THE DOPPLER EFFECT WITHOUT SRT

Historical note added at the end of this chapter 11-4-09

The Lorentz Transformation, and hence Special Relativity Theory (SRT), have been shown to be invalid. It follows that the Doppler effect, without SRT, must be recalculated. Cosmology relies heavily on the Doppler effect and without SRT we are led to a different vision as to the size, age, and nature of the universe than is currently in vogue.

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, 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. But the quantity z+1 represents the ratio of the sent to the received frequency. This ratio is here called the Doppler Factor. 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 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. 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.

COMPARING LIGHT AND SOUND

Suppose we start sending out pulses when the train 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, or is stationary with respect to the air or the aether, 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 problem! 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 rather than 1.5 seconds.

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!

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 space-craft 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.

TRAINS AND DOPPLER

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 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. To repeat: we call the ratio z+1 the Doppler factor.

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

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, the recession velocity under the Newtonian view is less than 0.5, whereas under relativity, the recession velocity is 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 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, 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, 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.

The easiest case to analyze is the case when source and destination are approaching at the speed of light, v = 1. If the source starts sending out pulses at one second intervals when the distance is 30 light seconds, the pulses will arrive spaced 1/2 second apart. The initial pulse will arrive after 15 seconds, when the separation is 15 light seconds, and the 16th pulse, sent when the separation is 15 light seconds, arrives 7.5 seconds later when the source is 7.5 light seconds from the detector. The color of the pulses would change in the direction of higher frequency.

If for example we had v = 0.5 we would get a contraction in the separation of pulses by the factor 1/1.5 or about 0.67. 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.

The problem is to demonstrate this using straightforward arithmetic. If a spacecraft begins sending out radar pulses when the two spacecraft are one light second apart, the first pulse reaches the other spacecraft when they are separated by 2 light seconds. At that point the signal bounces off and returns to the first spacecraft and by the time it returns the distance has again doubled, i.e. it returns after an additional four seconds. Six seconds from the start of the experiment have elapsed.

A second pulse sent one second after the first pulse, when the two spacecraft are 1.5 light seconds apart, reaches the second spacecraft after 3 seconds when they are 3 light seconds apart. It returns to the first spacecraft after 6 seconds - a total of 9 seconds for the round trip. Adding one second for the delay in sending the second pulse yields 10 seconds from the start of the experiment. This is four seconds more than the clock time for the return of the first pulse. So the time between returned pulses is 10 - 6 = 4. Consequently each spacecraft can determine the time between returned pulses, and if it is 4 seconds the two are indeed separating at one half the speed of light. g During that time 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 four billion light years away! It is also true that if the relative movement of earth changed during the time the light is under way the Doppler effect is affected, since the factor is calculated on the assumption that the relative motion is unchanged. It follows that we must be careful and circumspect in drawing inferences.

An interesting case can be made for a receding source that also moves around another object so that those pulses that originate when the source happens to be moving towards us exhibit somewhat less relative motion, while alternate pulses, when the motion is away from us, exhibit greater relative motion. The result produces a split spectrum (or possibly a broader spectrum) with two red lines separated by the difference in relative motion of the emitting object, as it moves around the other object.

NOTE ON THE SPEED OF LIGHT

The Doppler effect can be effectively used to disprove Einstein's Assertion that the movement of the source is irrelevant to the speed with which light propagates.

To show that the movement of the source, relative to the inertial system in which the speed of light is to be measured, influences the outcome of the measurement, consider the following thought experiment:

The inertial system in which the detectors are located, and in which the distance between them is fixed, is a moving train. The detectors are located at the front and at the rear of the train. To insure that the source is in motion with respect to this inertial system, or vice versa, we let the train speed by a source fixed to the ground, at 1/2 the speed of light. The light pulse proceeds along the train and the detectors can be mounted on the side of the train.

The train is to be one light second long. As the rear moves by the light source, (think of it as a flashlight directed towards the front of the train) the source is triggered to send one pulse towards the front of the train. The detector at the rear registers the time T1 as soon as the rear of the train triggers the light source. The light pulse can be thought of as proceeding either in the inertial system of the ground or that of the train. According to Einstein it moves at the same speed in any inertial system.

But since the train is in motion the light pulse overtakes the train, and hence triggers the detector at the front of the train, at a time T2, which must be no less than one second.

To calculate the time difference T2 - T1 without assuming that Einstein's assumption is correct, we can add the infinite series 1 + .5 + .25 + .125 + ... = 1/ (1 - .5) = 2 [or for an arbitrary train velocity v, 1+v+v2+ = 1/(1-v)] representing the time for the chase that the pulse requires to reach the front of the train. So it takes the light pulse twice as long to cover the distance of from the rear to the front of the train as it needs when the source is stationary in the system in which the distance between the detectors is fixed.

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 just a few days ago. I realized with amazement how much of the article is simply wrong and 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 painfully 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.

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.

So Doppler was a prisoner of his time. His treatment of the Doppler effect for light was naïve and confused, 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.