special relativity - length contraction - time dilation
What it shows:
The Relativity Train is a realization of the famous Einstein gedanken experiments involving traveling trains carrying clocks and meter sticks. The demonstration is used to show how the preservation of the postulated constancy of physical laws and the speed of light in all inertial frames requires length contraction and time dilation in the train frame relative to the lab frame of reference. The demonstration is, of course, not a real experiment but rather a visual means of showing (without using any equations) how length contraction and time dilation are necessary consequences of Einstein's two assumptions.
How it works:
The Relativity Train is a large model train 1 on a 4.8 m long track (a schematic picture is shown below). Three large clocks 2 are mounted on it (one on top of the engine, the middle, and the last car) and move with the train. Two similar "station" clocks sit in front of the train track and represent the clocks in the laboratory frame, or rest frame. Two photons, 3 moving in opposite directions and parallel with the train, represent light emitted from an imaginary firecracker or lightning flash. The train moves at 3/5 the speed of light as defined by the speed of the two photons. Special "meter sticks" and arrow markers complete the props for the experiment. The obvious should be stated at the outset: the apparatus does not provide a real display of relativistic effects but rather mimics those effects in slow motion.
The normal procedure in the presentation is to (1) go through the exercise of synchronizing first the station clocks and then the moving train clocks, (2) perform some simple measurements in both the moving and stationary reference frames, (3) discover that the measurements in the two reference frames do not agree and hence violate one or both of Einstein's postulates, (4) invoke length contraction and/or time dilation on the moving train reference frame and, (5) repeat the measurements in (2) and discover that symmetry has been restored in agreement with Einstein's postulates, i.e. the measurements in the two reference frames agree. The details in these five steps will be discussed presently, but it is important to appreciate the logic and flow of the presentation. To summarize, one starts off assuming nothing except Einstein's two postulates: (a) the speed of light is constant (the same in all inertial frames, independent of the motion of the source and the same in all directions) and (b) physical laws are the same in all inertial frames. In other words, it is impossible to tell by any experiment whether you are "truly" at rest or moving with a uniform velocity. Step (3) provides a loop-hole or contradicts this last statement and thus we "fix things up" by invoking length contraction and/or time dilation on the moving frame. This is a qualitative exercise to show, without any mathematics, that length contraction and time dilation are necessary consequences of Einstein's postulates. The general scheme of the demonstration has been outlined and now we will enumerate the details:
In general, it is best to go slowly. The complete use of the demonstration can take two full class hours. Everyone must understand and accept each step as inevitably required by the two postulates initially agreed upon.
(1) synchronization of clocks: Begin with all clocks set to arbitrary times. (i) First measure the length of a meter stick in the station frame (no clocks are required for this) by simply placing arrow markers at the ends of the stick (at arbitrary times) and stating that the length of the stationary meter stick is equal to the distance between the two arrow markers. (ii) Repeat this "experiment" by measuring the length of a moving meter stick (the stick is on the train) from the station frame. This is done by dropping arrow markers (at arbitrary times) in the station frame at the place where the moving stick happened to be. The students will immediately recognize the "error" in this measurement and will tell you that the positions must be indicated at the same time. Repeat the experiment again, dropping the arrow markers at the same time as indicated by the station clocks. The class will scream at you when you do this, pointing out that the clocks are not synchronized. This leads very nicely into the reasons why clocks must be synchronized when you want to compare lengths. (iii) Now you go through the process of synchronizing the fixed station clocks. An imaginary firecracker is set off midway between the clocks resulting in two photons traveling (at the speed of light) in opposite directions. The clocks are then adjusted so they read the same time when the photons arrive. This will be defined as the way to do all synchronizing of clocks during all future experiments. As a check, measure the speed of light by measuring the time for a photon to go from one clock to the other. Do this in both directions to check postulate (a). This seems so simple to the point of being boring, but wait! It's time to (iv) synchronize the moving train clocks; with the train moving, go through the same procedure as in (iii) in adjusting the train clocks. There is obviously a need to fiddle with the clocks - the one at the front must be set back in order for the photons to reach each clock at the same reading. Suddenly the class wakes up again. They claim that the clocks are not synchronized. But you remind them that you are just repeating the same procedure for synchronizing that you used before and, by the principle of relativity, you can't tell if you're moving or not. With the clocks synchronized thusly, measure the photon's speed each way to show the same time for light to travel forwards or backwards. Now that we know how to synchronize clocks, we can return to length measurements.
(2) Length measurements in stationary and moving reference frames. Put the long (100 cm) meter stick on the train and measure its length in the station frame of reference by placing arrow markers simultaneously (simultaneously in the station frame, i.e. everyday simultaneity). There are only two station clocks and the audience is asked to picture an ensemble of station clocks, spread out in the station frame, all appropriately synchronized (this is easy to do in the station frame). The number of cm's between the arrow markers is the station's measurement of the train's meter stick. Now measure the station's meter stick from the train in similar manner. Again, it should be emphasized to the audience that they are to imagine a continuum of synchronized clocks on the train (even though only three are used) and that, to carry out this measurement, the people on the train have been instructed to place an arrow marker as follows: at some specified, predetermined time, whichever person is closest to either end of the meter stick will record a mark at that position. In actuality, the lecturer "simultaneously" (as defined by the train's clocks - here it becomes obvious that simultaneity is not absolute) places the arrow markers on the moving train at this predetermined time at the position where either end of the meter stick is relative to the train. This measurement is a bit tricky and the way it's easiest to accomplish is to place a marker on the train next to one end of the station's meter stick when it is at the last clock on the train, note the time on that clock, and then place a marker "simultaneously" at the other end of the meter stick (i.e., the other end is marked when the nearest train clock reads the time that was previously noted). The number of cm's between the arrow markers is the train's measurement of the station's meter stick.
(3) Violation of Postulates! A comparison of the two measurements in part (2) shows a discrepancy; in the station frame the train's meter stick is the same length as the station's meter but in the moving train frame, the station's meter stick is measured to be shorter than the train's. This asymmetry allows us to tell who is really moving, in violation of the second postulate.
(4) Length Contraction is invoked: We substitute a shorter meter stick for the train (4/5 of the length of the station's).
(5) Symmetry is restored: With the short meter stick on the train (length contraction), the measurements in section (2) above are repeated, the results of which are now symmetric; i.e. both measurements indicate that the meter stick in the other frame is shorter than the meter stick in the frame in which the measurement is made. The train will find the station's meter to be only 4/5 as long as its own and the station measures the train's meter to be 4/5 as long as its meter.
There are three different ways we can introduce time dilation at this point: (a) show that c (the speed of light) is different in different frames, (b) show asymmetry in measuring the rate of a moving clock or, (c) show that relative velocity is different when measured by different frames. For brevity, only (a) and (b) will be described in detail.
Having agreed on a consistent set of length standards, we are now equipped to measure the speed of light in each frame. All clocks (moving and stationary) are running at the same rate.
(1) Speed of Light Measurement: Using the station meter stick and the station clock, measure the speed of light (try both directions). Record the value. Note that you would really need two station clocks to do this. Repeat this measurement using the train meter stick and the train clocks (again, try both directions). The answers are different! This clearly violates the postulates, and since the units of length are already set consistently, the problem has to lie in the units of time.
(2) Time Dilation is Invoked: Since the station's value of time for light to travel one station meter is smaller than the train's value, we must slow down the train's clocks or speed up the station clocks (either would do equally well, but for purely electromechanical reasons we have chosen to speed up the station clocks rather than slow down the moving train clocks - the audience need not be bothered with this bit of information). In any case, the flick of a switch accomplishes the deed and the clock rates can be compared - the train's clocks run at 4/5 the rate of the station clocks, as measured from the station.
(3) Symmetry is restored: Now measure the rate at which the station clocks run as seen by the train's synchronized clocks. Measure on the station clock the time the (train's) length standard takes to pass by. Measure the same interval in the train frame by taking the difference between the first train clock at the start of the interval and the trailing train clock at the end of the interval (as each clock passes the station clock in turn). In the train frame the station clocks run at 4/5 the proper rate! (note this is the same factor as that for length contraction) We can now repeat the speed of light measurements in step (1) above and find that that result too is consistent with the relativity postulates. Thus, having introduced time dilation (and length contraction) one can now demonstrate that (a) the velocity of light is the same in both frames, (b) each frame measures the other's clock as running slow and, (c) relative velocities are the same in both frames.
Setting it up:
The apparatus occupies the whole space in the front of the lecture hall. A hand-held remote control allows the lecturer to operate the train without having to constantly walk back to the main control box. Once in the hall, put it through its paces before lecture time. The apparatus has its idiosyncrasies and what can go wrong, will. It is advised to practice extensively before using this demo. Because of its size, the apparatus can not be wheeled into Lecture Hall A and advance notice is necessary to make appropriate arrangements for its use.
The relativity train allows the student to see changes in train position, clock positions, and wave (photon) position proceeding simultaneously. This all happens sufficiently slowly that one has time to notice all the important aspects unfolding and can at any instant halt the development of the physical process to examine in detail the prevailing situation. The size and visual clarity of this demonstration makes it especially useful for large audiences with little mathematical background as well as students more sophisticated in mathematics and/or physics and any variety of audiences. The original design was conceived by Prof. Costas Papaliolios (Harvard University) in 1971 and is predated by a more complex device by J. Streib. 4 It has been very popular through the years and is highly recommended if the lecturer is willing to invest the time (both in class and preparation). Rating ****
Some technical details for those wishing to reproduce a similar demonstration: Obviously this demonstration can be scaled
down to suit your particular needs and constraints. We present here some of the design features which may be generally
useful, regardless of size.
motor drive: It was discovered early on that the train's own engine motor is not sufficiently powerful to drive the train, clocks, and photons without serious slippage problems. We pull the train by a loop of Posi-Drive™ chain using a Bodine (series 200) motor whose power supply is in the main control box mounted on the tressel. It is remotely controlled by a hand-held master switch on a 20-ft cable and has three positions: brake and forward positions (both with a positive lock) and a spring-loaded reverse position. A safety switch, tripped by the engine cowling, is mounted on the trestle at the far forward end and will halt forward, but not reverse, motion. This is important as the lecturer will undoubtedly be so absorbed in the demonstration as to let the train crash into the bumper track. The train travels the two meter length in about 27.5 seconds.
photons: At the rear end of the track, mounted on the same shaft as the Posi-Drive idler pulley, are two pulleys (2.75 cm and 4.53 cm dia.) which serve to drive the photons. The photon drive pulley (1.30 cm dia.) is mounted on a post meant to look like a utility pole and is coupled to the track pulley via a rubber band. The front end of the track has a similar post with an idler pulley on top. A fishing line (5-lb test monofilament) tightly loops around the photon pulleys and the photons themselves are loosely suspended from this line. The top/bottom pulley diameter ratios determine the speed of the photons to be 5/3 the speed of the train.
clocks: We found that controlling the speed of simple DC motors by merely changing the DC supply voltage was not reproducible enough for our clocks. AC synchronous motors are cheap and very reliable, but a switching power supply must be used to slow down (or speed up) the motor. All the clock motors (train and station) are Cramer™ 110 VAC, 2 RPM motors. The station clocks are set up so that they may be run either (1) at the same speed as the train clocks or, (2) 25% faster than the train clocks. To provide option (2), a special circuit is used to supply 110 VAC at 75 Hz. This is accomplished by using a 12 VAC, center-tap, step-down transformer in reverse: the 12 V winding is used as the primary and the 110 V winding feeds the clock motors. A 12 VDC power supply feeds the center-tap of the primary. The DC voltage is switched in a push-pull mode by power transistors (2N3055) which are gated at 75 Hz by a 555 oscillator. A Darlington pair of transistors (2N2213) act as a buffer between the oscillator and the power transistors.
trestle: Rather than assembling and setting up the train on the lecture benches, a dedicated 4.8 m long cart was constructed on which the train and all accessories are stored. It was designed to look like a trestle (in keeping with the train theme) and the Dexion™-type angle iron used in the construction looks very much like steel girders. Model railroading "grass" and "gravel" on the trackbed add the finishing touches.
G-gauge, LGB train and track, manufactured by Lehmann, Gross and Bahn of Germany (available through toy or model
stores such as F.A.O Schwartz or Eric Fuchs Inc., Boston).
2 30 cm square smoked plexiglass clock faces with bright yellow circles in place of numbers on each face. The clocks have a single hand whose position can be changed by the lecturer and are driven by Cramer type 117, AC, 2 RPM motors.
3 The photons are yellow cardboard disks, 4.5 cm in diameter, suspended from 5-lb test monofilament. The photons hang without slipping as they are moved along by the monofilament until they reach the end of their travel, at which time they do slip.
4 John F. Streib, Am J Phys 31, 802 (1963).