When moving clocks run fast

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1 When moving clocks run fast Allen D. Allen a) Physics Division, New Terra Enterprises, Glorieta (Santa Fe), New Mexico , USA Abstract: An asymmetry in Lorentz contraction can be created with a point like object in uniform motion relative to another object that has a finite, linear, spatial extent along the line of motion (extended object). Let Δt be the time that elapses as the point like object flies by the extended object in the frame of the latter. Let Δt be the time that elapses as the extended object flies by the point like object in the frame of the latter. These elapsed times depend upon the frame of reference. However, because the extended object is Lorentz contracted and the point like object is not, the relationship between the elapsed times is interframe consistent according to Δt/Δt =. In the frame of the extended object, time runs slower in the moving frame, as expected. But in the frame of the point like object, time runs faster in the moving frame. In both frames, the duration of the flyby is shorter in the frame of the point like object. The twin type paradox is eliminated without acceleration. Cosmic ray showers provide a natural example. Analogous to measurement in quantum mechanics, an experimentalist can create a point like object by choosing to measure the uniform motion of an extended object relative to a point in the laboratory frame. Key words: Special Theory of Relativity; Symmetry; Lorentz Contraction; Time Dilation; Twin Paradox; Cosmic Rays; Flybys; Measurement a) allend.allen@yahoo.com 1

2 Résumé: Une asymétrie en contraction de Lorentz peut être créée avec un objet réduit à un point en déplacement uniforme par rapport à un autre objet qui a une dilatation spatiale linéaire finie le long d'une ligne de déplacement (objet dilaté). Soit Δt le temps qui s'écoule quand l'objet réduit à un point vole près de l'objet dilaté dans le référentiel de ce dernier. Soit Δt le temps qui s'écoule quand l'objet dilaté vole près de l'objet réduit à un point dans le référentiel de ce dernier. Ces temps écoulés dépendent du référentiel. Toutefois, parce que l'objet dilaté est contracté de Lorentz mais pas l objet réduit, la relation entre les temps la écoulés est «entre-référentiel cohérent» selon l'équation Δt/Δt =. Dans le référentiel de l'objet dilaté, le temps s'écoule plus lentement dans le référentiel en mouvement, comme prévu. Mais dans le référentiel de l'objet réduit à un point, le temps s'écoule plus vite dans le référentiel en mouvement. Dans les deux référentiels, la durée du survol est plus courte dans le référentiel de l'objet réduit à un point. Le paradoxe des jumeaux n'a plus cours sans accélération. Un exemple naturel nous est fourni par les gerbes de rayons cosmiques. De façon analogue à une mesure en mécanique quantique, un expérimentateur peut créer un objet réduit à un point en choisissant de mesurer le déplacement uniforme d'un objet dilaté par rapport à un point dans le référentiel de laboratoire. 2

3 I. INTRODUCTION Symmetry in the special theory of relativity (STR) is of classical and contemporary interest. 1 6 As should be clear from the relativistic addition of velocities, speed is a symmetric relationship in STR such that a v b = b v a (1) where μ v β is the velocity of μ with respect to β and μ v β is the speed of μ with respect to β. An additional symmetry arises from the postulate that all inertial reference frames are on an equal footing. This leads to an apparent twin paradox 7 9 because each of two observers in uniform relative motion will see that the other is ageing more slowly. The purpose of the present paper is to show that nature breaks the symmetry of Lorentz contraction with surprising results. Despite the occasional claim that Lorentz contraction has been experimentally observed, 10 this is not easily done and the phenomenon is still considered theoretical or even speculative by some Nonetheless, Einstein 13 tells us that Lorentz contraction follows directly from the physical postulates of STR so we should accept it as fundamental to that theory. 14 II. ASYMMETRIC LORENTZ CONTRACTION RESULTS IN THE ABSOLUTE TRANSFORMATION OF RELATIVE TIME Consider a point like object in uniform motion at constant relativistic speed v relative to another object that has a finite, linear, spatial extent of proper length L along the line of motion (extended object). In the frame of the point like object, the extended object has been Lorentz contracted such that its length becomes where L = 1 L, (2) 3

4 γ = (1 v 2 c 2 ) ½, (3) and c is the speed of electromagnetic radiation in a vacuum. According to (2), the elapsed time needed for the two objects to flyby one another as observed in the frame of the point like object is Δt = L v 1 = γ 1 Lv 1. (4) The elapsed time (4) is easily measured in the frame of the point like object. A timer in that frame starts when the front most part of the extended object encounters the point like object and stops when the back most part of the extended object encounters the point like object. In the frame of the extended object that object has its proper length L. Objects at rest in the frame of the point like object that have a finite, linear, spatial extent parallel to the line of motion are moving and are Lorentz contracted in the frame of the extended object. However, the point like object cannot be Lorentz contracted because it does not have any appreciable finite extent in any direction and would hardly be point like otherwise. More to the point, the speed of the point like object relative to the extended object is still v as per (1). Thus, the elapsed time for the flyby in the frame of the extended object is Δt = Lv 1. (5) There are several ways an observer in the frame of the extended object can measure how long it takes the point like object to sweep by the extended object as in (5). For example, a thin laser beam emanating from the point like object that is perpendicular to the line of motion can interact with a photoelectric surface that runs all along the length of the extended object. When the photoelectric surface is illuminated by the laser beam, it sends a current to a timer on the extended object. The timer runs only when the laser beam is incident on the extended object. From (4) and (5) we have 4

5 Δt/Δt =. (6) Needless to say, this does not mean that an observer in the frame of the extended object observes himself to be ageing at a rate that is faster than his proper rate simple because a point like object happens to fly by. Rather, he sees that the elapsed time for the moving point like object runs at a slower rate, as expected. Likewise, an observer in the frame of the point like object does not see that he is ageing at a rate that is slower than his proper rate simply because an extended object happens to fly by. Rather, he sees that the elapsed time runs faster for the moving extended object. In short, the extended object ages more during the flyby and the point like object ages less during the flyby. This precludes the twin type paradox without an appeal to acceleration. The time transformation is interframe consistent as shown by (6). Equation (6) is the chief result of the present paper. The author is indebted to an anonymous reviewer for pointing out that the same result may be obtained using the event calculus. While this may be less intuitive, it may be helpful to use an approach based on Minkowski s spacetime coordinates since this is the familiar formalism for STR. Let the flyby be point events (x,t) and (x t ) in two frames. In the unprimed frame the extended object is stationary with proper length L and the point like object flies by at speed v. In that frame, the events are (0,0) and For the primed frame this becomes (0,0) and (x t ), where (x,t) = (L,Lv 1 ). (7) x = (x vt) = (L vlv 1 ) = 0, (8) t = ( vxc 2 + t) = ( vlc 2 + Lv 1 ) = (1 v 2 c 2 )Lv 1 = 1 Lv 1. (9) Hence, whereas the elapsed time in the unprimed frame is 5

6 Δt = Lv 1, (10) in the primed frame it is Δt = 1 Lv 1 = 1 Δt, (11) as in (6). Analogous to measurement in quantum mechanics, the experimentalist can elect to create this scenario by choosing to measure the motion of an extended object relative to a point in the laboratory frame. However, when point like particles move through a long extent of space, the experimental setup is given by nature as illustrated in the next section. III. A REAL AND ONGOING EXPERIMENT A real example of (6) is provided by the point like ephemeral particles in a cosmic ray shower A detector on the surface of the Earth can detect the particles because, according to (6), the trip through the Earth s atmosphere lasts longer in the frame of the Earth than it does in the frame of the point like particles. This is tantamount to a slowing of the moving particles clocks. The shorter time for the trip through the atmosphere in the frame of the particles allows the ephemeral particles to survive for the duration of the trip without decaying. (Another example of how the correct frame is needed to predict the behavior of particles is found in high energy particle physics experiments. The equivalence of mass and energy only manifests in the zero momentum frame. 17 ) In the frame of the particles where their clocks run at the proper rate, Lorentz contraction of the Earth s atmosphere makes the duration of the trip through the atmosphere shorter than it is in the frame of the Earth, thereby allowing the particles to survive the trip without decaying. The reduced ageing of the particles during their trip through the atmosphere is absolute even when acceleration is negligible or ignored. 6

7 IV. CONCLUSIONS An asymmetry in Lorentz contraction can be created by considering a point like object in uniform motion with respect to another object that has a finite, linear, spatial extent along the line of motion (extended object). The elapsed time it takes for the two objects to flyby one another depends upon the reference frame. But the relationship of those elapsed times is interframe consistent according to (6). In other words, it is true in both frames that the point like object ages less than the extended object during the flyby. Cosmic ray showers provide a real example. Analogous to measurement in quantum mechanics, the experimentalist can create the point like object by choosing to observe the motion of an extended object relative to a point in the laboratory frame. 7

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