The Doppler Effect

OVERVIEW

To understand the Doppler effect we can use down to earth examples, using pulses of light or sound instead of waves, and a train speeding away from the origin of a coordinate system. We have still air to carry sound pulses, but in the case of light, no detectable medium to 'carry' the light pulses. We use pulses of light or sound, spaced one second apart at the source, and let the source be either at the origin of the coordinate system, or on a train moving away from the origin (where the receiver is then located). We calculate the difference in time of the pulses at the destination (which, of course, depends on the velocity of the train, the only velocity we need be concerned about) and since the distance between source and receiver is steadily increasing, the spacing of the pulses will be a constant that is greater than one second. It is the same for sound as for light pulses when the sender is at the origin, but not the same when the source is on the moving train and the receiver stationary at the origin. The Doppler effect for light is symmetric, for sound it is asymmetric. In the case of light we can use Einstein�s second principle. It states that, in the case of an experiment with light, source and receiver can be interchanged without affecting the result.

In the case of waves, instead of pulses, the effect of the increased distance between arriving wave crests implies an apparent increase in wavelength, which in the case of sound, makes for a lower tone - as can be noticed, from the sound made by the tires of a car that passes you on the highway, or a plane that dives towards you and then flies away.

For a receding supernova the Doppler effect produces a shift in the spectral lines of an atom, a red shift, as it is known � how large the shift, tells us something about how fast the supernova is receding, or was receding, from us. If the event originated well over a billion years ago, it represents ancient history, and we are not entitled to infer the present state of the star, or the present development, expansion or contraction, of the universe.

DETAIL

The Doppler effect is undoubtedly one of the most important phenomena in Cosmological and Astronomical research – but it has been misused, miscalculated, and misinterpreted. It pertains both to sound and to light – but it is not the same, and does not have the same mathematical formula - sound is carried and requires a medium, usually air, as carrier, while light can travel in vacuum and, as the Michelson-Morley experiment confirmed, light has no carrier.

The Doppler effect is the same for periodic pulses as for waves and manifests whenever source and destination are in relative motion – approaching each other or separating, while the pulses or waves are travelling from one to the other. It is most dramatic when we perceive a plane overhead coming towards us and then receding – the sound changes from a high pitch to a low one at the point of closest approach – and that is not due to any change in the inherent sound of the engine as some believe. If it were a monochromatic light source, instead of the sound of the engine, it would be a change from a high frequency to a low one – a shift from ‘blue’ to ‘red’ – to introduce the ‘Doppler red shift’ that we use, in connection with supernovae, to infer the expansion of the universe.

In cosmology we use the shift in spectral lines of a supernova explosion to determine how fast the supernova is, or better, ‘was’ receding from us when it exploded. ‘Was’ is the important word here. If the phenomenon of the explosion happened over a billion years ago, it has taken that long for the light to reach us. So we are inferring what happened then, not what is happening now – a mistake often made in referring to the expansion of the universe.

During the time that the signal is underway, over a billion years, the expansion of the universe could have slowed down or even reversed. Since the movement of the source after light is emitted does not affect the Doppler effect we can't know whether in fact the expansion continued - and we can't infer that the supernova occurred, say, four billion light years away (using a concept of ‘distance’ that implies our present time and measurements)! It is also true that if the relative movement of earth changed significantly during the time the light is underway the Doppler effect is affected, since the factor is calculated on the assumption that the relative motion remains unchanged. It follows that we must be careful and circumspect in drawing inferences.

The Doppler effect for light is symmetric (it doesn't matter if the source is moving or the receiver), and depends only on the separation rate between the source and receiver. In the case of sound, where there is a carrying medium, the effect is asymmetric. When source and receiver are moving apart this phenomenon is generally known as the Red Shift - if moving towards one another - the Blue Shift. The Doppler effect is symmetric under Special Relativity Theory, SRT, as well, but the effect is actually larger than predicted by SRT. Physicists usually use the relative shift, z, called the Doppler shift, which is defined as the difference between the sent and the received frequency divided by the received frequency. The quantity z+1 represents the ratio of the sent to the received frequency. This ratio is here called the Doppler Factor, DF. This is the quantity that can be treated under multiplication, when two successive events (such as a back and forth movement of sound or light) are considered. We can then take the square root, the geometric mean, as the average one-way effect. We will use pulses instead of waves in the discussion, as intuition is thereby simplified.

If the Michelson-Morley (M&M) experiment had found a difference in the two directions that light moves relative to the earth's movement, it would have confirmed the aether as carrier for light. With a carrier, such as air is for sound, there is always an asymmetry. The Doppler effect is different depending on the movement of source or destination with respect to the supposedly stationary carrier. But in the case of light, as M&M showed, there is no difference, the Doppler effect is not dependent on whether the source or the destination is in motion.

We must be careful not to push the analogy of sound and light too far. Sound is energy that is transmitted by molecules of matter, either air or solids. These molecules do not rush from source to receiver. They stay approximately in place, and mimic the vibrations impressed at the source. Light, on the other hand, is the radiant energy that moves from source to receiver. Each photon carries a signature that is its frequency, but that signature can be distorted through motion of the source or the receiver or through interaction with matter.

COMPARING LIGHT AND SOUND

To illustrate the Doppler effect, suppose we have a train moving at uniform speed along a track, and we let the engine emit pulses at one-second intervals. What matters is the ratio of the speed of the train relative to the velocity of either sound or light. We use v as the ratio and assume that in either case v = 1/2. Also, for the sake of simplicity, we use light-seconds, or sound-seconds instead of kilometers to express distance. One light second = 300,000 km, and similarly for sound.

Suppose we start sending out pulses when a train that is moving at half the speed of sound, is, say, 10 sound-seconds from the origin, where we have placed a stationary receiver.

If sound travels at, say, 2 km/sec (only for ease in calculations) the train starts to send out pulses when it is 20 km from the origin (20 km = 10 sound-seconds). A second later the train is 21 km or 10.5 sound seconds from the origin. At that point it sends out a second pulse, which now needs 10.5 seconds to reach the origin. We assume that the air, which carries the sound, is stationary with respect to the ground, so that the time of travel of the sound is still 2 km/sec (measured with respect to the stationary coordinate system, or the carrying air). Because the second pulse is sent out a second after the first pulse, the clock at the receiver reads 11.5 seconds when the second pulse arrives - a difference of 11.5-10 = 1.5 seconds. In general the spacing is (1+v) seconds at the receiver, when pulses are sent 1 second apart.

In the case of light, if we assume the train is moving at .5 light-seconds per second, we cannot argue that it will take 10.5 seconds for the second light pulse to reach the receiver. We know from the M&M experiment that there is no medium as carrier so the situation is different. But if the source is stationary, and the receiver is in motion, the magnitude of the Doppler effect will be the same for both light and sound, so we can analyze this case by putting the source at the origin and the receiver on the train. Consequently we let the light (or sound) pulses originate from a stationary source at the origin where the receiver was located, and ask how long it takes for the pulses to reach the moving train. If the first pulse is sent 10 seconds after the train passes the origin, the train is, at that point, 5 light seconds beyond the origin. The light requires 5 seconds to reach the place where the train was when the pulse was sent, but by the time the light gets there the train has moved on, an additional 2.5 light-seconds, we then repeat the process. It now takes the limit of an infinite series, 5 + 2.5 +1.25 +... = 10 seconds to reach the train - a case of the tortoise and hare! The second pulse is emitted 1 second later when the train is 5.5 light seconds further on. It now takes 5.5 + 2.75 + ... = 11 seconds to reach the train. We need to add one second for the delay in sending the second pulse, so the clock time is now 12 – an interval of 2 seconds between pulses.

The general formula now becomes 1+v+v2+...= 1/(1-v) for the spacing between pulses, that is to say, with v = 0.5 the spacing is doubled. The received frequency is one-half the emitted frequency. If blue light was emitted, red light is received. The same result applies to sound. To revert to the original problem, in which the source is on the moving train, we can, in the case of light, simply use the first principle of relativity. It allows us to interchange source and receiver. In the case of light we can use Einstein's first principle, to declare that we will get the same result if the source is on the train - but not so with sound! NOTICE THAT WE GET Z = 1 AT HALF THE SPEED OF LIGHT.(Since a doubling of the wavelength or spacing of pulses only requires a recession at 1/2 the speed of light.

To make the case for symmetry it is better to think of two spacecraft in outer space that try to determine whether the distance between them is fixed, or whether they are separating. Neither spacecraft can be said to be preferred. They can establish their relative motion by noticing if the signal sent out by the other is getting weaker, or by sending out pulses that bounce off the other and return.

If the spacing of the returned pulses remains constant and equal to the spacing with which the pulses were sent, the two spacecraft are at constant distance. If the spacing remains constant over time, but larger than sent, the spacecraft are separating at a constant rate. The Doppler effect, as inferred from the spacing, must be symmetric as long as there is no other body in terms of which motion or rest can be defined.

The result of these deliberations can be confirmed, theoretically, by using a train, a single observer, and no clock.

We need only station this observer half way between two points A and B on the ground, the distance between the rear and the front of the train. A trigger or trip wire can be used to generate a pulse of blue light at the points A and B on the ground, as the front of the train reaches B and the rear of the train simultaneously reaches the point A. Simultaneously a blue pulse can be generated on, and at the front of, the train. The train is presumed to be traveling at one-half the speed of light. What the observer should notice is that the two blue pulses from the ground reach him simultaneously, while the pulse from the train will be red in color and, as we can show, will reach him after the other two pulses have arrived.

The middle of the train will coincide with the position of the observer at the time the pulses from A and B are initiated. But the middle of the train will have moved on by the time the pulses arrive at the position of the observer. Since the pulse on the train requires the same time to reach the middle of the train as the pulses from A and B require to reach the observer, the pulse from the train must travel a longer distance, and therefore takes longer to reach the observer. We have complete symmetry here. If we put the observer at the midpoint on the train and emit blue light at the right time from the two end-points on the train, as well as from point A on the ground, we get the same result.

BACK TO THE THEORY

As already indicated, Physicists usually use the concept z, called the Doppler shift, and defined as the difference between the sent and the received frequency divided by the received frequency.
On the other hand, the quantity z+1, the Doppler Factor, call it DF, represents the ratio of the sent to the received frequency. This is the quantity that can be treated under multiplication, when successive Doppler effects occur. This product can be "averaged" (in the sense of a geometric mean) by taking the square root of the product, DF1 x DF2.

With v as the velocity of the body relative to the velocity of light, and z as the Doppler shift, we get that under SRT the relation between v and z is given by

v = [(z+1)2-1]/[(z+1)2+1]

(see Weigert & Wendker 1996, p. 266). Under the Newtonian view, we get for the Doppler factor

DF = z+1 =1/(1-v)

This formula implies that as long as z is less than 1, v is less than 0.5. In other words, for a given DF the recession velocity under the Newtonian view is less than 0.5, whereas under relativity, the recession velocity, v, must be larger.

COSMOLOGY AND DOPPLER

There are type 1A supernova data, published in 1998. The data show that z is always below one in the visual region.

This implies that even the most distant, and fastest moving, stars never attained a value z greater than one (or a DF greater than 2). So, under the analysis above, they never exhibited a velocity greater than one-half the speed of light. Apparently, the Big Bang wasn't all that big. This in turn suggests that matter existed before the Big Bang, and that the phenomenon may very well repeat.

A red shift can occur because of the increase in distance between the source and the receiver, but it could also occur because (Compton) energy is lost from a wave, or photon, in the course of its journey and interaction with a denser medium. There is no reason to believe that both phenomena can't occur simultaneously in the cosmos. We cannot easily unscramble these effects and arrive at a conclusion about astronomical distances based solely on a red shift that is the result of changes in frequency caused by both motion and energy loss. This is particularly true with respect to inferences in radio astronomy, where red shifts on the order of 5 or more have been observed, and which have defied sensible explanation.

Regarding Space Travel and Cosmology: The formula for what is here called the Doppler factor, DF, the ratio of the frequency sent to the frequency received, is either 1/(1-v) or 1/(1+v) depending on whether there is a steady increase or a decrease of distance, (a red shift or a blue shift) between source and destination, during light transmission. The factor applies to the spacing (as well as to the width) of pulses, as well as to the color of light.

As already mentioned, one possible application to space travel, or space intelligence, to determine relative velocities is the following.

Two spacecraft are moving apart, and we know (but they don't) that they are separating at one half the speed of light, c/2. Each spacecraft must determine the rate of separation using the Doppler effect. We know, and they know as well, that the Doppler factor is 1/(1-v). This means that for a separation velocity v = 0.5 the factor is 2. We also know that for a two-way trip the Doppler factors multiply, so for the round trip the factor must be 4.

HISTORICAL NOTE — CHRISTIAN DOPPLER, 1842

The original article was published by Christian Doppler in 1842, and was sent to me by Peter Marquardt in 2007. I only studied it carefully, and discussed it with Peter, when I visited him in Cologne a few years later. I realized with amazement how much of the article is misleading when judged from the perspective of 21st century physics and astronomy.

Doppler was a captive of his time: In 1842 it was generally believed that light could be conceived as totally analogous to sound — a wave that is carried by a medium conceived as analogous to air, acting on the eye just as a sound wave acts on the ear. It was still disputed as to whether the light waves undulated in or transverse to the direction of propagation, but Doppler needed the longitudinal belief to explain the perceived change in the body’s sensation of the tone or color that occurs when either the source or the receiver is in motion.

We can only speculate as to which phenomena were known in 1842 that were sufficiently fast so as to generate the ‘Doppler Effect’ at that point in history. It could have been a train whistle, since steam engines were already in use. It could have been a horse galloping by on the cobblestones of a street whose hooves would generate sound pulses that change in character as the horse approaches and then passes us by. In any case the Doppler effect for sound was known and he developed the correct formulas that show the effect to be greater when the receiver is in motion at a given rate, rather than the source. It was therefore natural for him to take the same formulas as applicable to light. That helps to explain why many still believe in the asymmetry of the Doppler effect for light.

Einstein, to his credit, rejected the need for an aether, as carrier of light, but he did not go so far as to believe in light quanta. He did not understand the need for the conclusion that the Doppler effect for light must be symmetric under his claim that receiver and sender can be interchanged as far as effects due to relative motion are concerned.
Doppler’s work was also tainted by the fact that black body radiation was not known, or understood until about fifty years later. That a bright burning fire gradually becomes red in color as it cools must have been observed, but the theory was not available until about the beginning of the 20th century. As a result, it was possible for him to confuse the red shift due to blackbody cooling with a red shift due to motion. In fact, astronomers generally could not have known whether the change in color of a star was due to motion or to cooling. But astronomers had a clue in the case of double stars where the change in color was periodic and due to the relative motion of one star around the other. That explains the title of Doppler’s work: “Über das farbige Licht der Doppelsterne”. But in reading his article, it is clear that the two types of reddening — due to motion and due to cooling — were not differentiated in his thinking.

This lack of differentiation becomes clear in his discussion of an astronomical phenomenon that was well known, and remembered, at that time. In 1572 there had appeared a star that quickly became brighter than any known star, and over the months and years that followed gradually became yellow and then red, and faded. He, along with most astronomers, was sure it would repeat this process in 150 or perhaps 300 years, and confidently treated it as a Doppler effect – on the assumption that he was dealing with two stars.

We know now that it was a supernova visible to the naked eye, and this strong an effect would not be seen again until 1987 — but it was no Doppler effect in the sense of a double-star, i.e. a periodic movement of one star with respect to another.
So Doppler was a prisoner of his time. His treatment of the Doppler effect for light was naïve, but his name is linked with the phenomenon. The lesson for us is clear. We are all prisoners of the belief systems that exist at a given historical moment. To what extent we can transcend these belief systems, I sometimes wonder. Faith in the teaching and lessons of the past — that is the ball and chain we drag with us.