 In this lecture, we will learn the following things. We will learn what is a muon. We will learn how to use the muon as a laboratory for making predictions with a Lorenz transformation. And finally, we will learn how the muon was the first direct test of the validity of special relativity. Let's begin by reminding ourselves one more time about the Lorenz transformation. If we make observations in a frame S' that we consider to be moving, and we want to convert them into observations in a frame S, we consider to be at rest, then the equations in the top left of the slide will do just fine. If on the other hand we have observations that are made in the rest frame S, and we want to convert them into observations in the moving frame, then all we have to do is change x to x' t to t' and v to negative v in the upper left equations and we get the necessary equations in the upper right. This function, gamma, that appears in all of these equations is a function that depends on the relative speed between the two frames, v, and it has the form of 1 over the square root of 1 minus v squared over c squared, where c is the speed of light. This can be expanded into a series representation of this function using the binomial expansion we've looked at this before, and so we get an expansion that looks like 1 plus a half v squared over c squared plus terms that are higher order in v, like v to the fourth, v to the sixth, and so forth. And for sufficiently low velocities, less than a few percent the speed of light, we can usually neglect some or all of these higher order terms in the series expansion and get a simpler representation of the gamma function at low velocity. Now we've looked at some consequences of the Lorentz transformation in special relativity, we've looked at length measurements and we've seen how they depend on frame of reference, we've looked at time measurements and seen how they depend on the frame of reference, and we've looked at the simultaneity of events in one frame and see that they are not necessarily guaranteed to be simultaneous in any other frame that's moving with respect to the one in which they are simultaneous. But is the relativity of time a real thing? Ideally what we would want to do is take two twins, put one in a spaceship that can accelerate very quickly up to speeds well in excess of half the speed of light. That's where the gamma function begins to take on values that are very much in excess of one. And then we would let the twin travel out on a journey of maybe 10 or 20 years and then bring them back, that's going to take another 10 or 20 years. And when the twin gets back from the journey at high speed, we would compare the two twins and see how they have aged from the perspective of observers on Earth. That would be a great experiment, except that it is really not easy to construct a vessel, not only that can hold humans for long duration space flights, but also that can accelerate up to speeds that are appreciably close to that of light. This is an engineering challenge that we as human beings have never really mastered. Instead, what we need to do to test the claims of special relativity is to identify a laboratory where such speeds can be readily achieved, but also one where there's a natural clock of some kind, a regular sequence of events whose time can be well predicted, so that we can compare those things when they're at rest to when they are in motion. Now tiny particles would be a great potential laboratory. Tiny particles have very small masses that are very easy to accelerate, and so it's possible that something like the electron, with a mass of 9.11 x 10 to the minus 31 kilograms, could be ideal for investigating fast moving objects and maybe even the relativity of time. For example, the ground state energy of the hydrogen atom, the lowest energy state that an electron can have when orbiting a single proton, is 13.6 electron volts. You can look back at the conversion factor for the electron volt, look it up on the web, whatever you want to do, but you'll find that this comes out to be about 2.19 x 10 to the minus 18 joules of energy. Now if you translate that into a corresponding kinetic energy for the electron in that state of hydrogen, what you'll find is that you can calculate the speed of the electron that is represented if you pretend that the electron is orbiting like a little planet around the proton under the influence of the Coulomb force. So let's do that. Let's imagine that the electron is orbiting the proton at the center of the hydrogen atom and let's use the energy of this state to estimate the kinetic energy and therefore the velocity of this electron. And if we do this using the classical kinetic energy, one half mv squared, and rearranging it to solve for the speed of the electron, we find that it should come out to be about 2.2 x 10 to the 6th meters per second or thereabouts. That's already pretty fast on its own. That's about 1% of the speed of light without having to do anything exotic except maybe study the electron in a hydrogen atom. Now of course we'd like to get the electron up to faster speeds than that, but if that's how fast it's already going in a hydrogen atom, then you can imagine it's probably not too hard to get it going faster. In fact, JJ Thompson, who's credited with the discovery of the electron, isolated them by ripping them off of parent metal atoms using the Coulomb force, using a strong electric field and a large electric potential. This has the effect of accelerating the electrons up to rather high kinetic energies for the tabletop experiments of his day, representing tens of thousands of electron volts of energy in the electrons. And that would equate to speeds roughly of the scale of 10 to the 8th meters per second. Perfect! Those are the speeds we want to investigate phenomena at. So a particle like the electron would be easy to accelerate, but there's a problem. The electron doesn't do anything. It's an extremely stable particle. In fact, left on its own, an electron will simply be for the rest of the history of the universe so far as we know. So it doesn't have any regular characterizable phenomena associated with it once you've isolated an electron. It lacks a kind of clock that it carries along with it that we could use to see whether or not the passage of time is affected by the motion of the electron. Well, are there any such clock-like phenomena in nature that are associated with very small particles? The answer is yes. Radioactive decay of atomic nuclei is exactly an example of a natural clock that ticks all the time in nature whether we're there to observe it or not and if we do observe it we can use it to measure the passage of time in a system. So for example, among her many discoveries, two-time Nobel Prize winner Marie Curie isolated the element polonium. It is highly unstable and the natural isotope of polonium, polonium 210, transforms spontaneously into a stable lead atom, lead 206, after emitting energy in the form of radiation. Specifically what it does is it spontaneously ejects two protons and two neutrons from the polonium nucleus. These two protons and two neutrons are bonded into the nucleus of a helium atom and this thing is known as an alpha particle. We'll return to alpha particles later in the course. The bottom line is there's some spontaneous phenomenon that happens with regular time intervals that we can use to actually track time in nature. Now polonium 210 has what is known as a half-life of 138 days but what does that phrase half-life mean? It means that if I were to isolate 100 atoms of polonium 210 in a sealed container and have some way of looking at those atoms and counting them every hour of every day, if I were to wait 138 days from the time I seal the container and then look in the container, on average I will find that after 138 days about 50 atoms of polonium 210 will remain in the container. The container will also now be home to 50 lead 206 atoms. They resulted from the spontaneous decay of the missing polonium 210. Now if I further wait another 138 days from that moment and look in the container again, on average I will find that I now have 25 polonium 210 atoms left. Half of the sample I had 138 days ago and correspondingly I'll find 75 lead 206 atoms in the container. A half-life is a regular interval of time and if you had some kind of equipment that could be used to establish the amount of a certain isotope present in a sample, you will find that after one half-life, after every half-life passes, you'll lose half of what was there the last time you looked. So unstable radioactive elements have a reliable built-in clock, a regular process that occurs at the same place that is the nucleus at regular time intervals. However, there's a problem and in the historical context what I'm talking about here is a problem in the early 1900s. Polonium and other radioactive elements were pretty hard to come by in the days when they were discovered and even in the decades after that. And even if you could isolate an appreciable sample of them, how would you know precisely how to count the numbers of those things whether they're at rest or whether they're in motion? And not only that, you got to put them in motion which means you need to accelerate them. And there are thousands or tens of thousands or hundreds of thousands of times heavier than electrons. Particle accelerators that are capable of bringing ions up to speeds approaching that of light of any decent quality and control were decades away in the early 1900s. They wouldn't emerge until the 1930s, 40s, and 50s. So it's nice that we have these regular clocks like radioactive isotopes, but you can't actually do practical experiments of the variety we're thinking about trying to do. That is attempting to see whether their clocks slow as they are put into a state of motion relative to the observing frame. If only we had a tiny particle that combined the lightness, abundance, and ease of acceleration of the electron with the regular instability of radioactive atoms. And it's into this part of our story enters the mu meson, or for short the muon. Now the mathematical description of unstable nuclear behavior and of the strong binding of things in the nucleus took decades to work out. But around the 1930s, with some experience now with other forces in nature, like electromagnetism, it was hypothesized that the forces inside the nucleus that both bind it together so tightly but also occasionally allow it to catastrophically break apart. That these forces were maybe of two different kinds and that they had particles like the particles of light that transmit electromagnetism that acted as intermediaries in the nucleus and transmitted these forces within the nucleus. And so these intermediaries were given a generic class of name, mesons, from the Greek word mesos, meaning intermediary. And by the 1930s, or certainly the 1940s, the hunt was on to find them. Now shown at the left here on this slide is an image that was taken by two physicists, Anderson and Netermeyer, and published in 1936. A previously unobserved electrically charged particle punches through the slab of lead that runs through the center of the photograph. These are two different views taken from different angles of the same particle interactions at the same moment in time. And the interactions are taking place in a lead target that runs through the center of the picture, roughly here in the picture. Now as this previously unidentified particle passes through the lead, it knocks apart nuclei, but in this process it barely loses any energy. This was a really strange beast in its day. It would come to be dubbed the mu meson, or muon for short, as the physicists of that day mistakenly thought that this has to be one of the sought-after nuclear force intermediaries. I mean what else could it be? This turned out to be a bit of a lack of imagination and experience on the part of physicists with the broader picture of nature, a good lesson for all of us of course, and this assumption turned out to be wrong. The particle was real, but its role in nature was not as originally assigned, and that wouldn't be fully understood until the 1940s, 1950s. Its electric charge, however, was pretty well determined from careful experimentation, and it was found to be identical to that of the electron, negative 1.609 times 10 to the minus 19 coulombs, so it carries with it the same elementary charge that the electron possesses. Its mass, however, was very unlike the electron. It weighed in at 207 times the mass of the electron, too light to be a proton, too heavy to be an electron, and crucially, unlike the proton and unlike the electron, it is also unstable. If you trap a muon nearly at rest, and there are some fairly straightforward ways to do this, on average you will find that it only lives about 2.2 microseconds, or 2.2 times 10 to the minus 6 seconds. Now, let me make an important aside while we're on the topic of unstable particles, about unstable particles, half-lives, and the characteristic lifetime of an unstable particle. The mathematics of unstable particles was developed in response to the discovery of radioactive decay of atomic nuclei, and it's a fairly straightforward application of algebra and calculus, and I find it's instructive to run through it here. Consider the kinds of systems we've been talking about so far. So, for instance, let's imagine you have a system of n-zero unstable objects, like 100 nuclei or particles, like the muon, and you observe them to have that number at time t. And then you wait a little bit. We consider some change in time, t plus delta t, at which point we then discover that the number of objects has decreased by negative dn. Now, here, dt and dn are differential units of time and number, respectively. And you find that the number of objects remaining after a time has passed, dt, since the original time measurement, is the original number, n-zero minus dn. But if you double the number of objects, so if you start with 200 unstable objects, for instance, and wait the same amount of time, you don't find that the dn is the same size as it was before it gets bigger. And if you triple the objects to 300 or quadruple them to 400, you again find that dn after the same dt is proportionally larger, and it's larger in proportion to the size of the starting sample. There is some proportionality between the change of number of objects, the change in time, and the original number of objects. And so we can express this observational relationship in a simple equation. Negative dn, the change or decrease in the number of objects, is equal to a constant, which we have yet to determine. We'll denote that with the Greek letter lambda, lowercase lambda, times the original number of objects, n times dt, the time that passed during which the time the number changed. Now you'll notice that this is set up to look like an equation of differentials, and so one could actually integrate both sides of this equation. You can put all the n's and dn's on one side, and all the constants in dt on the other side, and then you could integrate the side with the number stuff on it, from the original number, n0, to the final number, n, after waiting a time t. Now you'll notice that on the left-hand side we have the integral of negative 1 over n dn, or 1 over x dx, if that sounds more familiar to you. And so you should know from some experience with second semester calculus that the natural log of the argument n in this case will wind up being the answer to this integral. On the right-hand side we have a much simpler integral. We're just integrating a constant times dt from time 0 to time t whenever we observe the system later. And that's a very simple integral. You just wind up getting the time t back times lambda, and then you just evaluate it at the endpoints. So if you do that you should find that you get the following equation. The natural log of the original number of objects minus the natural log of the final number of objects at time t is equal to lambda times t minus 0. 0 is the original time at which you observe the system and see that it has objects n0 in it. Well if you rearrange now and try to isolate the number of objects n at the later time t on the left side, you wind up with this equation moving the logs and constants and signs around. And so finally you can solve for n as a function of time. And you find out that it's exponential in nature. If you start with a number of objects n0, the number of objects left after a time t is given by e to the minus lambda t times the original number of objects n0. Now let's talk about this constant of proportionality which we've been calling lambda. So in order to satisfy the requirement that the total argument of the exponential function be dimensionless, it must be true that lambda has units of inverse time, one over time or one over seconds per second hurts in the units of oscillatory phenomena. It's convenient to therefore define lambda as one over some characteristic time which I'll denote with the lowercase Greek letter tau where tau is known as the time constant of the phenomenon. Well what does it actually mean for t to reach tau, the time constant? Well if you allow enough time to pass that one time constant's worth of time goes by, you find that 63.2% of the original number of objects are gone. For unstable particles, this characteristic time is what is known as the lifetime of the particle, and you can actually show using some math we'll develop later in the course that mathematically tau is also equal to the average time that an unstable particle exists. So it has two meanings. One, it's the time after which 63.2% of the original n-zero objects have disappeared from the system, and two, it's the average time that any randomly picked unstable particle will exist. Now where does the half-life come into all this? Well you can show that the half-life of an unstable particle which we could denote as t with a subscript one-half is directly related to the time constant tau by the following simple equation. The half-life is equal to the time constant times the natural log of two. So when we say, quote, the muon has a lifetime of 2.2 microseconds, unquote, we're referring to the time at which there is a 63.2% chance that any single muon has decayed, vanished, gone away from the original sample of muons. Now let's talk about muons and observing them and their origins in the world around us. Muons are not naturally occurring in the same sense that atoms are naturally occurring. Atoms are generally speaking stable, they stick around for a long time, and they form large structures because they have a chance to bind to each other through chemical means, which is just electromagnetism in action. Muons on the other hand are a bit stranger, you have to make them, and because they don't live very long, you have a very limited opportunity to study them once they come into existence. Now thankfully, nature does make them all the time, and it does so because the Earth is constantly bombarded by particles from outer space that are smashing into the atmosphere at very high speed, very high kinetic energy, and these things are known as cosmic rays. And when cosmic rays slam into the Earth's atmosphere, they result in a whole bunch of particle interactions that ultimately spray muons down onto the Earth, among other things. So they do this by smashing into nitrogen or oxygen nuclei, having all kinds of nuclear reactions in the process that produce a whole bunch of other particles, and I'm not going to worry about what those are right now, but ultimately, muons can result from this, and the symbol mu with a minus sign next to it denotes the muon with its natural negative electric charge. There are also positively charged muons, and that's a subject we'll come to later in the course. Now Anderson and Nettermeier, who originally discovered the muon, did so using showers of particles from cosmic rays, so-called cosmic ray showers, and they did so by putting detectors at different altitudes in the Earth's atmosphere. So for instance, they did a bunch of experiments with a detector located on top of Pike's Peak, which is 4.3 kilometers above sea level, and they did a bunch of experiments at home base at Caltech in Pasadena, California, which is roughly at sea level. And it turns out in the decades we've been studying cosmic ray interactions and muon production, we've learned that most of the muons that are produced by cosmic rays are made roughly at a height of 15 kilometers above the Earth's surface. That's not the top of the atmosphere, but it does correspond to the place where the density of the atmosphere gets big enough that these interactions of cosmic rays and nitrogen and oxygen molecules or nuclei get very high in probability. And so we get a lot of muons that get produced at that part of the atmosphere. Now, based on the known instability of the muon, one might expect that if one counts a certain number of muons at a high altitude, say counting a number N1, then by the math of unstable particle decay and using the known lifetime of the muon when it's nearly at rest, that is, tau for the mu is roughly 2.2 microseconds, one could accurately predict the number of muons you should expect to see at a lower altitude, N2. Now, at that lower altitude, because particle decays had a chance to happen, we expect fewer total muons to be found. If we make 100 muons or 1,000 muons at 15 kilometers above the Earth's atmosphere and we go down a bunch of kilometers, we don't expect to find the same number of muons. We expect to find typically fewer. All right, now, this is very interesting here. Let me show you this. So here in the basement of Flandren Science, the physics department has a small experiment set up that allows us to capture muons created in cosmic ray showers above the Earth, trap them by trapping them in atoms and a material in this device over here on the left. And then after trapping them, we can wait and see how long they stick around until they decay. So all of this equipment is designed to establish the time at which a muon is trapped and then the time at which it then subsequently decays because it doesn't live forever. And if we take a look at the data here, what we find is that when we trap these muons and hold them nearly at rest in our reference frame, indeed we see an exponential falloff in the number of muons that survive after a certain amount of time, as predicted by the theory of particle decay. And we can see that after about 2.2 microseconds that there's a roughly 60 to 70 percent chance that any single muon will have already decayed exactly as previous experiments have determined. So this is our own little way of caging muon using atoms to trap them, then waiting to see them decay and measuring the time between those two events in a frame that's essentially at rest with respect to the muon. And indeed this is how we figure out, for instance, that the muon lifetime is about 2.2 microseconds. This experiment alone here in the time it's been operating, which looks to be about 2,300 hours or so, has trapped and observed the decay of about 1.6 million muon. So just think about the sheer number of muons that must be raining down on the surface of the Earth all the time. We're capturing just a tiny slice of all of those in their fantastic laboratory for looking at the little clocks that fundamental particles carry around with them so that we can study time using the tiniest building blocks of the universe. Now the muon's short lifetime should radically cut down its numbers as we go lower and lower into the atmosphere. And in fact, the effect is quite stunning. So let's imagine we give the muon the best possible chance of making it to a low atmospheric height. So close to the surface of the Earth. Now to do that, we're going to crank its velocity up to the fastest that anything that we know of can travel and that's the speed of light. So we're going to set the speed of a muon that's just been produced at 15 kilometers above the surface of the Earth. We're going to set its velocity aimed straight at the surface of the Earth to the maximum it can be 2.998 times 10 to the 8th meters per second. So if you crunch the numbers, you'll find out that in one lifetime a muon can travel just 0.66 kilometers, not even a kilometer. It doesn't even go 1.15th of the way down closer to the surface of the Earth. But at this point, it's already had a 63.2% chance of decay. There's a 63.2% chance that that muon won't exist anymore by this point. But let's imagine it survives and it makes it two lifetimes. After two lifetimes at the speed of light, it could have gone 1.3 kilometers, doubling the distance it's traveled into the Earth's atmosphere and now having made it a little more than 1.15th of the way into the Earth's atmosphere. But by this point, it pays the immense penalty of having a probability of 86.4% of having already decayed. 10 lifetimes will only bring a muon 6.6 kilometers into the atmosphere. That still leaves it about 8 kilometers above the surface of the Earth. But by then, it has had a 99.995% chance of decaying. There's really very little chance that muon really makes it this far. And if you take it twice as far, 20 lifetimes, the probability is even smaller. So the bottom line is that we don't really expect if we produce 1,000 muons at 15 kilometers to find really any of them down at sea level. So what actually is observed? Well, shown at the right is some data. It's real data taken from an experiment that really can count muons at different altitudes. And the graph shows the number of counts per minute versus the altitude where the measurements were taken. And these measurements were taken by high school teachers who were involved in a program called QuarkNet. This program engages teachers in K through 12, typically high school teachers in real physics research environments. And this data is actually taken from an experiment they did that was reported in the article that's listed in the footnote on this slide. Now, what they found was that if the experiment was run 3.5 kilometers above the surface of the earth, above sea level, they found about 300 counts per minute of muons at that height. Now, let's use the Galilean and Newtonian assumption of time, that time passes at the same rate for all observers. That is, whether the muon is moving or not, its clock and clock's on the ground tick at the same rate. Now, that's a total violation of the assumptions of special relativity and, of course, the conclusions that one would then draw from the Lorenz transformation, but we can make a prediction using the Newtonian or Galilean idea. And so we can basically estimate how many counts per minute we expect at half a kilometer, which is roughly the lowest height where the teachers took data. Now, what you observe is that at 3.5 kilometers, the number of counts is about 300 per minute, 300 muons per minute passing through the detector. And if we give the best case chance of all those muons making it down to half a kilometer above the surface of the earth, we find out that we should expect the yield to go as e to the minus y over c times tau, where c is the speed of light and tau is the lifetime. And so after a height change of just 3 kilometers going from 3.5 kilometers to half a kilometer here, we expect to find at most about 3.2 counts per minute from muons that are produced at this altitude of 3.5 kilometers. Is that what we actually observe? And the answer is, heck no. In fact, the teachers observed not three counts per minute, but 100 counts per minute at both sea level and about half a kilometer above the surface of the earth. It's pretty hard to tell the difference between those two sets of data. So why would that be? Why would it be that the Galilean Newtonian prediction, or at least its assumption that time is the same for all observers regardless of the state of motion, would not get this experiment right? It seems so simple. We know the lifetime of the muon when it's at rest. You know the height difference between where you make the first and second measurements. You just do some counting. It should be easy, right? And you don't even get close to the right answer. So why would that be? Well, I think we already know the answer. The answer is time dilation. Special relativity with its Lorenz transformation that's supposed to be valid for all speeds up to that of light will help us to understand this. So let's relate what's going on in the muon's reference frame, which we'll call S prime, and what's going on in the earth's reference frame, which we'll call S. So we choose the earth and the atmosphere to be at rest. We choose the muon to be moving. So it's viewed as a moving object with respect to the earth. And so we can call that the moving frame. Now in the reference frame of the muon, where it thinks it's at rest, its lifetime is 2.2 microseconds. Recall that this is the lifetime as observed when the muon is nearly or exactly at rest, and that would be its proper lifetime when it's exactly at rest. The proper lifetime is measured in the frame where all events happen at the same location in space. For the muon coming into existence and going out of existence, all happen at the same place itself. And so that's the frame in which proper lifetime is defined. That's also how we measure the lifetime of the muon as we stop it and we let it decay and we see how long that takes, typically. And so that's the 2.2 microseconds associated with how long the muon lives. Now the Lorenz transformation would predict that the time measured by an observer on the earth, the time that's passing in the muon's frame of reference, will be different from a person who would measure the time but ride along with the muon, thinking the muon is the thing at rest the whole time. So we can take observations in the frame of the earth, observations of, say, the clock ticks in the muon's frame at T2 and T1 and take the difference, the delta T between those ticks. And we can relate those to the spatial coordinates where all the events happen and the time measurements as observed in the frame of the muon, the s' frame. We're just using the Lorenz transformation one more time here. Now, all the events in the muon's frame of reference being created in the upper atmosphere, decaying later at a time T2 where the earth is closer to the muon, they all happen in the same place in the muon's reference frame. In other words, x2 prime is equal to x1 prime. Therefore, this equation simplifies and the time difference in the earth rest frame is relatable to the time difference in the muon's frame of reference by a factor of gamma times the time difference in the muon's frame of reference. Now, the lifetime of the muon and its frame of reference is 2.2 microseconds. So delta T prime, or T2 prime minus T1 prime, is 2.2 microseconds. So special relativity would predict that from the perspective of an observer on the ground, the muon would appear to live longer than would be expected if it were at rest as well. This is completely in accordance with the observational evidence. More muons, many more muons, are observed to survive to a lower altitude than would be expected from classical physics and its assumption of the absolute passage of time for all observers. So the data told us that of the, say, 300 muons per minute observed at 3.5 kilometers, roughly speaking, 100 per minute of those survive around half a kilometer above the earth's surface. In the reference frame of the earth, we can relate these numbers to the observed decay time of the muons in their rest frame, that is, tau, the proper time and also the lifetime of the particle, and the distance that they travel from the perspective of the earth and atmosphere rest frame y, as well as the typical speed of muons. So what we find is that taking the decay equation, n equals n0 times e to the minus T over gamma tau, then tells us that we can solve for the velocity and the gamma factor of the muons using the data. We know n and we know n0 from the data. We know why, because we know the height difference that the teachers made the measurements at. We can solve for this quantity gamma v, which is related, of course, ultimately to the speed of the muons in the atmosphere relative to the earth. So if we do the algebra here and solve for gamma v, we get the following equation. Now, I'm going to leave it as an exercise to the viewer. We want to solve ultimately for either gamma or v, but since they're each a function of the other, we have to do some algebra to isolate one or the other. And to help you along with this, recall that the gamma factor is defined as the one over the square root of one minus v squared over c squared. And that means that the velocity, if you solve for that, is equal to c times the square root of one minus one over gamma squared. And from this, you can take gamma times v and you can get a nice expression for that. So gamma v can actually be written entirely in terms of gamma, which is interesting. And if you use that trick, you can get to isolated expressions for either gamma or v from this equation here on the right-hand side. So go ahead and try that yourself as an exercise, but you should find the following things. You should find that the gamma factor for these muons, assuming that all the 300 that are created at or appear at an altitude of three and a half kilometers then could be counted or not at 0.5 kilometers above the Earth's surface. And you then, based on that assumption, estimate that the gamma factor is around 4.3. And if you solve for the velocity of these muons, they are radically close to the speed of light. They are 2.91 times 10 to the 8th meters per second in speed relative to the Earth and the atmosphere. Now, from the person on the ground's perspective, that journey from three and a half kilometers to half a kilometer above the surface of the Earth takes about 10.3 microseconds, which is way longer than one lifetime of a muon. So it's no wonder muons make a fantastic and early laboratory for tests of special relativity. Nature is readily making lots of them per second in the upper atmosphere. They can be measured using technology that was available in the early 1900s, at least the first half of the 1900s. They can be observed and to see when they decay and how often they decay and so forth, and all of that together can be used to assess the validity or not of special relativity. And of course, what we find is that special relativity wins the day. It is the correct description of space and time for inertial reference frames. And it's remarkable how well it actually works. Now, of course, the atmosphere is complicated. The production of muons and the atmosphere is complicated. If you really wanted to do a super thorough job of this, you would have to do a detailed simulation of the interactions of cosmic rays in the Earth's atmosphere, producing muons at various heights, and then see how many you count at various heights with and without special relativity. If you do that, we find that with special relativity, we can exactly produce the atmospheric data without special relativity. We utterly fail to reproduce the atmospheric data. It really is true that special relativity is the correct description of space and time and motion. Now, I find it's instructive to quickly take a look at this same situation, but from the muon's perspective. That is, if you could ride along with the muon at its ridiculous speed, what would you observe? Of course, in the muon's frame of reference, it's at rest, and the Earth and the atmosphere are rushing toward it, or in the case of the atmosphere, past it. So you and the muon come into existence very suddenly, three kilometers above the final Earth observation place. Now, that's in the perspective, that three kilometer statement, that's made in the frame of an observer on the Earth. We'll get to the distances in a second in the muon frame. What you can say for sure is you come into existence, the Earth is far from you, it's racing toward you at a speed of V, and it's getting closer to you all the time, and at some point, you'll go out of existence. And the question is, how far is the Earth and atmosphere going to move in the time between those two events coming into existence and going out of existence? So the perspective of the Earth observers on the left in this cartoon, and the perspective of the muon observer is shown on the right in this cartoon. We don't know the distance between the surface of the Earth and the muon in this picture. We only know it from the original experiment in the Earth rest frame. But here we are confident that the Earth is rushing toward us at the opposite velocity that's measured in the Earth frame for the muon heading toward the Earth. Now, from the muon's perspective, of course, it's standing still. In all events coming into existence and decaying, they happen at the same location in its frame of reference. Therefore, the time it typically is going to stick around is going to be 2.2 microseconds in its frame of reference. It sees the Earth below it when it comes into existence, and it sees that surface of the Earth rushing toward it at a velocity of negative v. So how far does the muon have to go to make it to its destination? From its perspective and its reference frame. Time is ticking away at whatever rate it goes at for the muon, and ultimately it can measure time using its own lifetime, which is about 2.2 microseconds. Nothing funny with time in its rest frame. But of course, the distance where the muon was created above the surface of the Earth is three kilometers in the rest frame of the Earth. That's the frame where the Earth and its atmosphere appear to be at rest, and that makes that distance three kilometers the proper length or proper height above the surface of the Earth. That is the longest distance that any reference frame would measure between where the muon is created and say the surface of the Earth. The muon, on the other hand, will see the Earth-atmosphere system as moving, and therefore distances in that system contracted along the direction of flight. And the length or height above the surface of the Earth that it will measure will be the proper length, three kilometers, divided by gamma. And this comes out to be about 23.3% of the proper length, or 0.699 kilometers, 699 meters. That's the distance that the muon perceives between where it comes into existence and that final measuring point, which was three kilometers away in the frame of the Earth and the atmosphere. So from the muon's perspective, we conclude that it observes that the distance it will travel is contracted compared to what observers on the Earth are seeing, and that contraction factor is one over gamma. From the muon's perspective, the distance between the place and the atmosphere where it came into existence and where it ultimately decays is greatly shortened, requiring only a time of delta T prime of about 2.4 microseconds to make the trip because the Earth atmosphere system is contracted and moving relative to it. And so, you know, given that it lives about 2.2 microseconds in its reference frame, it's absolutely plausible that it could make it that full distance that people on Earth said it went. It's just that the people on Earth are confused because the distance is shorter than they claim from the muon's perspective. So while observers on Earth and an observer moving with the muon would disagree on the reason for the muon reaching the lower measurement point, they all agree that it's very likely to happen. The Earth observer argues that the reason it makes it is because time in the muon's frame is passing more slowly than they claim because the muon is moving. And so that takes longer to decay as a result of that. It's able to cross the 3 kilometer gap, even though it should have only lived 2.2 microseconds because time has slowed down for it while it's in motion. The muon observer says, no, our clocks are working just fine. What's going on is that because the Earth and atmosphere are rushing toward us, they're in motion, and so they seem contracted along the direction of motion. And as a result of that, we don't have to go that far to make it to, say, the surface of the Earth. And we're definitely going to make it in about 2.2 microseconds or so. That's why we made it so far. Now, they're both right. Even if they have different reasons for what happens, they both observe the same outcome, the muon makes it to the surface of the Earth, but they disagree on the space and time reasons for that. And that's okay because the Lorenz transformation allows them to relate their perspectives and put their measurements into the other person's frame to see what's going on and resolves the paradox in that sense. So to review, in this lecture we have learned, first of all, what is a muon? It's a subatomic particle. It's about 200 times heavier than the electron. It's about five times lighter than the proton. And it has the same elementary charge as the electron. So it's its own thing, and it would take decades after it was originally discovered to finally fit it into the sort of final picture of nature that we've reached today. The muon, regardless of what it really is, is an outstanding laboratory for testing predictions that are made using the Lorenz transformation. Specifically, about whether or not muons, given their very short lifetimes, should be able to travel the vast distances from where they're created in the Earth's atmosphere to where they can be measured down on the surface of the Earth. And in fact we find that muons in vast numbers make it from where they're produced in the upper atmosphere to the ground. But they're not supposed to if time passes at the same rate for all observers in all frames of reference. So it may seem weird that time doesn't pass at the same rate when you're moving, but it's the truth. We have direct tests of this, not only with muons, but with many other systems as well. And in many ways, the muon wound up being the very first direct test of the validity of special relativity, and it held up against that test beautifully to live another day and make more predictions, which is what makes it such a spectacular theory of space and time.