Special Relativity in Week One: 1) The Principle of ... - Loreto-Unican

went up in the air over Boston, the Earth turned for threehours under the plane, and then the plane came back down inSan Francisco. It was the Earth that moved, not the plane.Definition of the principle of relativityThe stage is now set for a more precise definition of theprinciple of relativity. Imagine that you are in a capsule andyou may have any equipment you want inside the capsule. Theprinciple of relativity states that:There is no experiment that you can performthat will allow you to determine that thecapsule is moving with uniform motion.We point out that our classroom is such a capsule. Due tothe rotation of the Earth, the motion of the Earth about theSun, the fact that the Sun is carried around by rotation ofthe Milky Way galaxy, and then there is the expansion of theuniverse, all these contribute to the motion of our classroom.But only the rotation of the Earth, which carries us aroundin a circle every 24 hours, has been directly observed. Whenyou are moving in a circle, you are not moving in a straightline. This slight deviation from uniform motion can bedetected with a Foucault pendulum.Speed of wavesThere is one experiment that was expected to allow us todetect uniform motion. That experiment involved the measurementof the speed of a light wave.To illustrate why such an experiment might be important,we start with an analogous experiment that is easier to observe.It is the measurement of the speed of a compressionalpulse on a stretched Slinky. Figure 2 is a photograph of astretched Slinky suspended from a steel pipe. There is a supportingthread from the pipe down to every fourth coil of theSlinky.If you pull back on one end of the Slinky and let go, you geta compressional pulse like the one shown in Fig. 3, that travelsdown the Slinky, reflects at the far end, and returns. Althoughwe had demonstrated these Slinky pulses for many years, onlyrecently did we do a careful analysis of the speed v pulse of acompressional pulse. The result was v pulse= KL / µ , where Kis the Hooke’s law spring constant, L the length of the Slinky,and m its mass per unit length. 3,4A phenomenon observed by David Keeports 5 is that thetime it takes for the pulse to travel down the spring and comeback does not depend on how much you stretch the spring. Tosee why, write the mass per unit length as m = m/L, where m isthe mass of the spring. Then our formula for v pulse becomes2vpulse = KL = KL = LK .m / L m mThe time T it takes the pulse to go down and back, a distance2L, is T = 2L/v pulse . The factors of L cancel, with the resultthat T should not depend on L. Figure 4 shows our test of thisfeature of the pulse speed formula. We tripled the length ofthe Slinky from 83 cm to 260 cm, and the time the pulse tookto go down and back changed by only one video frame, 2.83 sto 2.87 s.The point of our Slinky experiment is that we have a formulafor the speed of the Slinky wave pulse, and that we canuse a video camera and simple algebra to precisely test thatformula.Relative motionThe next experiments are where we study my motion relativeto the Slinky. As shown in Fig. 5(a), I hold a meterstickSunSunFig. 2. Slinky suspended by threads from a pipe.supersonic jetsupersonic jetBostonSanFranciscoSanFranciscoBostonFig. 3. Compressional pulse.rotatingEarthrotatingEarthFig. 1. A hypothetical trip from Boston to San Francisco.Fig. 4. We tripled the length of the Slinky and the pulse tookone more video frame to go down and back.The Physics Teacher ◆ Vol. 49, March 2011 149

pulse pass by me at twice the predicted speed, I had experimentalevidence that I was moving relative to the Slinky. If Isee the light pulse pass by me at twice its predicted speed, Ihave experimental evidence that I am moving relative to themedium through which light moves.But light moves through empty space. Observing the lightpulse passing by at a speed 2c allows me to detect my motionrelative to empty space. It allows me to violate the principle ofrelativity.If anybody observes a light pulse moving throughempty space at a speed different than the predicted speed 7c = 1/ µ ε , 0 0 he or she will be able to violate the principle ofrelativity. Thus, if the principle of relativity is correct, we havethe result of Einstein’s second postulate, that the speed of lightmust be the same to all observers.This second postulate is counterintuitive. It does not workfor familiar objects like automobiles. There is no such thingas a car that travels at 100 km/hr relative to everybody. Ifyou stand beside the road and a car passes you at 100 km/hr,someone driving in the opposite direction at the same speedsees the first car pass him or her at 200 km/hr. How can it bethat light pulses behave differently? The answer, we point out,is that light moves at an incredible speed that is well beyondour realm of experience. One of the most important lessonsof physics is that when we look in a realm we have not seenbefore—the realm of high speeds, huge distances, or tiny distances—unexpectedthings happen.What does the realm of high speeds, speeds near the speedof light, look like? We use thought experiments such as a lightpulse clock, and experiments involving the lifetime of muonson a mountain top, to find out. We learn that clocks moving ata high speed run slow and that mountains moving by us contract.That is the subject of the next articles.5. David Keeports, “Demonstrating wave speed on a spring,” Phys.Teach. 34, 460–461 (Oct. 1996).6. In the LC experiment, the resonant oscillation frequency f of anelectric current moving between an inductor L and a capacitorC is f = 2π / LC . If L is a solenoid of length h, area A L , with Nturns, and the capacitor C is a parallel plate capacitor of area A Cand plate separation d, then using the formulas L = μ 0 N 2 A L andC = ε 0 A C /d, we can derive the resultLC.This result depends only on simple lab measurements that donot involve light.7. We have been asked if the numerical value of (μ 0 ε 0 ) couldchange due to the motion of the observer. If the principle of relativityis correct, the answer is no. Imagine a spaceship passingby us on the way to colonize a planet. Suppose students in thatspaceship did the LC experiment and got a different value for1/ µ0ε 0 . Reading Earth-based physics texts, the students couldconclude that the change in the value of (μ 0 ε 0 ) was due to theuniform motion of the spacecraft, and thus the LC experimentcould be used to violate the principle of relativity.Elisha Huggins is professor emeritus at Dartmouth College. He is alwayslooking for effective ways to teach physics concepts.29 Moose Mt. Lodge Rd., Etna NH 03750; lish.huggins@dartmouth.eduReferences and Notes1. For an interesting discussion on the history of the principleof relativity as a physical law that applies to everything, seeCharles Scribner Jr., “Henri Poincaré and the principle of relativity,”Am. J. Phys. 36, 672–678 (Sept. 1964).2. We have written two introductory physics textbooks, Physics2000and Physics2000 Non-Calculus, that begin with specialrelativity. Our main purpose was to explicitly demonstrate thatsuch a course could include basic modern physics topics in acomfortably paced course, with no need for an extended edition.(The texts are discussed at www.physics2000.com.)3. Elisha Huggins, “Speed of wave pulses in Hooke’s law media,”Phys. Teach. 46, 142–146 (March 2008).4. In contrast, the well-known formula for the speed of a transversewave is vwave = T / µ , where the tension T equals thespring constant K times the length (L – L 0 ) that the spring orSlinky has been stretched. Thus the speed of the transversewave differs from that of a compressional wave by replacingthe L by (L – L 0 ) in Eq. (1), where L 0 is the unstretched length.Since L 0 is usually much smaller than L for a stretched Slinky,the compressional and transverse Slinky waves have about thesame speed.The Physics Teacher ◆ Vol. 49, March 2011 151