses the similarities between Einstein s length ontration and the FitzGerald, Lorentz, and Larmor length ontration. The FitzGerald, Lorentz, and Larmor length ontration was originally deried for only the ase of a frame moing relatie to the ether frame, and not for two moing frames. When extending the FitzGerald, Lorentz, and Larmor length transformation to any two frames, we will learly see that it is di erent than the Einstein length ontration. Under the FitzGerald, Lorentz, and Larmor length transformation we get both length ontration and length expansion, and non-reiproality, while under Einstein s speial relatiity theory we hae only length ontration and reiproality. Howeer, we show that there is a mathematial and logial link between the two methods of measuring length. This paper shows that the Einstein length ontration an be deried from assuming an anisotropi one-way speed of light. Further, we show that that the reiproality for length ontration under speial relatiity is an apparent reiproality due to Einstein-Poinaré synhronization. The Einstein length ontration is real in the sense that the preditions are orret when measured with Einstein-Poinaré synhronized loks. Still we will laim that there likely is a deeper and more fundamental reality that is better desribed with the extended FitzGerald, Lorentz, and Larmor framework, whih, in the speial ase of using Einstein-Poinaré synhonized loks gies Einstein s length ontration. The extended FitzGerald, Lorentz, and Larmor length ontration is also about length expansion, and it is not reiproal between frames. Still, when using Einstein synhronized loks the length ontration is apparently reiproal. An enduring, open uestion onerns whether or not it is possible to measure the one-way speed of light without relying on Einstein-Poinaré synhronization or slow lok transportation synhronization, and if the one-way speed of light then is anisotropi or isotropi. Seeral experiments performed and published laim to hae found an anisotropi one-way speed of light. These experiments hae been ignored or ridiuled, but in our iew they should be repeated and inestigated further. Key Words: length ontration, length expansion, speial relatiity theory, ether theories, atomism, reiproality, anisotropi one-way speed of light. Introdution George FitzGerald was the first to suggest that the null result of the Mihelson Morley experiment ould be that the length of any material objet (inluding the Earth itself) ontrats along the diretion it moes through the ether, or, as explained in his own words: Iwouldsuggestthatalmosttheonlyhypothesisthatanreonilethisoppositionisthat the length of the material bodies hanges aording as they are moing through the ether or aross it, by an amount depending on the suare of the ratio of their eloity to that of light. FitzGerald, May 889 We should bear in mind that FitzGerald was a supporter of the ether theory and that here he is talking about the speed of light relatie to the ether; in this ontext, the speed of the material bodies is

2 the speed against the ether. Lorentz proides a 89 mathematial formalization of length ontration. The Lorentz length ontration formula is gien as p () V where p is the eloity of the Earth relatie to the ether, and V is the eloity of light relatie to that of ether. The Lorentz formula is the first term of a series expansion of the more aurate formula r p V p () V that in todays notation would be written as = p and = V ;thisgies r (3) Lorentz was mainly interested in experiments related to the motion of the Earth against the ether, and it was assumed the eloity of the Earth against the ether was lose to the Earth orbital eloity around the sun, that is approximately 30 km/s. In this ase the first term of the series expansion would be more than aurate enough. What is often ignored in modern literature is that in the FitzGerald and Lorentz length ontration originally was the eloity of the Earth against the ether and that was the speed of the light against the ether. In the Einstein length ontration formula the eloity is the relatie speed against another frame as measured with Einstein-Poinaré synhronized loks. Under Einstein s theory speeds against the ether are not een onsidered, as he assumed that the ether did not exist. Joseph Larmor (900) ombined the FitzGerald and Lorentz length ontration with time dilation and was the first to deelop a onsistent spae and time transformation whih, after rewritten in modern notation, at first sight looks idential to that of Lorentz and Einstein. Howeer, it is atually di erent, as the eloity,, in the Larmor transformation is learly meant to be the eloity against the ether. In 905, Einstein also published a length ontration formula that he related to the Lorentz transformation later on. Larmor held on to the idea that the true one-way speed of light was anisotropi relatie to a moing frame. Larmor proed that the null result in the Mihelson Morley experiment was fully onsistent with the anisotropi one-way speed of light. Still, Larmor neer ame up with a suggestion of how to detet this true one-way speed of light, whih is a way to detet motion against the ether (oid). In 905, Poinaré assumed that it was impossible to detet the Earth s motion against ether: It seems that this impossibility of establishing experimentally the absolute motion of the Earth is a general law of nature: we are, of ourse, set towards admitting this law that we will all the Postulate of Relatiity and to admit it without restritions. Henri Poinaré To measure the one-way speed of light, we need to synhronize two loks, and to synhronize two loks we need to know the one-way speed of light. This is why Poinaré assumed that we ould neer measure the one-way speed of light and therefore we ould neer detet motion against the ether. After Poinaré (905) simplified the Lorentz spae and time transformation to a ertain extent, Lorentz seems to hae beome heaily influened by Poinaré s iew that motion against the ether not ould be deteted. It is natural to beliee that after 905 Lorentz assumed the eloity,, in his transformations was a relatie eloity between frames rather than a eloity against the ether, but Lorentz himself does not seem lear on this point. In this ase the Lorentz transformation and the Lorentz length ontration would indeed be the same as under Einstein transformation, but with a somewhat di erent interpretation. In the Lorentz world, as influened by Poinaré, the ether still existed, but motion against it ould not be deteted. In the Einstein world the ether simply did not exist or at least it was not neessary to take it into aount if it was not possible to measure it. Larmor, on the other hand, seems to be holding on to the onept that the one-way speed of light is anisotropi (exept in the ether frame) and that by using length ontration and time dilation, based on the eloity of the Earth against the ether, this was also fully onsistent with the Mihelson Morley experiment. Howeer, we will soon see that Larmor s theory was inomplete, as it only dealt with a moing frame against the ether frame and not two moing frames. The Einstein Length Contration The Einstein length ontration is gien by If anyone has found material where they feel he is lear on this, please point me towards it.

3 3 e L e,, = L e,, r (4) where e is the speed of the moing rod two relatie to frame one as measured with Einstein-Poinaré synhronized loks. I use the notation e rather than, as we we will resere for the eloity of a rod (a referene frame) against the ether as measured from the ether. Further L e,, is the length of the a rod in frame two as measured from the same frame two. L e,, is the length of the rod at rest in frame two as measured from frame one. The speed of frame two as measured from frame one is e and the speed of frame one as obsered from frame two using Einstein synhronized loks is e. That is to say that the relatie eloities between referene frames are reiproal when using Einstein-Poinarè synhronized loks, a point we fully agree on. Further, we hae r e L e,, = L e,, = (5) This just shows that Einstein length ontration is reiproal. Frame one an laim that the rod at rest in frame two is obsered to be ontrated and frame two will at the same time laim the rod at rest in frame one is ontrated. That the length ontration is reiproal is somewhat of an unsoled mystery, not mathematially, but oneptually. Exatly what is the reason for this? As we will see, the reiproality has to do with the lok synhronization proedure. 3 Length Contration: Is It for Real or Is It an Illusion? Surprisingly there are still seeral di erent interpretations of the length ontrations from speial relatiity among mainstream physiists. Physiists writing and leturing about modern physis do not seem to be in omplete agreement on whether length ontration is a real physial e et or if it is merely an illusionary e et, or atually some type of optial illusion. For example, Lawden (975) laims The ontration is not to be thought of as the physial reation of the rod to its motion and as belonging to the same ategory of physial e ets as the ontration of a metal rod when ooled. It is due to hanged relationship between the rod and the instruments measuring its length. Patheria (974) orretly points out that there is a major di erene between FitzGerald-Lorentz transformation on one side and Einstein length transformation: It must be pointed out here that the ontration hypothesis, put forward by FitzGerald and Lorentz, was of entirely di erent harater and must not be onfused with the e et obtained here. That hypothesis did not refer to a mutually reiproal e et; it is rather suggesting a ontration in the absolute sense, arising from the motion of an objet with respet to the aether or, so to say, from absolute motion. Aording to a relatiisti standpoint, neither absolute motion nor any e et aruing therefrom has any physial meaning. 3 In partiular, this holds true for the FitzGerald length ontration and the original Lorentz ontration and also the way Larmor uses length ontration. For the Poinaré influened Lorentz ontration one ould argue that it is mathematially idential to the Einstein length ontration, as the eloity in the formula probably should be interpreted as relatie eloity measured with Einstein-Poinaré synhronized loks and not the eloity against the ether. Shadowitz (969) laims If the measurements are optial, then in order to aoid an inorret result, the light photons must leae the two points at the same time as measured by the obserers: they must leae simultaneously. It is lear that the proess of length measurement is di erent from the proess of seeing. Amazingly, this distintion was not notied until 959, when it was first pointed out by James Terrell. 4 This point is also a rather important one. Further Rindler (00) writes Page. 3 See Patheria (974) page 4. 4 See Shadowitz (969) page 6.

4 4 This length ontration is no illusion, no mere aident of measurement or onention. It is real in eery sense. A moing rod is really short! It ould really be pushed into a hole at rest in the lab into whih it would not fit if it were moing and shrunk. 5 Freedman and Young (06) laims Length ontration is real! This is not an optial illusion! The ruler really is shorter in the referene frame S than it is in S. 6 Further Harris (008) writes It is a grae mistake to dismiss length ontration as an optial illusion aused by delays in light traeling to the obserer from a moing objet. This e et is real. 7 Uniersity physis students are not always taught the same way though; for example, Krane (0) seems to laim something di erent, at least in part: Length ontration suggests that the objets in motion are measured to hae shorter length then they do at rest. The objets do not atually shrink; there is merely a di erene in the length measured by di erent obserers. For example, to obserers on Earth a high-speed roket ship would appear to be ontrated along its diretion of motion, but to an obserer on the ship it is the passing Earth that appears to be ontrated. 8 As we see there are a number of opinions about length ontration and learly not all seem to agree. They may agree on the formula, but not on the interpretation of it. 4 Bak to the FitzGerald, Lorentz, and Larmor length ontration The length ontration gien by FitzGerald, Lorentz, and Larmor is L 0, = L, r (6) where L, is the length of the rod in the moing frame as obsered from the same moing frame and L 0, is the length of the moing rod as obsered from the ether frame. It is now of great importane to be aware that here is the eloity of the rod against the ether as measured from the ether. Lorentz (89) is lear that this is the speed against the ether. Larmor does not point out this diretly, but it is obious from reading his whole theory that this is intended to be the speed against the ether. This means that the length ontration formula aboe is not the same as the Einstein length ontration formula. Furthermore, the formula aboe only holds for the speial ase of a moing frame against the ether frame. Assume that we hae two frames moing against the ether; the length ontration for eah frame would be L 0, = L, r (7) And a rod at rest in the oid frame L 0,0 will, from the frame moing at speed relatie to the oid frame, appear to be expanded relatie to an idential rod at rest in frame one. That is to say, L 0, = L0,0 (8) The subsript in L 0, indiates that the rod is at rest in frame zero (the ether or oid frame) and is obsered from frame one (the frame moing relatie to the ether). The length of the rod at rest in frame two as obsered from the oid frame must be L,0 = L, r (9) and a rod at rest in the oid frame L 0,0 will, from the frame moing at speed relatie to the oid frame, appear to be expanded. That is to say, 5 See Rindler (00) page 6. 6 Freedman and Young (06) page 9. 7 Harris (008) page. 8 Krane (0) page 35.

5 5 L 0, = L0,0 (0) This also means that if we hae a rod at rest in frame two, it must appear to hae the following length as measured from frame one: L, = L, () A rod at rest in frame one, as obsered from frame two, must be obsered as haing a length of L, = L, () This is learly di erent than the Einstein length ontration. The length ontration formula aboe is when using loks with no lok synhronization error. In the ether frame, Larmor (and likely Lorentz and FitzGerald) assumed that the one-way speed of light is isotropi and the same as the round trip speed of light. In this frame we an atually synhronize loks based on the Einstein and Poinaré proedure without introduing a lok synhronization error. Further, we must hae and L, = L, L, = L, (3) (4) In euation 3 and 4 we need to measure the length of a rod from a frame that itself is moing against the ether. In this ase, using the Einstein and Poinaré method of synhronizing loks will gie a synhronization error. We will see that formula 3 and 4 atually are fully onsistent with the Einstein length ontration formula. The di erene is that my generalization of the Larmor length ontration relies on loks without any synhronization error, while the Einstein length ontration ontains an embedded lok synhronization error. 5 The One-Way Speed of Light In general, we need to hae two synhronized loks to measure length ontration. To synhronize the two loks we need to know the one-way speed of light. We will soon see that een if the true one-way speed of light is anisotropi and we derie the length ontration from this standpoint, we will get the same result as Einstein when using Einstein-Poinaré synhronized loks. If we hae a stati ether and light is moing through the ether, then the one-way speed of light relatie to the ether must be isotropi, and we an all this speed. Relatie to a moing frame, the one-way speed of light as obsered from the ether frame when the signal is sent in the same diretion as the laboratory moes relatie to the ether must be (5) The aboe one-way speeds of light are for a signal sent in the parallel diretion to the moing frame. Larmor (900) was the first to show that the one-way speed for light relatie to a moing frame as obsered from the ether frame must be ab = p sin( ) os( ) (6) where is the angle of the light signal relatie to the diretion of the moing frame relatie to the ether. Prokhonik (967) gies the same result 67 years later without referring to Larmor, still Prokhonik moes forward onsiderably in understanding ether theories and speial relatiity theory as well. Still, Prokhonik (967) theory is inomplete, but this is understandable, as a full understanding of a topi

6 6 often takes many steps by many researhers oer a long period of time. We will here onentrate on the length ontration in the parallel diretion. The one-way speed of light in ether theories is and isotropi against the ether, and ± relatie to a moing frame as obsered from the ether frame. When the one-way speed of light is obsered from the moing frame itself it is 9. and in the opposite diretion we hae ĉ ab = (7) ĉ ba = + (8) What we will answer in this paper is what type of length ontration will we get if the ether does exist and the true one-way speed of light is anisotropi. This is an interesting uestion een if one assumes that one annot detet the anisotropy in the one-way speed of light. In fat, we laim the anisotropy in the one-way speed of light already likely has been deteted by seeral independent experiments, Marino (974), Torr and Kolen (984), Cahill (006b), Cahill (006a), Cahill (0) but this has been ignored and een ridiuled by main frame physiists. 0. These experiments should be repeated and studied further instead of being ignored. 6 Clok Synhronization Error If the one-way speed of light truly is isotropi against the ether as measured from the ether frame suggested by Joseph Larmor, then if one synhronizes loks based on the Einstein-Poinaré method by assuming that the one-way speed of light is always isotropi, this will lead to a synhronization error in eery moing frame exept in the ether frame. Not only that, but the Einstein synhronization error will be di erent in eah frame, something that will make it appear as though the one-way speed of light is isotropi in eery frame, if measured from the same frame that the one-way light speed is measured against. This will lead to suh things as apparent reiproality for length ontration as long as we are measuring with Einstein-Poinaré synhronized loks. Whether or not this error an be deteted remains a partially open uestion, and the term error is possibly too strong at this point as if it is an error or not depends on repetition on some experiments before we finally onluds. Here we will be looking at the Einstein lok synhronization error for light signals sent parallel to the diretion of the moing frame. If we use Einstein synhronization, we also assume that the one-way speed of light is isotropi in any frame. This means that we assume that the time it takes for light to trael from A to B or from B to A is the aerage of the one-way time of light. This is what Einstein assumed and what today s physiists still naiely assume represents reality. The aerage of the one-way time of light in any frame as measured from that same frame the experiment is set up in is deried as ˆ = L + L, (9) ĉ ab ĉ ba where ˆ is the aerage one-way time of light that will only be idential to the true one-way time of light if the one-way speed of light is isotropi. Further, the true one-way time of light in the A to B diretion must be and in the B to A diretion it must be ˆt ab = L ĉ ab, (0) ˆt ba = L ĉ ba. () The Einstein synhronization error is the di erene between the assumed one-way time we arrie at based on the inorret Einstein assumption that the one-way speed of light is always isotropi and the true one-way time of light based on the true one-way speed of light. The Einstein synhronization error (or we ould all it the Einstein time error) as obsered from the moing frame for a light signal traeling 9 See Haug (04) for full deriation. 0 Seeral other independent methods gie indiretly the same speed of the solar system against the ether, see Monstein and Wesley (996) for an oeriew. It seems unlikely to be a oinidene that one-way speed of light experiments gies a speed lose to this.

7 7 from A to B must be the real time it takes for the light to trael that distane minus the time that Einstein assumed the light would take; that is : ˆ ab = ˆt ab ˆ ˆ ab = L ĉ ab ˆ ab = L 0 ˆ ab ˆ ab = ˆ ab = ˆ ab = ˆ ab = ĉ ab L L L ĉ ab + L ĉ ba L L ĉ ba L + L L A L A + ( + ) L ( ) A ( )( + ) ˆ ab = L. () The assumed Einstein time is more than assumed; this is the one-way time we will atually obsere when using Einstein-Poinaré synhronized loks (also aording to atomism). The point is that suh loks are synhronized based on the likely flawed assumption that the true one-way speed of light is isotropi, whih it is not, or alternatiely, that the true one-way speed of light an neer be deteted, as likely inorretly assumed by Poinaré. The Einstein synhronization error for light that traels in the opposite diretion of the diretion of the moing frame must be A ˆ ba = ˆt ba ˆ ˆ ba = L ĉ ba ˆ ba = L 0 ˆ ba ˆ ba = ˆ ba = ˆ ba ˆ ba = See also Léy (003) and Haug (04). ĉ ba L + L L 0 L L ĉ ab + L ĉ ba L ĉ ab L A L A + ( ) L ( + ) A ( + )( ) L A

8 8 ˆ ba = L. (3) The synhronization error from B to A is exatly the opposite of the synhronization error from A to B. This is rather obious if one thinks arefully about it. This symmetry, or more preisely, this perfet asymmetry (mirror symmetry) is extremely important. It means that when we measure the round-trip time of light, the Einstein synhronization errors in eah diretion will always perfetly anel eah other out to be zero, ˆ ab + ˆ ba = L L + = L L =0. (4) As long as we set up experiments that diretly or indiretly measure the round-trip time of light, we will not be able to detet a synhronization error in most experimental set-ups, een if it is there. 7 Bak to the Einstein Length Contration If we are using Einstein synhronized loks, then we will atually also obsere an apparent length ontration that di ers from the true length ontration. Assume two frames moing against the ether. Frame one is the Earth moing at a eloity of against the ether as obsered from the ether frame. Frame two is a train moing at a eloity of against the ether as obsered from the ether frame. On the ground of Earth we hae a rod with two lasers (plaed perpendiular to the length of the rod), one on eah end of the rod. The distane between the two lasers an be simply measured ia a ruler in the same frame as the ruler. Next we plae this rod on board a train at rest relatie to the Earth s ground. Then we aelerate the train up to a speed e relatie the Earth as measured with Einstein-Poinaré synhronized loks, and a speed of against the oid as measured from the oid. The speeds and hae not been obsered, as we assume we do not know how to detet and measure anisotropy in the one-way speed of light. And without finding the true one-way speed of light, we must synhronize loks with Einstein- Poinaré synhronization or through slow lok transportation that leads to the same synhronization error, see Léy (003), Haug (04). The easily obsered eloity of the train against the ground using two Einstein synhronized loks is linked to the Earth s eloity and the train s eloity against the ether in the following way (see Haug (04) for the full deriation). e =. (5) Bear in mind that this is an apparent eloity ontaining an Einstein speed error due to an embedded Einstein synhronization error. To be lear this is the eloity one will measure with Einstein-Poinaré synhronized loks in real experiments, but the eloity is measure with loks that are synhronized based on the assumption the one-way speed of light is isotropi. That is to say, it is based on the assumption that one annot detet motion against the oid. Bear in mind that the famous Mihelson Morley experiment is just a way to hek that the round-trip speed of light is isotropi, after one takes into aount length ontration and time dilation; it is not a test for the one-way speed of light. To find the length of the rod in the train from the ground, we only need one lok with a laser reeier. This single lok onneted to a laser reeier finds the time interal between the first and seond laser beam moing oer the detetor. Let s all this time interal ˆt. This time interal must be ˆt = ˆt = L, L, ˆt = L, ˆt = L,. (6)

9 9 In the deriation aboe bear in mind that L, = L, and that the time measurement is only done with one lok. So, this time measurement does not rely on any lok synhronization. Howeer, to find the rod s length as obsered from the other frame we also need to know the rod s eloity relatie to the ground where we try to measure the length ontration. The eloity of the rod is eual to the eloity of the train. And to measure the eloity of the train, we hae to use two Einstein-Poinaré synhronized loks (assuming we did not know the true one-way speed of light). The length of the rod when at rest in frame two (the train) as measured by frame one (from the ground of the Earth) using Einstein-Poinaré synhronized loks is therefore gien by L e,, = ˆt e ( ) L e,, = ˆt L e,, = L, ( ) L e,, = L, ( ). (7) The rod s length at rest in frame one as measured from frame two with Einstein-Poinaré synhronized loks is L e,, = ˆt e ( ) L e,, = ˆt L e,, = L, ( ) L e,, = L, ( ). (8) A rod as measured from the same referene frame in whih it is at rest is always the same no matter what frame it is plaed in. This is again beause measuring stiks shrink/expand in the same way as the rod. This means, L, = L,, whihalsomeans L e,, = L e,,. (9) This is a rather important result as it means length ontration when using Einstein-Poinaré synhronized loks will appear to be reiproal. Eah frame an laim it is the rods in the other frames that are ontrated. This is, howeer, apparent length ontration and not true length ontration. Alternatiely, if we had measured the train s speed ia loks without any lok synhronization error, we would hae found the train (rod) speed to be ˆ, =. (30) The rod s true length ontration as measured without any lok synhronization error is gien by L, = ˆtˆ, ( ) L, = ˆt L, = L, ( ) ( )

10 0 L, = L, L, = L,. (3) Einstein s speial relatiity theory length ontration euation is gien by L, = L, r When using Einstein synhronized loks the relatie speed is related to the absolute speeds (the speeds against the ether) in the following way e = e. Based on this we an rewrite the Einstein length ontration to L e,, = L, r u t L e,, = L, L e,, = L, s e ( ) L e,, = L, L e,, = L, L e,, = L, L e,, = L, L e,, = L, r ( ) (3). (33) In other words, the Einstein length ontration euation is idential to apparent length ontration under the assumption of the existene of the ether (preferred oid frame in atomism), euation 7, when using Einstein-Poinaré synhronized loks. This of ourse also means that the Einstein length ontration euation, as well as obserations with Einstein synhronized loks, yields an apparent length ontration and not the true length ontration. The true length ontration is atually di erent than predited by Einstein and as obsered by Einstein synhronized loks. Under ether theories, Einstein s length ontration euation is also onsistent with Einstein-Poinaré synhronized loks, but Einstein- Poinaré synhronized loks ontain an Einstein synhronization error that a ets the measurement of length ontration. I all it an Einstein synhronization error rather than an Einstein-Poinaré synhronization error, as Poinaré likely would hae laimed that the true one-way speed of light was anisotropi een if he laimed it not ould be measured. Furthermore, the Einstein length ontration is not idential to the length ontration initially suggested by FitzGerald and later by Lorentz and Larmor. The length ontration they introdued was, at least initially, the true length ontration related to motion against the ether (oid). In Einstein s speial relatiity theory, the length ontration is due to relatie eloities only and more importantly it ontains

11 an embedded Einstein synhronization error. Howeer, FitzGerald (889), Lorentz (89), and Larmor (900) are not ery lear when they talk about the speed against the ether; they do not explain how this eloity should be measured exatly. Lorentz formalized length ontration for objets moing against the ether in 89, where it is lear that the eloity in his length ontration formula was the eloity of the moing frame (in his examples Earth) against the ether (oid). In his later work, Lorentz was heaily influened by Poinaré, and it is no longer lear whether the eloity in Lorentz euations is just relatie eloity as measured with Einstein-Poinaré synhronized loks or atually the eloity against the ether. 7. Reiproal Length Contration? Einstein and his adherents beliee that length ontration is reiproal, where eah frame an laim it is the other frame that is moing and that a rod in the other frame will appear ontrated relatie to the frame we are obsering it from. For example, assume we are making two idential rods. Next, one rod is plaed on board a train, and the train then starts to moe against the Earth s ground. Obserers on the train and obserers on the ground will now, aording to Einstein s speial relatiity theory, be able to laim that it is the rod in the other frame that ontrats. This seems impossible and against logi; the rod annot shrink in both frames at the same time, an it? This is known as the length ontration paradox. Can length ontration really be reiproal? The orret answer is that the apparent length ontration we obtain when measured with Einstein- Poinaré synhronized loks indeed is reiproal. It is real in the sense that this is what we will measure with suh lok synhronization, but again there is a deeper and more real reality behind this when remoing the lok synhronization error. The true length ontration as measured with loks without synhronization error is not reiproal between frames. Assume one frame is moing at eloity and one frame is moing at eloity against the ether (oid). Two idential rods are made in frame one, and then one of the rods is moed to frame two. The length of the rod at rest in frame one as measured from frame one is L,, and the length of the rod at rest in frame two as measured from frame two is L,. Furthermore, we must hae L, = L,. The length of the rod in frame one as measured from frame two when using Einstein synhronized loks must be (34) L e,, = L,, and the length of the rod in frame two as measured from frame one when using Einstein synhronized loks must be This again means L e,, = L,. (35) L e,, = L e,,. (36) These are not the true rod lengths, but apparent length indiretly ontaining an Einstein synhronization error. Een if only one lok is used to measure the rod s length, we need to know the eloity of the rod relatie to the lok. When measuring the eloity of the rod relatie to the lok when using Einstein synhronized loks, we will indiretly build an Einstein synhronization error into our length measurement. The solution to the Einstein length ontration paradox is that not all the rods in the di erent frames are ontrating; one of them is atually expanding relatie to the other frame, but when using Einstein synhronized loks we only obsere apparent length ontration that indeed leads to reiproal length ontration. The di erent sizes of the Einstein synhronization error in eah frame makes it appear that the length ontration in the opposite frame is the same no matter if we obsere the rod in frame two from frame one or the rod in frame one from frame two. Length ontration is indeed reiproal when using Einstein synhronized loks, but this is an apparent length ontration. 8 Conlusion We hae shown that Einstein length ontration not is the same as FitzGerald and Lorentz length ontration. The FitzGerald and Lorentz length ontration was deeloped for a moing frame and the

12 ether frame. Here we hae extended the FitzGerald and Lorentz length ontration to hold between any two referene frames. The euations are then learly di erent than that of Einstein. Still, when deriing length ontration from Einstein-Poinaré synhronized loks then een if the true one-way speed of light is anisotropi we get exatly the same formula as Einstein. Howeer, we an see that the reiproality is a kind of illusion due to lok synhronization error. Referenes Cahill, R. T. (006a): A New Light-Speed Ansitropy Experiment: Absolute Motion and Graitational Waes Deteted, Progress in Physis, 4, (006b): The Roland De Witte 99 Experiment (to the Memory of Roland De Witte), Progress in Physis, 3, (0): One-Way Speed of Light Measurements Without Clok Synhronization, Progress in Physis, 3, FitzGerald, G. F. (889): The Ether and the Earth s Atmosphere, Siene, 3(38), 390. Freedman, R. A., and H. D. Young (06): Uniersity Physis with Modern Physis, 4th Edition. Pearson. Harris, R. (008): Modern Physis, Seond Edition. Pearson. Haug, E. G. (04): Unified Reolution, New Fundamental Physis. Oslo, E.G.H. Publishing. Krane, K. (0): Modern Physis, Third Edition. JohnWiley&Sons. Larmor, J. J. (900): Aether and Matter: A Deelopment of the Dynamial Relations of the Aether to Material Systems. Cambridge Uniersity Press. Lawden, D. F. (975): Introdution to Tensor Calulus, Relatiity and Cosmology. John Wiley & Sons. Léy, J. (003): From Galileo to Lorentz and Beyond. Aperion. Lorentz, H. A. (89): The Relatie Motion of the Earth and the Aether, Zittingserlag Akad. V. Wet., (), Marino, S. (974): The Veloity of Light is Diretion Dependent, Czehosloak Journal of Physis, pp Monstein, C., and J. Wesley (996): Solar System Veloity from Muon Flux Anisotropy, Apeiron, 3(), Patheria, R. K. (974): The Theory of Relatiity, Seond Edition. Doer Publiations. Poinaré, H. (905): On the Dynamis of the Eletron, Comptes Rendus. Prokhonik, S. J. (967): The Logi of Speial Relatiity. Cambridge Uniersity Press. Rindler, W. (00): Relatiity, Speial, General and Cosmology. Oxford Uniersity Press. Shadowitz, A. (969): Speial Relatiity. Doer Publiations. Torr, D. G., and P. Kolen (984): An Experiment to Measure Relatie Variations in the One-Way Speed of Light, Preision Measurements and Fundamental Constants II, ed. by Taylor B. N. and Phillips W. D. Nat. Bur. Stand. (U.S.) Spe. Pub., 376,