 In this lecture, we will learn the following things. We'll begin by learning what is the classical Doppler effect on an oscillatory phenomenon like a wave. We'll also learn about the effect of the motion of a light source on the characteristics of the light other than its speed. And finally, we will learn how to compute the so-called special relativistic Doppler effect on light and interpret the effect on observations of the world around us. Let's begin by recalling oscillatory phenomena from introductory physics. Specifically, let's look back at something called simple harmonic motion. This is a kind of repetitious motion that has a time and space structure that allows itself to be described using sine or cosine functions of space and time. So for example, depicted in the graph at the bottom of the slide, we have the vertical position of some object as a function of the horizontal position of the object. And we see that the vertical position varies gently upward, then downward, then upward again, and then repeats with the horizontal position. And this motion of the y-coordinate with respect to the x-coordinate lends itself to a direct description in this case using a sine function. Now a wave phenomenon such as a water wave or a sound wave can similarly be described using exactly this kind of mathematics. A sound wave is a region of high compressed air followed by low compressed air followed by high compressed air and so forth. It's a so-called density wave in air. A water wave is similarly an increase in the number of water molecules in one region of water and a decrease in the number of water molecules in another, arising in a falling of the surface of the water. These wave phenomena are oscillatory in nature and can be exactly described using the same kind of sine or cosine function approach that we apply on simple harmonic motion. Now the wave phenomena, just like oscillatory phenomena, have characteristic numbers that describe their spatial distribution. There's no one place where a wave is and where it is not. For instance, you could say that there's more of the wave in this region of y and less of the wave in this region of y. The wave is a structure that's spread out in space and it has both a spatial structure and because it can move in time, it has a time structure as well. We have to use certain quantities to characterize the overall macroscopic shape of an oscillatory phenomenon or a wave. And the wavelength, denoted by the lower case Greek letter lambda, is one such number. For instance, the wavelength of a wave, like the one depicted here, could be taken as the distance between crests of the wave, the locations of the maximum displacement from zero in the y direction, or it could be taken as the zero displacements of the phenomenon. Really picking any two similar points on the wave and drawing a line between those points horizontally will give you the wavelength. Now if we were to observe this phenomenon passing us by, by picking a reference point in space and just watching it go by that point, the time between maxima or minima passing the same spatial reference point is known as the period capital T of the wave. This is the time between the same thing happening over and over and over again in the wave phenomenon. Now the inverse of the period, one over T, is the rate at which, for instance, maxima passed that point, and it's known as the frequency. And it can be denoted in one of a couple of ways. For instance, the lower case letter F for frequency, which equals one over T, or it can be denoted using the Greek letter nu, which looks like a little curved V. That's also used often to describe the frequency of a wave. And again, that's just equal to one over the period, or one over T. Frequency have units of per second, or Hertz, H-E-R-T-Z, the unit of frequency. Now the speed with which waves move in space during some unit of time is actually given by a very simple product of frequency and wavelength. If you want to know the speed of a wave, you just take the wavelength lambda and you multiply it times the frequency F. And of course, for a light wave, the speed with which all light waves move is known to be C, the speed of light. And so this will just be, again, the product of the wavelength of the light wave and the frequency of the light wave. But that product will always yield C, 2.998 times 10 to the 8th meters per second, regardless of whether an observer is in motion relative to the source of the light or not. We know that already as one of the postulates of special relativity. Now summarizing, again, the gross properties of waves, it's helpful to pick a characteristic point on a wave and think about the repetition of those characteristic points as representative of the spatial or temporal distribution of the wave phenomenon. So for instance, we can think of waves of sound or waves of light as merely being represented by lines or planes. So for instance, the location of a line in two dimensions or plane in three dimensions could indicate a location in space of a maximum of the traveling wave. Using this picture at the right, you might imagine that each of the locations in space marked by one of these red planes is a place where you would find a maximum of the wave having been sliced through by the plane. This is a very common way to quickly and simply sketch a wave without having to draw the sine or cosine function. The distance between lines or planes is the wavelength. That's the sort of cartoonish way you represent that particular feature of a wave in the image. Now such a line or plane would be referred to as a wave front. And wave fronts can be used to characterize the location in space of a particular point on a wave and all of the repetitions of that point. The frequency of such a phenomenon can be thought of as how many fronts per second are emitted by the source. So if you think about one plane and then another plane and then a third plane being emitted by the source, the distance in time between those planar emission that would be the period of the wave and one over that would be the frequency, the rate at which it emits wave fronts. Now this brings us to the so-called classical Doppler effect. And I'm going to use sound waves to motivate this because most of you have probably at one point in your life or another actually experienced the Doppler effect with sound waves. The Doppler effect occurs when the source emitting a wave is itself moving relative to an observer. So if we're talking about a sound wave here, we're talking about a listener, someone who can receive the pressure changes in their ears. And when that observer is moving relative to the source, the Doppler effect can occur. And this is actually illustrated in this cartoonish animation below. A car starts emitting sound perhaps by the driver laying on the horn and the wave fronts are represented by those red circles. So everywhere you can locate a point on a red circle would be a wave front of the sound waves. And they're emitted at rest uniformly in all directions. But as the car accelerates forward, the wave fronts in front of the car begin to pile up. The sound waves get closer together. The wavelength shrinks. And behind the car, we see the wave fronts spread out. The wavelength gets bigger. So in this example using sound waves, we have the effect that in the direction of motion of the emitter, the car honking its horn, for instance, and ahead of the source, the wave fronts are pressed together more densely, shrinking the wavelength, and thus increasing the frequency with which waves will strike our ears. If we were to be standing on the backside of this moving object, while it's moving away from us, sort of against its direction of motion, the wave fronts are more widely spread apart than they would normally be. And this increases the wavelength and thus decreases the frequency with which these waves reach our ears. So to human ears, it's the frequency of waves that determines what we call pitch. High-pitched sounds are also high-frequency sounds. The wave fronts are striking our ears more frequently. And vice versa, low-pitched sounds are low-frequency sounds. The time between wave fronts hitting our ears is longer. Now let's think about the Doppler effect on light waves. There is a classical Doppler effect on light waves, but because time and space measurements are also relative to the frame in which you're making the measurement, there is a relativistic component that gets added to this kind of pitch shift even for light waves. So it's true that while all observers may agree on the speed of light, we know that special relativity leads to the conclusion that space and time measurements may differ between observers in different reference frames. Now wavelength is a space measurement and frequency is a time measurement. So couched in that language, it should come as no surprise then that while observers in relative motion all agree that a wave of light travels at sea 2.998 times 10 to the eighth meters per second, regardless of what frame of reference they're in, observers in different frames will disagree on the wavelength and frequency of that light wave. Now to measure wavelength, for example, is to be able to simultaneously locate maxima on a wave, think back to our discussion of measurement and how one measures distances on a moving object. There are different ways you can do it, but one of those ways is to simultaneously co-locate points on the object, in this case, one maximum and then the next maximum. But we know that simultaneity is a frame-dependent statement and in moving frames of reference, we know that the objects pinned to those frames appear contracted along the direction of motion from observers that are not in that frame and are, for instance, at rest with respect to that frame. Similarly, to measure frequency is to be able to measure the time displacement between two events at the same location in space, how often or how much time happens between wave fronts going by a single point. And we know that from the perspective of a frame that's at rest, the time in the moving frame passes more slowly and these two frames would disagree on the amount of time between two wave fronts. So the combination of the classical piling up of wave fronts or stretching out of wave fronts due to the motion of the source with these time or space effects that come from the special relativity postulates and the Lorenz transformation come together in what is known as the special relativistic Doppler effect. And we will derive it here using the Lorenz transformation applied on top of the classical Doppler effect calculation. And we will discuss the implications of this phenomenon for observing the universe. It has some extremely deep impacts on our ability to understand nature, even distant parts of the universe where we have no physical access to moving objects. So to motivate the derivation of the relativistic Doppler effect, I'm going to start by talking about just the sort of classical Doppler effect. And to do this, what I want to do is have you imagine a light emitting device illustrated here as a blue ball that is sending out wave fronts to the left along this coordinate axis. And it's doing so at regular intervals in its rest frame. So for instance, we might have a moment in time, T1 prime and in the frame of the source, where it emits its first wave front. Its first maximum is emitted from a point, say for instance 14 along this coordinate axis. And then at a later time, T2 prime, it emits the second front. So in the time between emitting the first front and the second front, of course, the first front has moved at the speed of light to the left. And it's now at 0.13 on the x-axis at the moment T2 prime that the second front is emitted by the source. And if these fronts are emitted regularly, as would be true in a simple harmonic oscillatory or wave phenomenon, then there will be a third front emitted at T3 prime and a fourth front emitted at T4 prime. And the distance in time between these successive emissions of fronts will all be the same corresponding to the period of the source, capital T prime in the frame of the emitting source. So T2 prime minus T1 prime will be the same as T3 prime minus T2 prime, all of those intervals between neighboring wave front emissions will be the period of the source as observed in the frame of the source. This defines the period of the light wave. Thus we have a regular frequency in the source frame. We can write that frequency in the source frame as F prime equals one over T prime. Now that was done with the source sitting at rest along this coordinate axis. And so in that picture, an observer sitting at zero on the coordinate axis would agree that the wave fronts all arrive with the time between them equal to what the source said it would be because the source and the observer were at rest with respect to each other. But now let's imagine that the light emitting device that's sending out those wave fronts at regular intervals and its own reference frame is moving with respect to the observer at zero in the above coordinate axis. So let the velocity of the source be plus V that is it's moving away from the origin to the right and entirely along the x-axis. Treat the observer as being at rest, we'll call that frame S and the source as being in motion we'll call that frame S prime. Let's think about what will be the distance between wave fronts arriving at the observer from the perspective of the source. So we're doing all of this from inside the source frame S prime. We will transform to the rest frame of the observer later. So here is our source. It's at location 12 along the x-axis and it emits its first front front one at time T one prime and the wave front moves toward the zero points on the x-axis at the speed of light C. Now the next time that the source emits a wave front it has itself moved to the right from point 12 to point 13. In the meantime, the light wave front that it emitted the first front front one has moved at the speed of light to the left and in this particular example it winds up being at point 10 on the coordinate axis. So in the frame of reference of the source the distance between the wave fronts would have been lambda prime, the wavelength. And that's depicted here on this cartoon showing where the original un-stretched wavelength of this phenomenon would have been if the source had been at rest. The source had been at point 12 when it emitted front one. Front one is now at point 10. And so the distance between those two would be lambda prime, the wavelength of the phenomenon in the frame of the source. But the source is now moved and so it's at this location 13 that it emits its second front. So by the time it emits the second wave front at time T two prime, the first wave front is a distance of C times T two prime minus T one prime or C times the period in the frame of the source. And that's just equal to lambda prime. That's the distance from where it was emitted. But the source is now farther from the observer by an amount of V times T two prime minus T one prime or V times the period when it emits the second wave front. And again, that's depicted up here. So this is the distance from the point of emission that front one has traveled which technically would have been one wavelength of the original emission but the source has moved back further along the x-axis by a distance VT prime, V times the period. And so the source will argue that as a result of this an observer who is looking at these wave fronts coming at them sitting at point zero should see the combined distance of lambda prime plus VT prime between the two wave fronts and this will actually be the observed wavelength of the phenomenon according to the person riding with the source. That will be what the observer sitting at zero should see. So this is all illustrated above. We take lambda prime and we add VT prime and that's going to give us the observed wavelength in the frame of the source. So this is all illustrated and we can write the equation down adding these two together. And then we can rewrite this in terms of frequencies by remembering first of all that period is equal to one over frequency or frequency is equal to one over period and that the speed of light is equal to the product of wavelength and frequency. Similarly the wavelength for instance in the frame of the source will be the speed of light divided by the frequency in the frame of the source. And the wavelength according to the observer from the perspective of frame S prime will be equal to the speed of light divided by the frequency according to the observer in that frame. So we then find again all of this from the perspective of frame S prime that the observed period at zero should be one plus V over C all divided by the frequency of emission F prime. So maybe pause here and try to work all this out for yourself but again keep in mind that we have not yet transformed this observation into the rest frame. That is the frame in which the observer is at rest. Right now we're calculating the observed wavelength or period according to what the observer should see if their time measurements were absolutely in agreement with the moving source. This is the classical Doppler effect the stretching or compressing of wavelength with the motion of the source. We haven't yet included for instance time dilation or length contraction in all of this. Now we're going to take that last step and to aid us in notation here we're going to begin by defining a very convenient symbol and that is the lower case Greek letter beta which is rather regularly used to represent the ratio of the speed of the frame for instance divided by the speed of light. Because speeds V never exceed the speed of light C and because speeds can never be any less than zero beta is a number that goes from zero to one. Zero for things that are at rest one for things that are moving exactly at the speed of light and can take all values in between. In terms of beta the claimed period of the phenomenon as observed by the observer in frame S prime should be one plus beta divided by the frequency of emission. But again so far all of this is in the S prime frame. This is what a person in S prime following along with the source would argue is what the observer should see. The original frequency of emission from the perspective of the source F prime which we can just call F with the subscript source and includes the relative speed of the source in the observer V and the period and frequency with which a person in the source frame would expect the observer to receive the wave fronts. T prime observed or correspondingly the observed frequency. However, if we now do the special relativity and use the Lorenz transformation and transform this stuff into the actual frame of reference of the observer we know that there's going to be another effect here and for instance we could summarize that by saying that it will be the relativity of time. Time in the source frame where all the emissions happen at the same location is proper time. The source always says that wave fronts are being emitted from its location in space and as a result of that that's the frame where proper time will be observed. But in any other frame moving relative to the source time dilation will be what is observed that is the passage of time on the moving object will appear to be slower than the observers on the moving object would claim. And it's given simply by taking the proper time and multiplying it by gamma. So the time, the period observed in the rest frame of the observer will be gamma times the period that the source claims the observer should have seen according to the classical Doppler effect. So we then finally arrive at a expression for the period of the phenomenon of the light wave as observed in the rest frame. So we start by just saying that the period observed in the rest frame will be equal to gamma times the period that should have been observed from the perspective of the moving frame T prime with the subscript OBS. We can substitute in with one plus beta over the source frequency F prime. And we can do some algebraic gymnastics to sort of rewrite this in a more pleasant looking form. We've got gamma and we've got beta. Of course, gamma depends on beta. Gamma has V over C all squared inside of it. That's beta squared. So it's nice to try to rewrite this all either in terms of just beta or just gamma. And so with a little algebraic gymnastics starting with writing gamma is one over the square root of one minus beta squared, you can then do a little work and show that a final neat looking expression for this is that the period observed in the rest frame is equal to the square root of the ratio of one plus beta over one minus beta, all times one over the source frequency, the frequency of emission from the perspective of the source itself in its rest frame. So we can transform this into an observation, of course, of the frequency in the rest frame by simply doing one over T observed. And that just flips the stuff in the square root upside down. And you wind up with this neat little relationship that the frequency of the light observed in the frame that's not moving. We'll see the frequency as emitted in the source frame where the source is at rest shifted by an amount given by the square root of one minus beta over one plus beta. So we've solved the problem now. We've derived the special relativistic Doppler effect, the shifting of the frequency due to relative motion between the source of emission and the observer of the light by considering the situation where the source is moving away from the observer. Now, this special relativistic Doppler effect is a combination of two effects. The classical Doppler effect of just the effect of the moving source that adds extra space between the wave fronts. But in addition to that, the dilation of time due to relative motion of the source and the observer, proper time and therefore proper frequency would be in the frame of the source. This is modified by a gamma factor to go into any other reference frame. So the special relativistic Doppler effect is a combination of the classical Doppler effect with the relativity of space and time measurements. And you actually would expect from just Newtonian and Galilean relativity that there's a Doppler effect on frequency and wavelength of light. But the special relativistic addition to that actually makes the effect even more extreme than expected from Newtonian and Galilean mechanics. And in fact, what we see in the universe is what is predicted by special relativity and not just the old mechanical Galilean and Newtonian approach to motion. Now, for a source that's moving toward an observer that is approaching the observer while emitting wave fronts, the sign of the velocity is all that needs to be changed. We go from having beta, the velocity moving away to negative beta, the velocity now moving toward the observer. And in fact, you can do the work yourself, but this formula up here for the source moving away from the observer can be transformed into the case for the source moving toward the observer by flipping the sign of beta. So taking beta and turning it into negative beta. And all that does is it takes the stuff into the square root and flips it upside down. So now we have the square root of one plus beta divided by one minus beta. That whole thing times the frequency of emission in the source rest frame. So I would recommend you practice this calculation by checking for yourself that this second equation for an approaching source is correct. But once you've convinced yourself of that, the shortcut is a really easy thing to remember. If you can remember one of these two formulas, you can get the other one simply by changing the sign of beta, not too bad. So let's talk about some expectations from the special relativistic Doppler shift. For example, if a light source is moving away from us or toward us, what do we expect to happen to the frequency of its light? So for a source that's moving away from us at speed beta along our line of sight, we expect to scale the source frequency by the following quantity, the square root of the ratio of one minus beta over one plus beta. Now if you play around with this a little bit, you'll notice that this thing is always less than or equal to one. It's exactly one when beta is zero. And if beta is anything other than zero, its value decreases from one. The frequency therefore that we should observe should always be lower than in the sources frame of reference, owing to the stretching of its wave fronts combined with the dilation of time. Now because as I said, beta is a number that's inclusively bounded between zero and one, we are taking the ratio of a number less than one and a number greater than one for beta that's anything other than zero. Now if instead the source and observer are moving toward each other, then we scale the source frequency by this quantity, the square root of one plus beta divided by one minus beta. And again if you play around with this, you'll find out that this is either always equal to one or greater than one. This means that the observed frequency is always greater than what is observed in the frame of the source. Since we're dividing a number greater than or equal to one by a number that's less than or equal to one. So to summarize all of this, for a source that's moving away from the observer of the light, the frequency that the observer sees will be lower than the frequency that's observed in the rest frame of the source itself. And similarly for a source that's moving toward an observer, the observer will always see that the frequency is increased over the frequency as observed in the rest frame of the source of the light emissions. These equations are all for frequency. But we can very quickly derive the equations for wavelength using the fact that the speed of light is equal to wavelength times frequency. So if we go through the brief algebra gymnastics for this, we find that we get the following equations for the observed wavelength depending on whether the source is moving away from the observer or if the source is moving toward an observer. So as expected, when a receding source is present, this gives us a lower frequency and a longer or greater wavelength. So the frequency has gone down. Therefore the wave fronts are farther apart from each other because the wave is still traveling at the same speed C. On the other hand, when the source is approaching us, we get the higher frequency, which means the wave fronts are coming at us more often. And that means a shorter wavelength will be observed for the phenomenon. So let's talk about the perception of light color due to the full relativistic Doppler shift. So I've rewritten here the equations for the wavelength that an observer sees depending on whether the source is moving away from the observer, in which case the wavelength is stretched by the motion, or if the source is moving toward an observer, in which case the wavelength is compressed by the motion. Receding sources of light are said to redshift compared to when they are at rest. And that's because longer wavelengths correspond to redder light than shorter wavelengths, which correspond to bluer light. I've illustrated this over here on the right using a spectrum and specifically I've isolated the visible or color spectrum of the electromagnetic frequency spectrum. So for instance, red light near the edge of where the human eye can detect the color red has a length of about 700 nanometers or 700 billions of a meter. Blue light or violet light, which is at the other end of our ability to see comes in at around 400 nanometers or 400 billions of a meter in length. Blue light has a shorter wavelength and thus a higher frequency than red light. So if a source is moving away from us, the wave fronts get stretched out and that would take something that's bluer and shift it more toward the red end of the light spectrum. And conversely in approaching sources said to be blue shifted because this results in shorter wavelengths which corresponds to the bluer end of the color spectrum. Now of course it's possible that if you have an object that's already say violet in color and it's moving toward you very rapidly at a significant fraction of the speed of light the shifting effect can be such that it actually shifts so blue that it goes outside the visible spectrum and then you'd have to look for it in ultraviolet or X rays or other similar very short wavelength electromagnetic phenomena. Similarly if an object is already very red and on the edge of the ability of the human eye to see it and the source of the light is moving away from you appreciably quickly this can result in a red shift that puts it into the infrared or even microwave or radio depending on the speed of the object that's emitting that light. You can imagine therefore that this has some strong implications for measuring our place in the whole cosmos. For example, without making physical contact with distant stars or galaxies which are collections of billions or trillions of stars it's possible to actually determine whether or not those objects are receding away from earth or approaching toward earth based on the degree of the color shift of their atomic spectra. Let's take a look at an example of this. Through long centuries of observation of distant objects by astronomers and especially by breaking down the light from distant objects into their component colors the so-called color spectrum or atomic spectrum. Astronomers have determined that the stuff that makes up everything out there is the same stuff that makes up everything down here on earth and that is at least for the luminous stuff the stuff that emits light or absorbs light that stuff is atoms. And the atoms that are out there appear to be the same as the atoms that are down here on earth. Iron has the same atomic spectrum whether it's found on earth or in the heart of a star. So as a result of that we can look at the light coming from distant objects figure out what atoms it's made from and knowing the pattern of light each atom gives off determine whether or not, first of all it's composed of certain atoms and second of all whether it's moving relative to us. So here's how you figure out the motion. The spectrum on the left over here on the slide is actually from our own star, the sun. The sun is not appreciably getting closer or farther away from us over the course of a day or a year. We're going around the sun and our distance is changing slightly as we orbit the sun every year. But it's not happening so fast that you get an appreciably different shift at least to the eye in the color of the sun. So we can consider the sun to be an atomic spectrum that represents a star at rest. On the other hand, on the right hand side is an atomic spectrum from a very distant so-called supercluster of galaxies. That's a cluster of galaxies of stars which are themselves clusters of stars. And it's named BAS 11. It's not so important what it's called. But if you stare at this for a few moments you'll notice that there's an interesting similarity in the pattern of the light between our sun and the light that's coming from all the stars that make up this distant supercluster of galaxies. There's a missing color line here in our sun and then there's a gap where there's lots of colors and then there's another missing line. And then there's another gap and then there's a missing line and then there's a small gap and there are two missing lines. And if you look over on the right at the light from the supercluster you see that, oh look, there's a line in the red that's missing and then there's that same gap and then there's a line in the yellow that's missing and then there's that same gap and then there's a line in the green that's missing. And then there's that same small gap and then the two dark lines that are next to each other. It's as if somebody took the pattern in our sun and shifted it toward the red end of the spectrum. And this is exactly what we would expect from special relativistic Doppler effect if the supercluster is moving away from us, thus stretching its emitted wavelengths of light longer from our perspective. So these black gaps, so-called absorption lines in the spectrum have the same pattern but in a slightly shifted location in the sun compared to this supercluster of galaxies. And the fact that those missing colors are red shifted means the galaxy supercluster is moving away from us. That's the conclusion from the special relativistic Doppler effect. And you can actually then use measurement differences of the wavelengths between where the missing wavelengths are present in the sun and where they're present in the galaxy supercluster and using some astronomy, you can actually estimate the relative velocity, beta equals V over C, with which the supercluster is receding from us. This is incredible. This allows us to measure velocities without having physical access to a material object. All we have to do is look at the pattern of light that comes from its atoms and knowing that those patterns are the same patterns that should be found here in the atoms on Earth, look at the shifting of those patterns to determine the relative velocity of us to the distant object. This kind of measurement is actually how we know that the universe as a whole is expanding. So far as we've been able to determine all distant objects appear to be receding away from the Earth as if carried along on the momentum of an initial explosion that set the whole universe in motion of all points expanding outward from every other point. This implies the universe as a whole and on the largest distance scale is expanding with time. So let's review what we have learned in this lecture. We've looked at the classical Doppler effect, both in a cartoonish way and using the example of a moving source emitting wave fronts along a horizontal axis. We've then considered the effect of the motion of a light source where observers all agree that the light waves are moving at the same speed. We've looked at the effect of the motion on characteristics of light other than that speed which isn't changing the wavelength and the frequency of that light. And by combining the classical Doppler shift with time dilation, we've seen how to compute the special relativistic Doppler effect on light and we've even looked at ways that you can interpret that effect on observations and take observations of the natural world and use those to infer relative velocities on the grandest scales of the cosmos.