Making Relativity Real

Teaching Relativity in High School

I am at Fermilab, my third day as a DOE Teacher Research Associate, listening to a lecture by one of the collaborators on NuTeV, a Deep Inelastic Nutrino Scattering experiment. I have not yet begun to think like a particle physicist, which is to say relativistically. Our lecturer, talking about the drift chambers in the detector, throws out a couple of numbers:

"The muon, traveling near light speed, ionizes some of the gas in the chamber. The freed electrons are attracted by the nearest positively charged wire and drift toward it at 5000 meters per second, reach it and give us a signal. Since we can time the arrival of the signal to a few nanoseconds precision, we can locate the muon's path within a few microns."

That did it! Only a particle physicist would refer to motion at 5000 m/s (faster than any non-astronaut has travelled!) as "drifting". That began for me a mental journey that brought me to a clearer understanding of one of the counterintuitive results of special relativity -- the relativity of simultaneity. I would like to share this with you in the hopes it will help you make relativity more real to your students. To understand this, you must accept that the speed of light in a vacuum is a universal constant; everything else can be derived from this point, by you or your students, with nothing more than simple algebra and the Pythagorean theorem. As for the constancy of the speed of light, if you lean toward theoretical physics, you can take Maxwell's word for it, or if you prefer experimental physics, look at the result of the Michelson - Morley experiment!

We won't be needing the full-blown Lorenz transformation for this, but can do nicely with just time dilation and length contraction, both of which are much easier to derive. Let's start with time dilation. For this we must build an imaginary "light clock" as shown in figure 1. (Any competent tinkerer could build a real one, but to get it to tick at a reasonably slow rate, say once a second, the mirror has to be 150,000 Km away!) A pulse of light striking the photodetector triggers the strobe, which sends a pulse out to the mirror, and as it is reflected back it triggers the strobe, ad infinitum. When this clock is working (in your mind), load it onto a space ship, so it can fly through your reference frame at a high rate of speed. Place another observer on the ship to keep an eye on the clock, and think about what each of you will see. The observer on the ship will see the clock ticking away just as before, the path of the light going straight to the mirror and back, but you will see the light traveling along two legs of an isoceles triangle, while the space ship travels along its base. (Figure 2.) If we let c represent the speed of light, v the speed of the space ship, and t the time it takes the ship to move from L to N in the figure (which is also the time it takes the light to move from the strobe to the mirror), then the lengths of the legs are l = ct, m = vt, n =ct.

But wait a minute! l and n can only be equal if v = 0, not in our case with a moving ship, so either the two values of c must be different -- not allowed by Maxwell, Michelson & Morley -- or the two values of t are different. The values of t in leg m and the hypotenuse n are the values observed by us on the ground, so let's keep them as t, and call the value of time observed by our travelling observer t'. Here is the little bit of algebra I promised you, starting with the Pythagorean theorem:

And that's time dilation: the time experienced by the traveller, t', is always less than the time experienced by the stationary observer. If v = 0.8 c, time is dilated by a factor of 0.6. For every ten years the stay-at-home ages, the traveller only ages six. This is the so-called twins paradox; not at all paradoxical, just counter-intuitive to beings who never encounter relativistic speeds. To particle physicists, this is routine. Muons produced by cosmic rays in the upper atmosphere could not reach the ground before decaying if they were stuck with our time frame, but thanks to their relativistic speeds and time dilation they reach the ground in droves! The same effect applies to muons produced in beamlines at Fermilab.

To get from time dilation to the Lorenz contraction, you must realize that both observers agree not only on c, the speed of light, but also on v, their speed relative to each other.(More formally, the relative speed - not velocity - of one reference frame with respect to the other.) Then consider two points on the ground - your reference frame - beneath the path of the space ship. If both observers start clocks as the ship passes over A and stop them as it passes over B, then you will say the distance from A to B is vt, whereas the traveller will say it is vt', always a shorter distance. There is nothing more to it than that. Distances in the direction of travel shrink (but distances in directions perpendicular to the direction of travel remain uneffected).

Lorenz contraction makes it possible for high energy physicists to investigate the structure of protons and neutrons (nucleons). Although the proton was once thought to be a fundamental particle, we now know it to be a complicated and ever-changing arrangement of quarks and gluons. To probe this structure a high energy particle is directed into a target, and scatters off a quark within a nucleon. In the rest frame of the nucleon, the nucleon looks rather like a blueberry muffin with lots of blueberries, and it is hard to imagine the probe particle picking out just one quark (blueberry) to interact with. However, in the rest frame of the probe, Lorenz contraction makes the nucleon look more like a chocolate chip cookie! The probe scatters off one quark (chocolate chip), and by measuring the scattering angle and energy exchange we learn about the underlying structure.

Now, with these tools at hand it's time to tackle the question of simultaneous events as seen from reference frames with different velocities. I confess I was always uneasy with the usual explanation, first given by Einstein himself, involving a light source in the middle of a train, and photons traveling toward both ends. I find this situation confusing because the train and the light are moving in the same direction, and am left with the uneasy feeling that the nonsimultaneous arrival of the light pulses in the stationary observer's frame of reference may be just an illusion caused by the differing distances the two reflected beams must travel to get back to the observer. (It is always tempting to explain counterintuitive events as illusions.)

So you can understand this next thought experiment, I will tell you a bit about one type of particle detector, the scintillation counter. A scintillation counter consists of a layer of some material, either oil or special plastic, which emits photons when charged particles pass through it, with photomultiplier tubes to detect the photons. On NuTeV, our scintillation counters are about three meters square and 2.5 cm thick, with four phototubes at the corners. To someone thinking non-relativistically, the charged particle, a muon in our case, passes through and the phototube sends out its signal. But the muon is travelling near light speed, so in the time it takes the photon to travel from where it is produced along the muon's track out to the phototube (several nanoseconds), the muon travels a comparable distance downstream. For our thought experiment, let's place a simple geiger tube downstream of the counter as shown in figure 3. We arrange the dimensions so that the muon will travel the distance x in the same time interval as the photon travels the distance y. For example if the muon is travelling at 0.8 c, and y is 1 M, then x will be 0.8 meters, and the whole event will take 10/3 ns. (In a real scintillation counter, there are time delays associated with the production of the photon and the index of refraction of the scintillating material, but for our thought experiment we can safely ignore these.) We make our two cables the same length so that the pulses from the phototube and the geiger tube arrive in the lab at the same time, and therefore call the two events simultaneous.

However, we shouldn't be too hasty here, but should see what the muon thinks about all this! In the muon's reference frame, all this equipment is travelling at 0.8c with respect to the muon, time is dilated and length in the direction of travel contracted by a factor of 0.6, so the muon is passed by the scintillation counter and struck by the geiger counter 2 ns later. The counter has travelled 0.48 m in this time. Meanwhile the photon, travelling at the speed of light, must have only travelled 0.6 M and has not yet reached the phototube, since distance perpendicular to the direction of travel is uneffected. Figure 4 shows the whole event in the muon's reference frame. The distances x' and y' (the distances travelled by the lab and photon) must be in the same ratio as x and y, since both observers agree on the values of both v and c. Hence simultaneity is relative.

To us mostly non-relativistic humans, the consequences of special relativity seem strange, but to particle physicists relativity is a way of life! I hope this discussion will encourage you to share these ideas with your students. As you can see, it isn't all that difficult to understand, it is just hard to accept, until you have to.

Len Bugel: