The doppler radar trap paradox on the path to gravity.

In summary, the police man fires a low wattage continuous wave radar gun at a craft passing him and measures the crafts speed with a beat frequency that he measures with the incoming and outgoing waves. The craft is made of a superconductor and acts as a perfect mirror, so the police man can measure the crafts speed accurately. The police man's original calculation of the crafts speed was incorrect because the photons have energy and the craft's distance from the reflector was 1.7329 meters.
  • #106
Max™ said:
Actually the 1905 paper wouldn't have had you calculate the clock as if it were accelerating, simply you would have moved the clock along the path from point A to point B at a uniform velocity, and upon arriving next to the rest clock it would have incurred a certain amount of time dilation. Which is trivially unphysical, objects don't just instantly launch off at set velocities without periods of acceleration.

Aye. I understood this all along -- I am just a poor at using language unambiguously.
I spend too much time re-reading and editing as it is...

It is just that when keeping track of time on a clock, one needs to know if it was accelerated historically.
Eg: that sudden jerk which goes from 0 to whatever in no time...

Once the jerk took place (dirac) -- the relative non-accelerating velocity will continue to make the clock tick slower relative to a clock which has never been accelerated at any time during the experiment.


The theory wasn't well suited to describing the full situation, so he strove to restrict it to situations where it was valid until he could produce a more suitable theory for all general situations.

You are right though, they aren't paradoxes upon investigation, merely upon a cursory reading do they appear to be such.

I feel smarter when I help others feel smarter, simply demonstrating knowledge is trivial, imparting knowledge to another shows a true understanding of the subject, teaching is it's own reward... luckily since it doesn't pay well.

Aye. Thanks Max.
 
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  • #107
andrewr said:
That (~2) would not even be *exactly* true in an inertial reference frame. ... I don't think it will work out to 4/3rds by gut reaction *Perhaps you mean just shy of 4/3? *
Correct, that is what I meant by the "~". The "~2" means "approximately 2" and the "~4/3" means "approximately 4/3". Sorry if the notation was unclear.

andrewr said:
I was taking the 2x momentum as a given from physics texts which specify that -- so, in fact, those texts are wrong .. ?
I wouldn't say "wrong", it is true, but only for the special case of a non-accelerating mirror at rest. If they didn't say that then they were being sloppy. My textbook (Serway) for example is sloppy, but uses it correctly in the context where there is a restoring force on the mirror and so the mirror's acceleration is 0.

andrewr said:
I don't follow you -- It seems you are speaking about the energy in the police frame -- but the change energy is 66.666...% in that frame. Since momentum is related to energy by p = E/c; I would expect momentum to change by 2/3rds as well.

eg: E**2 = p**2 * c**2 + m0**2 * c**4; but photons have 0 rest mass, therefore:
E**2 = p**2 * c**2
or
E = p * c
p = E/c
Don't forget that energy is a scalar quantity and momentum is a vector quantity. So although they are related to each other they are not the same. Also, notice that E²=p²c² has both a positive and a negative root for p (in one dimensional collisions). So the magnitude of the momentum changed by ~2/3 (i.e. from 1 to ~±1/3), but the direction also changed (i.e. from 1 to ~-1/3) resulting in a change in momentum with magnitude >1 (i.e. ~-4/3). This is why it is important to know that both energy (a scalar) and momentum (a vector) are conserved and why the problem cannot be worked entirely using just one or the other.
 
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  • #108
DaleSpam said:
Correct, that is what I meant by the "~". The "~2" means "approximately 2" and the "~4/3" means "approximately 4/3". Sorry if the notation was unclear.

It was clear -- I was tired -- eg: insomnia. I just missed it.

I wouldn't say "wrong", it is true, but only for the special case of a non-accelerating mirror at rest. If they didn't say that then they were being sloppy. My textbook (Serway) for example is sloppy, but uses it correctly in the context where there is a restoring force on the mirror and so the mirror's acceleration is 0.

My text is the same -- sloppy.

Don't forget that energy is a scalar quantity and momentum is a vector quantity. So although they are related to each other they are not the same.

Sure, although true conservation of energy guarantees conservation of momentum -- so that if one were to assign a direction to energy, all the information can be handled at that level.
Taking the derivative of energy (classic) -- where the energy is balanced, will give momentum.
That was the original incentive in Physics for computing a scalar field from vector ones -- eg: the E field can be integrated to arrive at a potential energy field (a scalar).

Also, notice that E²=p²c² has both a positive and a negative root for p (in one dimensional collisions). So the magnitude of the momentum changed by ~2/3 (i.e. from 1 to ~±1/3), but the direction also changed (i.e. from 1 to ~-1/3) resulting in a change in momentum with magnitude >1 (i.e. ~-4/3). This is why it is important to know that both energy (a scalar) and momentum (a vector) are conserved and why the problem cannot be worked entirely using just one or the other.

I'll have to think about that. But, my math teachers always told us to test ideas at the extremes -- because these are the most likely places for failures to occur, and mistakes to be made visible.
The issue, with respect to energy there -- is that if the mass of the mirror is allowed to go infinite -- the reflected photon has a momentum change -- but the mirror does not.
Only the sign of the momentum changes, though, and not the value.

Since no acceleration occurs on the mirror, even if moving, -- at infinite mass -- then one has the issue that the mathematics OUGHT to reflect there being NO energy transfer. I would expect, mathematically, that the energy transfer drops as the mass increases because of the limiting case.

A second thing that I notice, is that the wave solution to the Maxwell version of the problem, seems to assume that the mirror is immersed in a magnetic field -- which, from careful inspection, is not the case.
The mirror moves OUT of the field as the photon reflects. We can balance energy and momentum by assigning (mathematically) various scenarios -- but that will always leave open the question of experimental verification. I will still need to work the problem various ways -- and I have equipment coming to do some rudimentary tests to satisfy my curiosity.
 
  • #109
andrewr said:
Sure, although true conservation of energy guarantees conservation of momentum -- so that if one were to assign a direction to energy, all the information can be handled at that level.
I agree and will just point out that that is exactly what the four-momentum does.

andrewr said:
The issue, with respect to energy there -- is that if the mass of the mirror is allowed to go infinite -- the reflected photon has a momentum change -- but the mirror does not.
No, the mirror does have a momentum change which is opposite to the momentum change of the photon. It just does not have a change in velocity, but that is expected since a finite change in momentum (force) cannot change the velocity of (accelerate) an infinite mass. All of the above works for an infinite mass mirror (or a mirror that does not accelerate due to an external force), simply replace all of the "approximates" with "exactlys".
 
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