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Home , Length contraction, Proper length, Proper time, Spacetime, Theory of relativity, Time dilation

Open Access Journal of -4, Issue-2, 2020, PP 20-25 ISSN 2637-5826

A Oddity: Dilation and Contraction for the Amateur Enthusiast

Zion Elani* Institute for Nanotechnology and Advanced Materials, Bar-Ilan University, Israel *Corresponding Author: Zion Elani, Institute for Nanotechnology and Advanced Materials, Bar- Ilan University, Israel.

ABSTRACT is undoubtedly one of the pillars of where concepts such as and subtly play a role in various aspects of nature. Typically veiled under complex and difficult- to-fathom , the path to understanding these phenomena can leave a novice student lost and confused. In this lecture notes, we attempt to explain and arrive at these concepts using physically intuitive methods and elementary mathematics without the use of advanced mathematical knowledge to make it easier for high school students and amateur enthusiasts to comprehend.

INTRODUCTION predicts that the observer on would see the in the satellites ticking in a slower rate than It has been more than a hundred since the the ones on Earth because of the time dilation dawn of modern physics. Yet, even today, effect of their relative .[1] Time dilation surprisingly, high school chemistry and biology accounts that for an observer at an inertial frame of students have not been able to fully comprehend reference, the clock moving relative to him will the concepts of modern physics. One of the appear to tick slower than the one ticking in his main reasons for their failure to do so is the own inertial . The faster the utilization of difficult mathematical tools such relative , the greater the time dilation. as calculus for the derivation of these concepts Similarly, the phenomenon of length contraction which in turn intimidate and the students is observed in a moving object when it appears and amateur enthusiasts.[5] to be shorter than its actual length in its resting state (own inertial frame of reference), responsible Both important and basic concepts of modern in part for the working of electromagnets.[2] physics such as time dilation and length contraction play a significant role without us After encountering the role of special relativity being aware of it in electromagnets,[2],[13] in these simple -to-day examples, our minds cathode ray tubes,[8] the yellow color of Gold[11] are forced to delve deeper into understanding and liquidity of .[12] Ever wondered how these phenomena in order to seek their origin. GPS (Global Positioning System) technology But in the quest to find these answers, an amateur works and maintains its accuracy? When enthusiast or a novice student might find pondering this question, the significance of special themselves confused and lost in the complex, relativity usually is not taken into account. mathematically difficult derivations and analyses, However, in order for the GPS to be applicable ultimately serving nothing to their curiosity. in our day-to-day life, it is important for it to Moreover, lecturers and teachers often struggle in maintain higher precision by 20 − 30 nanoseconds their attempt to explain those concepts to such which can only be achieved by taking special students in a simple and heuristic way without relativity into account1 as the satellites carrying using advanced mathematical tools. It is high time the atomic are relatively in motion to the we develop simple, pedagogic, intuitive and observers on earth. That is, relativity easily comprehensible methods to satisfy the curiosity of young minds, further fueling their

desire to question the mysteries of nature and 1As a of fact, should also be while encouraging them to quest for taken into account. The effect of gravitational frequency more. In [3] the reader can find a pedagogic shift on the GPS is that a clock closer to a massive object exposition that supplements our approach. will be slower than a clock farther away.

Open Access Journal of Physics V4 ● I2 ● 2020 20 A Spacetime Oddity: Time Dilation and Length Contraction for the Amateur Enthusiast

In[5],[6] we have shown how to derive the • The speed of in a is the same – equivalence, 퐸 = 푚푐2, in a simple for all observers, regardless of the motion of and pedagogic way. Here we continue our the light source or observer. pedagogical approach by presenting the derivation The first postulate upon which Einstein based of time dilation and length contraction. In this note the theory of special relativity relates to we explain a simple, easily understood and reference frames. One should keep in mind that physically intuitive method based on a thought the motion of a body can only be described experiment that uses relative to an observer, or a set of -time equipped with pictorial representations to derive coordinates. These are called frames of reference. the concept of time dilation and length contraction For example, a car’s motion is measured relative in a student-friendly way with the only to its starting . An inertial frame of reference assumptions here being the constancy of the speed is a reference frame in which a body at rest of light, and that the magnitude of relative remains at rest and a body in motion moves at a velocity, 푣, is the same in every reference frame, constant speed in a straight unless acted on i.e., between two observers (though we do by an outside . While driving in a constant introduce the concept of inertial frame and the velocity (on a straight line) you cannot tell how fast two postulates of special relativity). We will show your car is travelling without looking out the how the derivation of time dilation and length window. These are the simplest frame of contraction can be intuitively reached using a references. In such a reference frame the physical simple light-clock. These lecture notes derive from laws should be the same, but that they may vary the fact that they may aid the minds of students across non-inertial ones. The assumption of and amateur enthusiasts for an easy grasp. While special relativity is that light always travels at the they provide no original contribution to the same speed for every observer, regardless of that that has been widely discussed over the years, observer’s velocity. We write the constant speed of we hope that they may be of some help to light in 푐. The invariance of the leads teachers and curious students alike. to counter-intuitive phenomena, such as time The next section briefly explains the theory of dilation and length contraction. More precisely, the special relativity. It is not necessarily crucial to time interval in which two events occur at the same understanding fully the contents of this section point in space appears longer to a moving for one to follow the derivation explained in the observer. Thus, a moving clock appears to run latter sections. All we need to keep in mind is slow. And an object appears shorter to an that the speed of light is a constant regardless of observer who is moving relative to the object. In the motion of the light source or observer and the next two sections we will show how these that the magnitude of , 푣, is the phenomena can be intuitively explained. same in either reference frame. The following section describes the phenomenon of time LIGHT-CLOCK AND PYTHAGOREAN THEOREM dilation using an easily perceivable derivation - A DERIVATION OF TIME DILATION based on elementary mathematics. In the later As we explained in the introduction to this note, sections we explain the origin of length special relativity indicates that time dilation contraction, again using a simple derivation means that observers moving at a relative followed by a subsequent and concise summary. velocity do not measure the same elapsed time for an . More precisely, for an observer in an SPECIAL RELATIVITY IN A NUTSHELL inertial frame of reference, a clock that is moving Special relativity explains how space and time relative to him will be measured to tick slower are linked for objects that are moving at a than a clock that is at rest in his frame of reference. constant velocity.[4], [14], [7] This beautiful Put simply, ’moving clocks run slow’. theory describes some remarkable features about space and time where moving clocks run slow and To understand how time dilation comes into , let us first build a light-clock consisting of two moving meter sticks are short (along the direction of motion).[15],[9] These phenomena have been mirrors, 푀1 and 푀2, placed perfectly to one proven by a lot of experiments. Special relativity is another as in Fig. 1 and separated by a 푑. based on two postulates which can be simply This clock will measure time in terms of the written as: distance travelled between the mirrors by a light pulse at speed 푐. One period in this clock would • The laws of physics are (identical) be completed when the light pulse travelling in all inertial frames of reference (i.e. non- from the mirror 푀1 is reflected by 푀2 (’tick’) accelerating frames of reference).

21 Open Access Journal of Physics V4 ● I2 ● 2020 A Spacetime Oddity: Time Dilation and Length Contraction for the Amateur Enthusiast and reaches back to 푀1 (’tock’) as shown in Fig. 1. The distance between the two mirrors is 푑 and the clock period is 훥푡0 = 2푑/푐. Hence, the time taken to complete one period covering twice the length 푑 is given by: 푑𝑖푠푡푎푛푐푒 2×푑 Δ푡 = 2 × = = 1 푠푒푐표푛푑 (1) 0 푠푝푒푒푑 표푓 푙𝑖푔ℎ푡 푐 Consider the same light-clock placed in a train moving at a constant velocity 푣 relative to a stationary observer on the platform (it does not matter whether the motion is to the right or to the left). Since the light-clock in the train is in a relative motion to the stationary clock at the Figure1. The light clock consists of a light pulse platform, the light pulse in this case travels in a bouncing between two parallel mirrors 푀1 and 푀2. way as shown in Fig. 2.

Figure2. Relative to an observer on the platform, the light-clock moves at velocity 푣. For the observer on the platform that measure time with sides given by 푑 and 푙. The light pulse’s on this light-clock, one period is again given by the path from mirror 푀1 to mirror 푀2 and back to distance travelled by the light pulse from 푀1 to 푀2 mirror 푀1 is given by 2 × ℎ (see Fig.3). Hence, and back to 푀1 divided by the speed of light, 푐, but the time period, Δ푡measured by the observer on 2×ℎ since the light-clock is moving with the train at the the platform is givenby, Δ푡 = . (2) velocity 푣, the distance covered by the light pulse is 푐 not the same as in the case of the stationary light- From the point of view of an observer on the clock. The light pulse’s path is actually longer and platform, the distance the train covers as the light is described by the of the right goes across it is given by 푥 = 푣Δ푡, (see Fig.3):

Figure3. Pythagorean theorem: For a right-angled triangle the of the hypotenuse is equal to the sum of the of the other two sides, ℎ2 = 푙2 + 푑2. 푑𝑖푠푡푎푛푐푒 푡ℎ푒 푡푟푎𝑖푛 푐표푣푒푟푠 ( ) = the distance the train covers as the light goes ⏟푎푠 푡 ℎ 푒 푙𝑖푔 ℎ 푡 푝푢푙푠푒 푔표푒푠 푎푐푟표푠푠 𝑖푡 across it (see Fig.3): 푥 푟푒푙푎푡𝑖푣푒 푡𝑖푚푒 푎푠 푚푒푎푠푢푟푒푑 푏푦 푎푛 푣Δ푡 ( ) = ( ) (3) 푙 = . (4) ⏟푣푒푙표푐𝑖푡푦 ⏟표푏푠푒푟푣푒푟 표푛 푡ℎ 푒 푝푙푎푡푓표푟푚 2 푣 Δ푡 The hypotenuse, ℎ, of this right triangle can thus From this we see that the side 푙 is equal to half be calculated using well-know Pythagorean

Open Access Journal of Physics V4 ● I2 ● 2020 22 A Spacetime Oddity: Time Dilation and Length Contraction for the Amateur Enthusiast theorem applied to our case as in Fig. 3, We should pay attention that for much smaller than the speed of light 푐, 푣 << 푐, the 훾 ℎ2 = 푙2 + 푑2. (5) factor (eq.11) is almost 1. As such, in order to feel Thus, applying Pythagorean theorem, we can time dilation, the velocity 푣 in the 훾 factor should find the relation between the time Δ푡 as be large enough, close to 푐. But this cannot be measured by the observer on the platform (who achieved in day-to-day life. Thus, time dilation is observed the moving light-clock) and Δ푡0, the infinitesimal and hardly noticeable by current observed on the train (where the scientific instruments.[16] However, satellite clocks light-clock is at rest): make use of this equation in order to maintain accuracy in Earth clocks. Due to their constant 푣Δ푡 2 ℎ = √푑2 + 푙2 = √푑2 + ( ) , (6) motion relative to the clocks on Earth, the atomic 2 clocks on the satellites fall 7 microseconds where we used (eqs.2-4). (10−6) behind per day than the ones on Earth. The above equation enables us to relate Δ푡 This difference must be rectified in order to (eq.2), the time measured by an observer on the maintain the accuracy of the GPS system.[1] platform, to the proper time, Δ푡 , (eq.1) as 0 This equation for Δ푡 is truly remarkable. The measured by the observer who stands on the states that time is not train. By taking the square of these two absolute and being relative as per coordinates equations we get expressions for (Δ푡)2(Δ푡 )2: 0 and speed, it undergoes dilation. Most intriguingly, 2 2 2 2 4푑 푣 2 2 4푑 time-dilation is valid not only for a light-clock but (Δ푡) = 2 + 2 (Δ푡) (Δ푡0) = 2 (8) 푐 푐 푐 for all other clocks in the universe, be it biological 2 From this we get the relation between (Δ푡) and clocks, sand clocks and likewise, thus extending its 2 (Δ푡0) , namely observance to every asset encapsulated by the 푣2 (Δ푡)2 − (Δ푡)2 = (Δ푡 )2. (9) universe. Time itself is not absolute, time passes 푐2 0 slower for an observer who is moving relative to The above equation can be simply written as, another observer. 푣2 (Δ푡)2 (1 − ) = (Δ푡 )2. (10) 푐2 0 LENGTH CONTRACTION - DISTANCE DEPENDS ON AN OBSERVER’S MOTION By taking the square root of this last equation and introducing the 훾 factor In the last section, we showed that clocks measure different time between two different 푣2 훾 = √1 − ,(11) observers that are in relative motion. Even 푐2 though our light-clock measures different time we finally get: in relative frames (the train relative to observer Δ푡0 on the platform), the relative speed, 푣, i.e., the Δ푡 = = 훾Δ푡0 (12) 푣2 distance divided by time elapsed, remains √1− 2 푐 constant. How is that possible? What this means In plain English, (eq.12) tell us, is that the other quantity, i.e., distance or length 푡𝑖푚푒 푎푠 푚푒푎푠푢푟푒푑 푏푦 푎푛 must change as well for it to be possible. ( ) = 훾 ⋅ ⏟표푏푠푒푟푣푒푟 표푛 푡ℎ 푒 푝푙푎푡푓표푟푚 Suppose a rod of 퐿0 is placed at Δ푡 푝푟표푝푒푟 푡𝑖푚푒, 푎푠 푚푒푎푠푢푟푒푑 푏푦 푎푛 rest on the platform and the light-clock is placed ( ) (13) ⏟표푏푠푒푟푣푒푟 표푛 푡ℎ 푒 푡푟푎𝑖푛 at rest on the train moving parallel to the rod Δ푡0 with speed 푣, see Fig. 4.

Figure4. A rod of proper length 퐿0 is placed at rest on the platform and the light-clock is placed at rest on the train moving parallel to the rod with speed 푣.

23 Open Access Journal of Physics V4 ● I2 ● 2020 A Spacetime Oddity: Time Dilation and Length Contraction for the Amateur Enthusiast

The states that the velocity the two observers. By using the expression for 푣 is the same for the two observers. The time dilation (eq.12), i.e., observer on the platform sees the train moving Δ푡0 Δ푡 = = 훾Δ푡0 (18) at speed 푣, while the observer on the train sees 푣2 √1− the the platform (and the rod) passing in the 푐2 opposite direction at speed 푣. we can see that the ratio between the length on the platform,퐿 , and the length on the train, 퐿, is What will be the travel time of the light-clock 0 given by, (at rest on the train) from one end of the rod to 퐿 (Δ푡 )푣 1 1 the other? From the point of view of the = 0 = = , (19) 퐿 (Δ푡)푣 푣2 훾 0 √1− observer on the platform, due to time dilation, it 푐2 will simply be, where we used the expression for the 훾 factor, Δ푡 = 퐿0/푣 = (eq.11). We therefore conclude that the observer proper length of the rod as measured on the platform (14) on the train measure the length of the moving train′s velocity rod to be The observer on the train (where the light-clock 퐿0 퐿 = = 퐿0/훾. (20) is at rest) does not see any time dilation and his 푣2 √1− 2 light-clock time period is (eq.1). As such he will 푐 measure travel time of the light-clock (at rest on In words, (eq.12) tell us, 푙푒푛푔푡ℎ 표푓 푡ℎ푒 푟표푑 1 the train) from one end of the rod to the other as, ( ) = ⋅ ⏟푎푠 푚푒푎푠푢푟푒푑 표푛 푡ℎ 푒 푡푟푎𝑖푛 훾 퐿 Δ푡 = = 퐿 0 푣 푝푟표푝푒푟 푙푒푛푔푡ℎ, 푎푠 푚푒푎푠푢푟푒푑 length of the rod as measured on the train ( ) (21) (15) ⏟푏푦 푎푛 표푏푠푒푟푣푒푟 표푛 푡 ℎ 푒 푝푙푎푡푓표푟푚 rod′s velocity 퐿0 We used to think that the length of the rod, According to (eq.20), the rod from the point of measured either on the train or on the platform, view of the train is shorter than the rod on the should be one and the same. But as we see from platform. Length, then, is not an absolute the above equations, the elapsedtime is different! The expression for the length of the concept, it depends on the observe. Since the rod (the two end points of the rod) on the light-clock’s travel time across the rod’s platform (where the rod is at rest) is given by, endpoints is longer on the platform than on the train, the rod’s length is also longer on the 퐿0 = Δ푡 × 푣 (16) platform than on the train (length contraction However, from the point of view of the observer observed on the train). As a matter of fact, the on the train, the length of the moving rod is: observer on the platform will see the two parallel mirrors of the light-clock shorter than 퐿 = Δ푡 × 푣 (17) 0 an observer on the train where the light-clock is Surely the length of the rod is different between at rest, see Fig. 5.

Figure5. An observer on the platform will see the two parallel mirrors of the light-clock shorter than an observer on the train (where the light-clock is at rest).

We need to keep in mind that length contraction it is not felt in the day-to-day speed, being only is observed in the direction of the relative perceivable at speeds closer to the speed of light motion of the body. Also, as with time dilation, 푐.[10]

Open Access Journal of Physics V4 ● I2 ● 2020 24 A Spacetime Oddity: Time Dilation and Length Contraction for the Amateur Enthusiast

SUMMARY , 322(10):891–921, 1905. https://doi.org/10.1002/andp.19053221004 In this lecture note we continue our pedagogic [5] Zion Elani, “A Brief Pedestrian Derivation of 퐸 = presentation of introducing concepts in special 푚푐2 for the Amateur Enthusiasts”, Open Access relativity for amateur enthusiasts.[5],[6] We Journal of Physics, 4(2), 2020, pp 17-19. have introduced a simple, memoryless and https://www.sryahwapublications.com/open-access physically intuitive line of approach to derive -journal-of-physics/volume-4-issue-2/# the concepts of time dilation and length [6] Zion Elani,“Standing on the Shoulders of Giants: contraction to enable amateur enthusiasts and Derivations of Einstein’s 퐸 = 푚푐2 from beginners to fully comprehend and appreciate Newtonian Laws of Motion”, Open Access Journal modern physics. Firstly, a simple derivation for of Physics, 4(2), 2020, pp 1-8. https://www time dilation with a pictorial presentation was .sryahwapublications.com/open-access-journal-of- given. This method used basic Pythagorean physics/volume-4-issue-2/# theorem to derive the concept of time dilation [7] G. Farmelo. It Must be Beautiful: Great Equations along with the fact that the speed of light is of Modern . Granta, 2003. constant. The following section explained in a [8] A.P. French. Special Relativity. Taylor & Francis, similar pedagogic way the concept of length 1968. contraction using the time dilation derivation [9] D. Halliday, R. Resnick, and J. Walker. Funda- and the fact that the magnitude of relative mentals of Physics. John Wiley & Sons, 2010. velocity, 푣, is the same in either reference frame [10] Oleg D. Jefimenko. On the experimental proofs of (i.e., between two observers). We avoid the use of relativistic length contraction and time dilation. the relativity of simultaneity concept and didn’t Zeitschrift für Naturforschung A, 53(12):977–982, introduce ; we have December 1998. https://doi.org/10.1515/zna-1998- discarded their use completely. We hope that our 1208 approach will help mathematically ill-equipped [11] Donald R. McKelvey. Relativistic effects on students to understand better such complex chemical properties. Journal of Chemical concepts without the prior knowledge of advanced Education, 60(2):112, Feb 1983. https://doi.org mathematical tools and analysis techniques. We /10.1021/ed060p112 believe that the treatment presented here has a great [12] Lars J. Norrby. Why is mercury liquid? or, why do pedagogic merit as students will be able to grasp it relativistic effects not get into chemistry textbooks? easily and gracefully. We also hope that such kind Journal of Chemical Education, 68(2):110, Feb 1991. https://doi.org/10.1021/ed068p110 of treatments to explain and derive complex will keep satisfy students curiosity and [13] R. G. Piccioni. Special relativity and in an introductory physics course. The Physics awaken their thirst for more. Teacher, 45(3):152–156, March 2007. https:// REFERENCES doi.org/10.1119/1.2709673 [14] J. Stachel, M.J. Klein, and D.K. Buchwald. The [1] Neil Ashby. Relativity in the global positioning Collected Papers of : corre- system. Living Reviews in Relativity, 6(1), January spondence, May-December 1920, and supplemtary 2003. https://doi.org/10.12942/lrr-2003-1 correspondence, 1909-1920, English . - [2] B.H.Chirgwin,C.Plumpton, and C.W. Kilmister. 2006. - xxv, 373 pp. Volume 10: The Berlin years. Magnetic Fields, Special Relativity and Potential Princeton University Press, 1987. https://einstein- Theory: Elementary Electromagnetic Theory. papers.press.princeton.edu/vol2-doc/ Elsevier Science, 2013. [15] E.F. Taylor and J.A. Wheeler. Spacetime Physics. [3] Ira Mark Egdall. Teaching special relativity to lay W.H. Freeman, 1992. students. The Physics Teacher, 52(7):406–409, [16] Alexandra Witze. Special relativity aces time trial. October 2014. Nature, September 2014. https://doi.org/10.1038 [4] A.Einstein. Zur elektrodynamik bewegter körper. /nature.2014.15970

Citation: Zion Elani, “A Spacetime Oddity: Time Dilation and Length Contraction for the Amateur Enthusiast”, Open Access Journal of Physics, 4(2), 2020, pp 20-25. Copyright: © 2020 Zion Elani. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

25 Open Access Journal of Physics V4 ● I2 ● 2020