Critial Refletions on the Hafele and Keating Experiment W.Nawrot In 1971 Hafele and Keating performed their famous experiment whih onfirmed the time dilation predited by SRT by use of marosopi loks. As it had already been shown [1], the ontinuation of reasoning applied by Hafele and Keating leads to the absurd onlusion that the Earth is not rotating around the Sun. Hafele and Keating derived a proper formula starting from false reasoning and this is the origin of the paradox. They tried to derive the formula from SRT, while the proper derivation an only be obtained from GRT []. There were also serious doubts onerning the experimental part of their work [4,5], but it does not matter now beause today the GPS system onfirms what H&K wanted to prove. Finally, H&K wanted to onfirm SRT but their experiment onfirmed, in fat, the GRT. If we take a loser look at other experiments onfirming SRT, we will ome to the onlusion that all the experiments in fat onfirm GRT similarly to the H&K experiment, beause in order to ompare times in two referene frames we have to disturb motion of one of the frames and this brings the problem to problems desribed by GRT. The pure inertial motion makes observation from two observed eah other frames fully symmetrial, and we are not able to define in whih of the frames the time flies slower. In this ase we an draw the onlusion that the slowing of time in pure inertial motion an be only an observational effet. Only the hange of speed of one of the partiipants transforms the observed time dilation into the real one. Therefore we an ask the questions are there any serious experimental evidenes onfirming SRT? And - if in order to register the shortening of time the hange of speed is neessary an we assume, as it is done in SRT, that time slows down as a funtion of veloity.? Maybe the slowing of the time is rather a funtion of the hange in veloity than the veloity itself? In my previous papers [1,], I was writing about the paradox resulting from the experiment preformed by Hafele and Keating. The idea of this experiment was to ompare indiations of atomi loks positioned in three oordinates systems moving in relation to one another: in the jets traveling around the world to the west and east, and on the Earth s surfae. Hafele and Keating wanted to prove the preditions of SRT with the help of marosopi loks. Solution of any problem with the help of SRT is possible only in the frame of inertial observer. Sine all the loks taking part in the experiment are performing a rotational motion around the Earth s axis, Hafele and Keating deided to aept a risky assumption that the inertial frame should be positioned on the Earth s pole. The assumption was risky beause aording to SRT the oordinates system should be bound with one of the loks taking part in the experiment [3]. The binding of the inertial system with the Earth s pole was onneted with adding the veloity of rotational motion of the Earth around it s axis to the veloity of eah of the onsidered loks. This should hange the predited results of the experiment and thus, aording SRT, suh assumption should not be allowed. However, Hafele and Keating argued that what is not allowed for linear motion an be allowed for the rotational one. They started from the SRT formula desribing the time dilation as a funtion of veloity:
(1) t 1 ω r ( ωr) ( ωr ) 1 1 time of the lok on the Earth s surfae time indiated by the moving lok t time of the hypothetial lok on the Earth s pole ω - Earth s angular veloity r radius of the Earth Finally, the hange of time resulting from the differene of height of the loks - added, and the formula took a following form: ( ) () ± ω ν gh ν - veloity of plane in relation to the Earth s surfae h- height of the flight in relation to the Earth s surfae gh / - was The formula, although based on the risky assumption, is orret (see appendix for detailed derivation). However, binding the inertial oordinates system with the enter of rotation allows to aept, as the inertial frame, also another enter of rotation; for instane, the pole of the Sun. Suh an assumption will hange the veloities of all of the loks taking part in the experiment, by a great omponent (approx. 1 times higher than the veloities of the loks in relation to the Earth s axis), and this should hange the predited indiations of the loks. In fat, suh hange of the rotation s enter brings the hange of predited results by several thousand perent, giving a very strong dependene of the predited results on the hour of the start and landing see formula (3). (3) {( ± ω ν ) ( ω r ± ν ) V os[( ω ± ν / r)( t t )] ω rv os[ ω ( t t )] gh} / V veloity of the Earth rotation around the Sun t time of the plane s takeoff t time (h:min:se) of the day The absene of suh dependene, during the experiment, together with an assumption that the reasoning of Hafele and Keating is orret, leads to the absurd onlusion that V (in eq. 3), whih is equivalent to the statement that the Earth is not rotating around the Sun. Moreover, binding, in turn, the referene frame with the enter of the galaxy, we are oming to the analogous onlusion that the galaxy is not rotating either et., et.
The orret solution of the problem an only be obtained if we are onsidering the problem aording to the GRT. We are starting from the desription of motion in the onstant gravitational field. V (4) g 1 t MG g 1 r M mass of the Earth r radius of the Earth time indiated by the moving lok t the time at an infinite distane from the Earth in the referene frame in whih the gravitational field of the Earth is onstant. Substituting for V in formula (4) respetively the veloities of loks in jets and the lok on the Earth s surfae in relation to the hypothetial referene frame on the Earth s pole, we are also obtaining the formula (). The formula derived on the basis of GRT is idential with the one derived by Hafele and Keating. However, following the assumption of the onstant gravitational field, the referene frame onneted with the Sun or with the entre of the galaxy must rotate together with the Earth (rotating around the Sun or the enter of the galaxy) and it prevents the addition of any veloity, resulting from the rotational motion, to the veloities of the loks. Hene, the problem of the loks is not the problem of the inertial motion any more, but the problem of the non-inertial motion whih an be properly solved only with the help of GRT. Therefore, the Hafele and Keating experiment did not onfirm the preditions of SRT, whih was the authors intention. Apart from the doubts onerning their experimental method and the method of interpretation of the experimental results [,4,5], their experiment onfirmed only the preditions of GRT. On the other hand, the fat of the time dilation during the flight around the Earth is onfirmed every day by the GPS system. However, the formula desribing this time dilation is derived on the basis of GRT in a way similar to the one shown in the Appendix. The paper [1] proving, on the basis of the Hafele and Keating experiment, that the Earth is not rotating around the Sun, ould be treated as a joke or a ridiule. However, the problem that Hafele and Keating enountered, seems to be more serious that it appears. The analysis of the Hafele and Keating experiment and its onlusions enouraged me to take a loser look at other experiments onfirming preditions of SRT. It proved that, similarly to the Hafele and Keating experiment, other experiments have proved, in fat, GRT while experimental onfirming of SRT seems to remain impossible. I would like to start the onsideration from a disussion of a ompletely symmetrial problem: two twins in two idential rokets are in empty spae. In a ertain moment they are taking off in opposite diretions and are moving with idential veloities. After reahing a ertain distane, a randomly seleted twin is turning bak and after some time he athes up with the seond twin. Then they are omparing their loks. Let s notie that in the desribed ase both twins were mutually observing the time dilation in their frames, but finally the hange of the veloity was the main phenomena, deiding whih of them was younger. As
long as the twins were moving with a uniform motion, they were mutually observing the time dilation in their frames but it was not possible to hek if their times were really flowing slower or whether the slowing of time was only a seeming result being the effet of the observation. The ondition whih is neessary to register the real time dilation in the observed frame is the hange of the line, along whih the events are simultaneous, being the result of the hange of the veloity [6]. Hene the randomly hosen twin, whose roket has turned bak, will register a shorter time than the twin still moving with the uniform motion. During the analysis of the twin paradox, or other similar phenomena, only the time dilation resulting from the SRT is taken into aount, while the moment in whih the veloity had been hanged is ignored. Meanwhile, as was mentioned before, if the observed time dilation is to beome the real dilation, it is neessary to take into onsideration the moment in whih the hange of veloity took plae. It seems that this moment an not be ignored during the analysis of the phenomenon. The above remark is not related to the quantitative desription, beause the results obtained with the help of SRT are in aordane with the measurements. We are talking now about the qualitative desription whih, when the point in whih the veloity is hanged is ignored, gives us the false justifiation of the hange of the time. Finally, suh a false justifiation hinders further attempts at understanding suh enigmati physial variable as time and, as it has been shown on the example of the Hafele and Keating experiment, an lead to various paradoxes. Now let us ome bak to the Hafele and Keating experiment desribed above, in whih the attempt at desribing non-inertial motion with the help of SRT led to a series paradoxial onlusions. If we are desribing a non-inertial motion (in ase of the Hafele and Keating experiment it is a rotational motion around the Earth s axis) with a help of differential equations, then in fat we are desribing series of infinitesimally small segments, along whih the body is moving with uniform motion aording to SRT, and a series of points, between this segments, in whih the lines, denoting events that are simultaneous, are hanging. Hafele and Keating in their theory, treating the problem only with the SRT apparatus, took into aount only the motion along these segments, while the points between the segments in whih the lines, denoting events that are simultaneous, are hanging, had been negleted fig. 1. Finally, they obtained a orret solution similarly as it has been obtained in the ase of the twins paradox desribed above but the result they obtained was based on a false reasoning and this allowed for drawing a series of absurd onlusions. Fig.1 No- inertial motion in the differential notation onsists of a series of infinitesimally small segments on whih the inertial motion takes plae. Let s notie that ignoring, in the desription of the twins paradox, the point at whih the line denoting events that are simultaneous is hanging, we are doing quite the same mistake as
Hafele and Keating did in their experiment. In the ase of the twins paradox, drawing any suh spetaular absurd onlusion, as in the ase of the Hafele and Keating experiment, has not sueeded yet. However, the mehanism of desription of the twins paradox and desription of the loks motion in the Hafele and Keating experiment are idential the mistakes are made, but due to their agreement with the experimental data, it is diffiult to disover them. Finally, we are still assuming that the hange of the time flow in the moving frame is only the result of the veloity, while the true physial reasons for this phenomena an be quite different. The relationship between the veloity and the hange of time flow is unquestionable; however, there exists additionally, a still ignored effet of hanging of the lines, denoting events that are simultaneous, whih is neessary to hange the observed time hange into the real time hange and whih deides whih partiipant of the experiment will age at a slower rate. Just the analysis of the influene of the moment in whih the line, denoting events that are simultaneous, is hanging, on the flow of the time, an be a lue for understanding the nature of suh an enigmati physial variable as time. Hene, analogously as in the ase of the Hafele and Keating experiment, the problem of the twins paradox and all other similar problems should be desribed with the help of GRT, taking into aount the point in whih the hange of the line denoting events that are simultaneous takes plae, and not only with the help of SRT. Conlusions. As it has been shown in this paper, the Hafele and Keating experiment proved the GRT theory, while it was intended to prove the SRT. All the other experiments intended to onfirm SRT, in reality probably also onfirmed GRT instead of SRT. The experimental omparison of times indiated by loks, lifetimes of partiles et., requires, in pratie, the disturbing of motion of one of the frames taking part in the experiment. Hene, regaress of the fat that while performing the experiment we an get an impression that we are proving SRT, we are in fat proving GRT. It is not possible to ompare loks without suh a disturbane, therefore we will never be able to analyze the pure SRT problem. The only proof for pure SRT is the onstany of the speed of light in all oordinates systems. However, reently, in the model of Eulidean Reality alternative to Relativity Theory [7,8], the onstany of the speed of light has been justified without the neessity of assuming the deformation of the spae-time. Therefore, it seems to be reasonable to take a more detailed look at the problems of the moving bodies and analysis of the influene of the moment when the veloity is hanging on the fat of the slowing of time in moving frames. Appendix Derivation of the H&K formula on the basis of SRT and GRT Inorret derivation on the basis of SRT (Hafele and Keating) Hafele and Keating assumed that the time indiated by the lok positioned on the Earth s surfae should be desribed by the formula:
(5) t 1 ω r ( ωr) ( ωr ) 1 1 time of the lok on the Earth s surfae time indiated by the moving lok t time of the hypothetial lok on the Earth s pole ω - Earth s angular veloity r radius of the Earth Next, the time indiated by the lok in the plane should be desribed by the formula: (6) ( ω r ν ) ± t 1 : ν the veloity of the plane in relation to the Earth s surfae. When we divide formula (5) by (6) we get: ( ω r ν ) ± 1 1 ω r and finally: ± ω ν (7) ( ) for the obtained time hange, the hange of time resulting from the differene of height of the loks - gh / - was added : h height of the flight of the plane g the Earth s aeleration The final formula has the form: ( ) (8) ± ω ν gh
Corret derivation on base of GRT If one would like to obtain the formula in the orret way, it would be neessary to start from the GRT, from the formula desribing the time of the body moving in onstant gravitational field: (9) g V t g 1 MG r M mass of the Earth r radius of the Earth time indiated by the moving lok t the time at an infinite distane from the Earth in the referene frame in whih the gravitational field of the Earth is onstant. V veloity of a lok in relation to the enter of the Earth whih is equal to ω r for the lok on the surfae of the Earth and ω r ± ν for the lok in the jet. Here the time of the lok on the surfae of the Earth is desribed with the formula: MG (1) ω r t 1 1 r while the time indiated by the lok in the jet is desribed by the formula: ( ) (11) MG ω r ± ν t 1 1 ( ) r h hene, dividing (11) by (1) we obtain for week gravitational field, low veloities and h<<r: ( ω r ν ) MG 1 ± 1 ( ) r h MG 1 ( ) (1) MG ω 1 r 1 r h r MGh 1 1 ( ± ω ν ) r and from the above we instantly obtain the formula (8) ( ) (13) ± ω ν gh ( ω r ± ν ) MG r ω r Referenes 1. W.Nawrot, Phys. Essays 17, 518 (4). W.Nawrot Critial Refletions on the Hafele and Keating Experiment Pro. of Conf. Mathematis, Physis and Philosophy in the Interpretations of Relativity Theory Budapest, 7-9 Sept. 7. 3. R. Shlegel, Nat. Phys. Si. 9, 37 (1971). 4. A.G. Kelly, Phys. Essays 13, 616 ()
5. D. E. Spener, U.Shama Pro. International Sientifi Congress Fundamental Problems of Natural Siene and Engineering Saint-Petersburg, Russia 6. E.S.Lowry, Amerian Journal of Physis 39, 59 (1963) 7. W. Nawrot, Proposal for a Simpler Desription of SRT, Galilean Eletrodynamis 18, 43-48 (7) available also in http://www.asterity.net/~witnaw/eng1/thenewmodelofreality.html 8. W.Nawrot Four Dimensional Eulidean Reality The basi properties of the model Pro. International Sientifi Congress Fundamental Problems of Natural Siene and Engineering Saint-Petersburg, Russia 8 9. J.C. Hafele, Nature 7, 7 (197) 1. J.C. Hafele and R.E. Keating, Siene 177, 166 (197) 11. J.C. Hafele, Nat. Phys. Si. 9, 38 (1971).