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- Thread starter nakurusil
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Authors: Rothenstein, Bernhard; Popescu, Stefan

The uniformly accelerated reference frame described by Hamilton, Desloge and Philpott involves the observers who perform the hyperbolic motion with constant proper acceleration gi. They start to move from different distances measured from the origin O of the inertial reference frame K(XOY), along its OX axis with zero initial velocity. Equipped with clocks and light sources they are engaged with each other in Radar echo, Doppler Effect and Radar detection experiments. They are also engaged in the same experiments with an inertial observer at rest in K(XOY) and located at its origin O. We derive formulas that account for the experiments mentioned above. We study also the landing conditions of the accelerating observers on a uniformly moving platform.

Comment: 15 pages, 8 figures, includes new results on radar detected times and distances

If you consider arXiv among "such" then delete my thread. If not please have a critical look at my paper you can download from arXiv.

sine ira et studio

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Chris Hillman

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Hi, nakurusil,

I am not sure I know what you mean by "

So let me restate the question as: "what are some mainstream treatments of frequency shift phenomena, involving a source and an receiver, when either the source or the receiver (or both) are accelerating"?

Some obvious places to begin are the first edition of Taylor and Wheeler,

Then you can see "Frame fields in general relativity", "Rindler coordinates", "Born coordinates", "Ehrenfest paradox", "Bell's spaceship paradox" in the versions listed at http://en.wikipedia.org/wiki/User:Hillman/Archive; [Broken] these have citations to published review papers where you can find many more references. Books like Nakayama,

And don't overlook the obvious: MTW itself has a huge bibliography which you can use to find some older but still important, useful, and relevant papers.

Unfortunately, I must add a specific caveat. I and others have noticed that the arXiv is particularly uneven in the case of papers on alleged "foundational issues" centering around relativistic physics, and I emphatically intend to include the treatment of accelerated observers as a known "problem area" in the arXiv where eprints have a better than even chance of being partially or completely incorrect. It is crucial to understand that most physicists (at least those who often deal with relativistic physics), upon being asked to pick out the bad eprints, would have little trouble doing so; there is fact wide concensus on right and wrong ways to treat these problems, but a small group of noisy dissidents came sometimes create an incorrect impression which might fool casual observers unfamiliar with standard mathematical techniques. Let there be no mistake: the appropriate mathematical techniques are very well known and widely used outside of these particular problems; the issues in questions come down to computations and there is no ambiguity about the fact that the incorrect claims are in fact mathematically incorrect.

[EDIT: having just noticed another post in this thread, perhaps I should caution against "guessing games" about which specific arXiv eprints I might have in mind. That would be profitless and in self-defense I will not respond to queries of that sort.]

You can also search PF for a recent thread (last two weeks or so) in which I posted some computations of frequency shift phenomena for various pairs of observers, including some accelerating receivers, in the Schwarzschild vacuum solution.

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Hi, nakurusil,

I am not sure I know what you mean by "general relativisticDoppler effect"; one major point I have tried to emphasize in several threads here is thatthis stuff really has nothing to do with general relativity or even curved versus flat spacetime, but rather with the geometry of congruences (especially null congruences), whether in flat or curved spacetimes.

So let me restate the question as: "what are some mainstream treatments of frequency shift phenomena, involving a source and an receiver, when either the source or the receiver (or both) are accelerating"?

Some obvious places to begin are the first edition of Taylor and Wheeler,Spacetime Physics, or another book which treats the k-calculus (IIRC, there is one by Rindler which does so), then Section 2.8 and Chapter 6 of Misner, Thorne & Wheeler,Gravitation(MTW) for two scenarios featuring accelerating observers and or sources.

Then you can see "Frame fields in general relativity", "Rindler coordinates", "Born coordinates", "Ehrenfest paradox", "Bell's spaceship paradox" in the versions listed at http://en.wikipedia.org/wiki/User:Hillman/Archive; [Broken] these have citations to published review papers where you can find many more references. Books like Nakayama,Geometry, Topology, and Physics, might be useful in following the first article; there is also some material on frame fields in MTW, and you can find these techniques treated in standard monographs such as Hawking & Ellis,The Large Scale Structure of Space-Time.

Note well!Wikipedia is inherently unstable, and that articles in this specific area have been dogged in the past year by a single cranky dissident who managed to chase out of WP least one contributor with a Ph.D. in physics, plus myself (Ph.D. in mathematics), who incorrectly maintains in the face of all evidence that the mainstream view is his view. For this reason, current versions of the articles I mentioned may be better than the last ones I contributed to, or they may be much worse, so I recommend that you start with the ones I cited and compare carefully with subsequent editions. To repeat something I find myself saying with distressing frequency at PF:I will not "discuss" physics with specific cranks or their supporters; it should not be neccessary to for me to explain why not and I will not do so.Ultimately, you are on your own in terms of evaluating specific versions of specific WP articles; it will not always be easy to tell simply from obvious clues whether or not the article faithfully and accurately reflects the current scientific mainstream. Ultimately, only someone who has read the literature with adequate insight and understanding may be able to tell.

And don't overlook the obvious: MTW itself has a huge bibliography which you can use to find some older but still important, useful, and relevant papers.

Note well!Everyone should be aware that published papers vary widely in quality; as you yourself obviously already appreciate (good!), some "journals" seem to function as trashcans which collect papers rejected by more rigorous journals. In addition,while the arXiv is an invaluable resource, it does not yet function as a refereed journal. The "endorsement" system is only analogous to "moderation" in a newsgroup; it cannot and does not prevent cranky eprints from being posted to the arXiv. Thus, the quality of eprints posted there varies even more widely than does the quality of papers in the published literature. So you should be cautious about anything you read outside a highly reputable and widely used textbook such as MTW until you know more.

Unfortunately, I must add a specific caveat. I and others have noticed that the arXiv is particularly uneven in the case of papers on alleged "foundational issues" centering around relativistic physics, and I emphatically intend to include the treatment of accelerated observers as a known "problem area" in the arXiv where eprints have a better than even chance of being partially or completely incorrect. It is crucial to understand that most physicists (at least those who often deal with relativistic physics), upon being asked to pick out the bad eprints, would have little trouble doing so; there is fact wide concensus on right and wrong ways to treat these problems, but a small group of noisy dissidents came sometimes create an incorrect impression which might fool casual observers unfamiliar with standard mathematical techniques. Let there be no mistake: the appropriate mathematical techniques are very well known and widely used outside of these particular problems; the issues in questions come down to computations and there is no ambiguity about the fact that the incorrect claims are in fact mathematically incorrect.

[EDIT: having just noticed another post in this thread, perhaps I should caution against "guessing games" about which specific arXiv eprints I might have in mind. That would be profitless and in self-defense I will not respond to queries of that sort.]

You can also search PF for a recent thread (last two weeks or so) in which I posted some computations of frequency shift phenomena for various pairs of observers, including some accelerating receivers, in the Schwarzschild vacuum solution.

"general" in the sense of arbitrary orientation (i.e. arbitrary angle [tex]\theta[/tex] and arbitrary relative motion between source and receiver). Sorry about confusing you. I will look for your posts on "some computations of frequency shift phenomena for various pairs of observers, including some accelerating receivers, in the Schwarzschild vacuum solution.". A pointer from you would be welcome.

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I was interested in the problem because in the case of the Doppler Effect with accelerating observers we should take into account the nonlocality in the period measurement by accelerating observers. The results of my investigations are presented in "Radar echo, Doppler Effect and Radar detection in the uniformly accelerated reference frame"

Authors: Rothenstein, Bernhard; Popescu, Stefan

The uniformly accelerated reference frame described by Hamilton, Desloge and Philpott involves the observers who perform the hyperbolic motion with constant proper acceleration gi. They start to move from different distances measured from the origin O of the inertial reference frame K(XOY), along its OX axis with zero initial velocity. Equipped with clocks and light sources they are engaged with each other in Radar echo, Doppler Effect and Radar detection experiments. They are also engaged in the same experiments with an inertial observer at rest in K(XOY) and located at its origin O. We derive formulas that account for the experiments mentioned above. We study also the landing conditions of the accelerating observers on a uniformly moving platform.

Comment: 15 pages, 8 figures, includes new results on radar detected times and distances

If you consider arXiv among "such" then delete my thread. If not please have a critical look at my paper you can download from arXiv.

sine ira et studio

The paper you quote does not appear to be correct since it is an attempt to plug in variable [tex]v[/tex] into the relativistic Doppler formulas. The oobjections I have to the paper are that:

1. It does not derive the relativistic Doppler effect from base principles.

2.Instead, it uses the formulas derived for non-accelerated motion and it plugs in the values computed for hyperbolic motion

Looking thru your papers, I think this one is better

The objections are smaller:

1. It is not clear what are the improvements over reference [3]

2. It deals only with source-receiver angle of motion [tex]\theta=0[/tex], i.e. is not general enough.

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While no particular papers come to mind, the required technique seems fairly straightforwards to me. Since Chris has already given you some suggestions for papers, I'll just add in my $.02 on the techniques I'd suggest.

You have a source, moving through some specific path through space-time, that emits signals as a constant time interval [tex]d\tau_1[/itex].

The signals themselves follow null geodesis

You have a destination, moving through some specific path through space-time, that receives the signals. We are interested in the interval between pulses (null geodesics) as a function of time. Note that the notion of simultaneity will, in general, depend on the coordinate system one uses, of course.

If you happen to be in flat space-time, regardless of whether or not your observers are acclerating, the answer is easy enough to compute using inertial coordinates, for the null geodesics will be straight lines in any inertial coordinate system.

This approach will basically demonstrate, when carried to its conclusion, that there is no doppler shift due to acceleration as long as one works in an inertial frame. There is only SR doppler shift in inertial frames in flat space-time, any other doppler shift comes from the choice of coordinates.

I.e. pick a frame momentarily comoving with the transmitter. Draw the null geodesics in this frame. The time interval between reception will depend entirely on the velocity of the receiver relative to the inertial frame we defined, the "angle" at which the receiver's wordline "cuts across" the congruence of null geodesics.

By computing the null geodesics for other geometries (say the FRW expanding universe - I've seen this partricular case done in textbooks for cosmological redshift and could probably dig up a reference in MTW), one can compute the doppler shift for those non-flat geometries as well. Rescaling the metric to "conformal time" can help this process a lot, greatly simplifying the geodesic equation.

[tex]

\frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{~\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0

[/tex]

It's a lot easier to pick suitable coordinates to simplify the above equation than it is to solve it in the general case. Most particularly one can often simpolify this equation greatly by by rescaling time so that T = f(t) - i.e. "conformal time". This will often make light cones "straight lines", at least if one starts out with spatial coordinates that are the same in all directions (i.e. isotorpic coordinates for the Schwarzschild metric).

The next most useful trick in dealing with geodesics is to take advantage of conserved quantities - every Killing vector in the source geometry generates a conserved quantity.

The least useful technique is to attempt to solve the differential equations above by "brute force".

You have a source, moving through some specific path through space-time, that emits signals as a constant time interval [tex]d\tau_1[/itex].

The signals themselves follow null geodesis

You have a destination, moving through some specific path through space-time, that receives the signals. We are interested in the interval between pulses (null geodesics) as a function of time. Note that the notion of simultaneity will, in general, depend on the coordinate system one uses, of course.

If you happen to be in flat space-time, regardless of whether or not your observers are acclerating, the answer is easy enough to compute using inertial coordinates, for the null geodesics will be straight lines in any inertial coordinate system.

This approach will basically demonstrate, when carried to its conclusion, that there is no doppler shift due to acceleration as long as one works in an inertial frame. There is only SR doppler shift in inertial frames in flat space-time, any other doppler shift comes from the choice of coordinates.

I.e. pick a frame momentarily comoving with the transmitter. Draw the null geodesics in this frame. The time interval between reception will depend entirely on the velocity of the receiver relative to the inertial frame we defined, the "angle" at which the receiver's wordline "cuts across" the congruence of null geodesics.

By computing the null geodesics for other geometries (say the FRW expanding universe - I've seen this partricular case done in textbooks for cosmological redshift and could probably dig up a reference in MTW), one can compute the doppler shift for those non-flat geometries as well. Rescaling the metric to "conformal time" can help this process a lot, greatly simplifying the geodesic equation.

[tex]

\frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{~\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0

[/tex]

It's a lot easier to pick suitable coordinates to simplify the above equation than it is to solve it in the general case. Most particularly one can often simpolify this equation greatly by by rescaling time so that T = f(t) - i.e. "conformal time". This will often make light cones "straight lines", at least if one starts out with spatial coordinates that are the same in all directions (i.e. isotorpic coordinates for the Schwarzschild metric).

The next most useful trick in dealing with geodesics is to take advantage of conserved quantities - every Killing vector in the source geometry generates a conserved quantity.

The least useful technique is to attempt to solve the differential equations above by "brute force".

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The paper you quote does not appear to be correct since it is an attempt to plug in variable [tex]v[/tex] into the relativistic Doppler formulas. The oobjections I have to the paper are that:

1. It does not derive the relativistic Doppler effect from base principles.

2.Instead, it uses the formulas derived for non-accelerated motion and it plugs in the values computed for hyperbolic motion

The Doppler Effect is simple as long as the relative motion is uniform and takes place along the line which joins source and observer (longitudinal Doppler Effec? If the Doppler experiment is longitudinal but accelerated we should take into account the fact that during the reception of two successive wavecrets the velocity changes. Taking into account that fact (nonlocality?) we can derive formulas that account for that fact. In the case of the nonlongitudinal Doppler Effect it is the angle under which the observer receives two sucessive wavecrests changes. Taking into account that fact we can derive adequate formulas. That is what I have tried to do with my humble tools.

Looking thru your papers, I think this one is better

The objections are smaller:

1. It is not clear what are the improvements over reference [3].

In my papers, in order to simplify the approach I consider the problem from the rest frame of the source. I consider not only the first and the second emitted wavecrests but all of them, deriving a Doppler formula for all the succesively emitted wavecrests D((n-1,n)) n the order number of the emitted wavecrest.

2. It deals only with source-receiver angle of motion [tex]\theta=0[/tex], i.e. is not general enough.My papers consider the nonlongitudinal Doppler Effect as well

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You have a source, moving through some specific path through space-time, that emits signals as a constant time interval [tex]d\tau_1[/itex].

That is the problem. What you propose is known as

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I don't see it as any more problematical than in any other case where one takes a limit.You have a source, moving through some specific path through space-time, that emits signals as a constant time interval [tex]d\tau_1[/itex].

That is the problem. What you propose is known asvery small period assumptionMost papers I know treat the problem making that assumption which obscures some peculiarities of the Doppler Effect.

One way of looking at doppler shift geometrically is that there is some function that maps [itex]\tau_1[/itex], the proper time of the emitter of some signal when that signal is emitted, into [itex]\tau_2[/itex], the proper time of the receiver when that same signal is received.

The slope of that curve can be regarded as the doppler shift, ie

[tex]\frac{d\,\tau_2}{d\,\tau_1}[/tex] (or perhaps the inverse).

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You know, even theI allways say "read the paper and not the place where it is published. Not all of us have access to top journals."

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Thank you, I'll give it a try.While no particular papers come to mind, the required technique seems fairly straightforwards to me. Since Chris has already given you some suggestions for papers, I'll just add in my $.02 on the techniques I'd suggest.

You have a source, moving through some specific path through space-time, that emits signals as a constant time interval [tex]d\tau_1[/itex].

The signals themselves follow null geodesis

You have a destination, moving through some specific path through space-time, that receives the signals. We are interested in the interval between pulses (null geodesics) as a function of time. Note that the notion of simultaneity will, in general, depend on the coordinate system one uses, of course.

If you happen to be in flat space-time, regardless of whether or not your observers are acclerating, the answer is easy enough to compute using inertial coordinates, for the null geodesics will be straight lines in any inertial coordinate system.

This approach will basically demonstrate, when carried to its conclusion, that there is no doppler shift due to acceleration as long as one works in an inertial frame. There is only SR doppler shift in inertial frames in flat space-time, any other doppler shift comes from the choice of coordinates.

I.e. pick a frame momentarily comoving with the transmitter. Draw the null geodesics in this frame. The time interval between reception will depend entirely on the velocity of the receiver relative to the inertial frame we defined, the "angle" at which the receiver's wordline "cuts across" the congruence of null geodesics.

By computing the null geodesics for other geometries (say the FRW expanding universe - I've seen this partricular case done in textbooks for cosmological redshift and could probably dig up a reference in MTW), one can compute the doppler shift for those non-flat geometries as well. Rescaling the metric to "conformal time" can help this process a lot, greatly simplifying the geodesic equation.

[tex]

\frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{~\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0

[/tex]

It's a lot easier to pick suitable coordinates to simplify the above equation than it is to solve it in the general case. Most particularly one can often simpolify this equation greatly by by rescaling time so that T = f(t) - i.e. "conformal time". This will often make light cones "straight lines", at least if one starts out with spatial coordinates that are the same in all directions (i.e. isotorpic coordinates for the Schwarzschild metric).

The next most useful trick in dealing with geodesics is to take advantage of conserved quantities - every Killing vector in the source geometry generates a conserved quantity.

The least useful technique is to attempt to solve the differential equations above by "brute force".

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I think you missed my point: you cannot substitute v=variable in the formula for the realtivistic Doppler effect that was derived for inertial motion and expect to get a correct result.Thank you for the attention you payd to my papers. I allways say "read the paper and not the place where it is published. Not all of us have access to top journals." My comments are inserted in your message. I would be happy to continue the discussion with you and let me know please the results you obtain in the field. Arxiv is a very democratic place where physicists could present the results they obtain.

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Chris Hillman

Science Advisor

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Just wanted to point out that this is most often employed in connection with "effective potentials", but is most successful in spacetimes with several independent Killing vector fields.The next most useful trick in dealing with geodesics is to take advantage of conserved quantities - every Killing vector in the source geometry generates a conserved quantity.

However, techniques from computational commutative algebra involving Groebner bases, which can be roughly summarized as analogous to triangularization of a system of linear equations via Gaussian reduction, often are applicable, even in the case of systems of nonlinear ODEs (as is generally the case here, for reasons which should be obvious from the form of the geodesic equation as written out by pervect above). See Stephani,The least useful technique is to attempt to solve the differential equations above by "brute force".

Another technique which can be useful is group analysis, in which one starts with a system including some undetermined function, and looks for particular choices which ensure that the system acquires extra symmetries, which often leads to exact solutions of special cases. Stephani has some remarks about this. Further introducutions to symmetries and differential equations include:

1. Peter J. Olver,

2. Bluman and Kumei,

3. Brian J. Cantwell,

4. Nail H. Ibragimov,

I'd recommend reading all of these.

Incidently, Killing equations are very simple to attack by CAS. In Maple, the key command is "casesplit", which carries out the triangularization mentioned above (as in linear algebra, this can be tedious for humans, but computers enjoy such mindless tasks).

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I think you missed my point: you cannot substitute v=variable in the formula for the realtivistic Doppler effect that was derived for inertial motion and expect to get a correct result.

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However, techniques from computational commutative algebra involving Groebner bases, which can be roughly summarized as analogous to triangularization of a system of linear equations via Gaussian reduction, often are applicable

....

Another technique which can be useful is group analysis, in which one starts with a system including some undetermined function, and looks for particular choices which ensure that the system acquires extra symmetries

Interesting, thanks for the info and references. I've done enough with the problem to appreciate how messy solving the geodesic equations can be, so I'm glad to hear about new techniques and methods.

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You are still missing my point, you can't stick v= variable into the original relativistic Doppler effect. The relativistic Doppler effect is derived for v=constant , using the Lorentz invariance of [tex]k,\phi[/tex] . If you are working with accelerated motion, you cannot use the original formulas anymore, you need to derive things from base principles, as you did in your second paper. Nothing to do with "high frequency".I did not. My humble oppinion is that the formulas which account for the Doppler Effect, in all its scenarios, we find in the literature of the subject, hold only in the case of the very high frequency assumption assuming that the observer receives two successive wave crests from the same point in space. In the particular case of the longitudinal Doppler Effect with constant relative velocity it is not necessary to make that assumption, because it is independent from the magnitude of the period. As you can see in my papers devoted to the subject, I derive formulas which are free of the very high frequency assumption accounting for all successive pair of wavecrests, in accordance with your point of view. I would be pleased to continue our discussion on that fascinating subject.

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You are still missing my point, you can't stick v= variable into the original relativistic Doppler effect. The relativistic Doppler effect is derived for v=constant , using the Lorentz invariance of [tex]k,\phi[/tex] . If you are working with accelerated motion, you cannot use the original formulas anymore, you need to derive things from base principles, as you did in your second paper. Nothing to do with "high frequency".

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I don't see it as any more problematical than in any other case where one takes a limit.

One way of looking at doppler shift geometrically is that there is some function that maps [itex]\tau_1[/itex], the proper time of the emitter of some signal when that signal is emitted, into [itex]\tau_2[/itex], the proper time of the receiver when that same signal is received.

The slope of that curve can be regarded as the doppler shift, ie

[tex]\frac{d\,\tau_2}{d\,\tau_1}[/tex] (or perhaps the inverse).[/QIUOTE]

Taking the limit you propose we arrive at the conclusion that the involved frequencies are infinite and I would not look at such a Doppler signal

sine ira et studio

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I was just trying to help you see the error in the paper, I guess that you want to cover your ears ans shout : "la-la-la". Your choice....You are still missing my point, you can't stick v= variable into the original relativistic Doppler effect. The relativistic Doppler effect is derived for v=constant , using the Lorentz invariance of [tex]k,\phi[/tex] . If you are working with accelerated motion, you cannot use the original formulas anymore, you need to derive things from base principles, as you did in your second paper. Nothing to do with "high frequency".

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thI don't see it as any more problematical than in any other case where one takes a limit.

One way of looking at doppler shift geometrically is that there is some function that maps [itex]\tau_1[/itex], the proper time of the emitter of some signal when that signal is emitted, into [itex]\tau_2[/itex], the proper time of the receiver when that same signal is received.

The slope of that curve can be regarded as the doppler shift, ie

[tex]\frac{d\,\tau_2}{d\,\tau_1}[/tex] (or perhaps the inverse).[/QIUOTE]

Taking the limit you propose we arrive at the conclusion that the involved frequencies are infinite and I would not look at such a Doppler signal

sine ira et studio

I'm not understanding your objection. The doppler shift has to be measured over a very small time interval if it is changing with time to get a "sharp" value. In the limit, this requires very high frequencies.

It sounds like you are perhaps worried about some of the engineering details of FM modulation and demodulation. This can get tricky, because FM is nonlinear, but it's not really very relevant.

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th

I'm not understanding your objection. The doppler shift has to be measured over a very small time interval if it is changing with time to get a "sharp" value. In the limit, this requires very high frequencies.

It sounds like you are perhaps worried about some of the engineering details of FM modulation and demodulation. This can get tricky, because FM is nonlinear, but it's not really very relevant.Thank you again. I am on the forum for learning and to know the oppinion of people busy in the same field as I am with my humble forces. The frequency of a wave (acoustic or electromagnetic) can be very small or very high. According to a definition of the Doppler Effect I know, we have to compare the proper period at which the source emits successive signals measured in the rest frame of the source and the proper period at which the observer receives them measured in its rest frame. In the case of accelerated motion or in the case of oblique incidence between the reception of two successive wave crests velocity changes and the incidence angle changes as well. Taking the limit is a mathematical operation,whereas measuring the period is a physical operation and that is what I have in my mind and I think there is a big differfence between the two approaches. Physicists speak about non-locality in the period measurement by accelerating observers. Non-locality is favoured by high periods and high propagation velocities and that makes that the successive periods measured by an accelerating observer change. What I mentioned in my thread was a joke considering that infinitesimal periods we consider high frequencies i.e. high energies and so I would avoid to detect them.As allways

sine ira et studio

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E = A cos(w_e*t + theta)

for a plane wave, where w_e is the emitted frequency

If an accerating receiver is measuring this wave, he will recieve a wave that looks like this: (eq 2)

E = A cos(w_r(t)*t + theta)

The process of recovering w_r(t) from E(t) is the process of FM demodulation. If w_r is very high, we can get a very good approximation of the FM demodulation by taking the inverse of the period as you suggest. But I would not say that the inverse of the period is the defintion of the "instantaneous frequency" (as it is called by electrical engineers).

You'll probably find much more about this in the electrical engineering literature, for instance if you FM a sine wave with another sine wave the resulting spectrum is well known and can be described by Bessell functions.

But the important thing is that an emitted wave of the form of eq(1) implies a recieves wave of the form of eq(2). This could probably be derived formally and more completely by transforming the electromagnetic field tensor F_ab for an accelerated receiver, and assuming a plane wave at the transmitter.

Getting too involved in the details of the practical aspects of FM demodulation is a not-very-productive path that obscures the physics. The simple route is similar to geometric optics, where one also assumes that radiation is being emitted at a very high frequency, so that the details of the wave-like nature of the radiation (the carrier frequency) are not important.

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