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Light SPhere Paradox

  1. Jul 5, 2010 #1
    Hi
    The setup for this conundrum is like the light sphere illustration.

    Two frames F and F' ------- F' moving ---->+x at v=.8c

    At the point of colocation of the origens: x=0 and x'=0 t=0,, t'=0 a light burst is emitted .

    Looking at two points at opposite sides of the expanding sphere

    P+ and P-

    P+ at x=10. t=10 , x'=3.334, t'=3.334

    P- at x=-10,t=10, x'=-30,t'=30

    As these locations in both frames are large photosentive panels. Lensless photodiode
    arrays directed at the origens and orthogonal to the motion so it can be assumed they are the same actual sizes. These measure the number of photons intersecting them over the whole area and output a luminosity value ( L)
    The question is: What would the respective relative values be?

    Assumptions :

    Any photon in the sphere at a point in its worldline is actually simultaneous with all other photons at that time and all are absolutely equidistant from the origen.

    Because of this the sphere can be viewed for convenience as having a single worldline.

    The spheres luminosity will diminish along the worldline. Any local segment will have an absolute luminosity at any given point.

    A point on the worldline may intersect relative frames at different points on their worldlines but it would be assumed that colocated observers at those points would be measuring the same point of the sphere.

    1) Based on this,, it would seem that at any colocated point, the measurements would have to be equal.

    Both frames at P+ would measure equal values:
    P+ L+=L'+
    P- L-=L'-

    2) Based on the 1st Postulate and the fact that luminosity must diminish as a function of distance D ,,,dec reasing exponentially with increasing distance

    it would mean that L+ with D=10 must equal L- with D=10

    L'+ [D=3.334] must be greater than L'- [ D=30]
    ________________________________________________________________________

    A) If we assume ----#1)

    that the measurments must agree between both frames colocated at a point then the L values cannot be a function of distance in both frames because the distances are equal in one frame [F] but greatly unequal in the other [F']


    B) If we assume -----# 2)

    that the values must be a function of D then the frames cannot agree on colocated measurements.
    The measurements must be equal in F but cannot be equal in F'


    If A) then there is a violation of the 1st Postulate.

    If B) then it has to be explained how two colocated measurements of the same point on the light cone could possibly be different???

    Or alternately how two observers measuring two different points on the cone [with different luminosities ] could possibly be colocated ???.

    As far as I can see:

    ---------- [Both A and B must be true]---- AND[---- Both A and B cannot be true.]------------

    So any solutions or insights would be great.
    Thanks
     
  2. jcsd
  3. Jul 5, 2010 #2

    Dale

    Staff: Mentor

    Did you account for the Doppler shift?
     
  4. Jul 5, 2010 #3
    I thought about it but concluded it was not relevant because

    1) it would be reciprocal in F'
    the decrease in energy (L+) at x'=3.334
    equal to the increase (L-) at x'=-30

    Whereas the difference in magnitude of L due to distance would be exponential I would imagine by a factor of 4 pi(r2) ???

    2) There are only two events. So just on principle whichever frame you choose to analyse it from the other frame will be the one dopplered. So that is reciprocal and self cancelling in this case as far as resolving the question.

    I* also played with other ideas like a photon not being localized but existing from the point of emission through the whole path until absorbtion but this didn't seem to lead anywhere.
    Thanks
     
    Last edited: Jul 5, 2010
  5. Jul 5, 2010 #4

    Ich

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    No. If it's extended, it can not be viewed as having a single worldline. Never.
    Acoordingly: no.
     
    Last edited: Jul 5, 2010
  6. Jul 5, 2010 #5

    Dale

    Staff: Mentor

    Then luminosity must be frame variant, not absolute. This really should not be too surprising since energy is frame variant and luminosity is related to energy.
     
  7. Jul 5, 2010 #6
    Energy is relevant within the context of the doppler effect. But unless you disagree with me about it not being enough to explain the difference in luminosity
    then it is purely a question of how many photons reach a given area.

    SO are you are saying that the number of photons in a given section of an expanding spherical surface at a single point must be frame variant?

    or do you mean the electrical energy generated by the photons being frame variant?

    Wouldn'y that be inconsistent with 1st Postulate??
     
  8. Jul 5, 2010 #7
    I did say for convenience. But you are right . Spatially it is a collection of worldlines but temporally at a given moment, all parts share a common time and distance as measured simply as distance from origen.
    You do understand I am not talking about it in coordinates of an inertial frame but viewing it as an absolute entity in reference to its point of origen.
     
  9. Jul 5, 2010 #8

    Ich

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    Sorry, I had a longer post but lost it. Here's the relevant thing:
    Exactly. Concerning angular size and total photon number per area, a moving source looks like a stationary source in the distance it had at the time of emission in the observer's frame. This effect is called aberration.
    Doppler shift additionally increases/decreases photon rate and energy per photon.
     
  10. Jul 5, 2010 #9

    Dale

    Staff: Mentor

    I neither agree nor disagree. I would have to see the math, but I am certainly not convinced either way at this point and an argument that ignores something as ubiquitous as the Doppler effect seems pretty weak to me.

    The number of photons is often indeterminant even in a single frame, and it is also frame variant. See the Unruh effect for one example, I don't know if it is a more general principle.

    Why would any of this be inconsistent with the 1st postulate?
     
  11. Jul 5, 2010 #10
    Ostensibly, the panels have a non-zero area. In order to solve your "paradox" (I don't think that there is any), you will need to calculate the flux falling on each panel.

    This question has been answered long ago by Einstein in his 1905 paper.

    The ratio of amplitudes as viewed from frame F and F' is:

    [tex]\frac{A'^2}{A^2}=\frac{1-v/c}{1+v/c}[/tex] (end of paragraph 7)

    The ratio of energies is:


    [tex]\frac{E'}{E}=\sqrt{\frac{1-v/c}{1+v/c}}[/tex] (paragraph 8)

    The above formulas look a lot like the relativistic Doppler effect because they are derived based on the Doppler effect:

    [tex]\frac{\nu'}{\nu}=\sqrt{\frac{1-v/c}{1+v/c}}[/tex]

    So, DaleSpam was right all along. Now, I did not read your post to the end , what is the paradox you are trying to figure out?
     
  12. Jul 6, 2010 #11
    Hi Actually it was through studying aberration and pinhole cameras that I got onto this in the first place.
    An emission event such as a large flat light panel emitting a single flash and subsequently observed in two frames should have different angular sizes depending on their motion relative to the flash. The flash of course occurs at a specific location in each frame and is stationary in both.

    Because of this they must both agree on the angular size of the panel,, yet both see the other in motion relative to the image as it propagates to their receptors.
    A pinhole camera explains why this this is possible.

    From the frame where the other is perceived as moving toward the source it appears as if the received image must appear larger [greater subtended angle]
    But in the moving camera the focal plane is moved closer to the aperture through length contraction and the motion of plane between entering the camera and hitting the plane.

    This causes the size of the image on the focal plane to shrink and appear farther away.
    Like wise if the camera is moving away from the source.

    Doppler

    Agreed ----D shift causes a decrease in photon rate and energy.

    But in the current scenario photon rate is irrelevant. It is a singular, very short interval reception and it is just the total number of photons that arrive that is important not how many per unit time

    On the energy increase/decrease through Doppler;
    1) Remember the source is stationary in both frames.
    It should be considered a spontaneous emmission from the quantum field,,, of unknown frequency.
    I used one frame as a rest frame but the two events I derived must occur in both frames.. So Doppler can be applied from either frame in which case it will reciprocal and purely relative unlike the two events in my problem where boith frames agree.

    SOrry you lost the rest. Thanks
     
  13. Jul 6, 2010 #12

    Ich

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    That's impossible.
     
  14. Jul 6, 2010 #13
    Fair enough. Your riight , simply because I convinced myself it was not relevant is not sufficient. So I am in the process.

    Assuming we apply the Doppler from the perspective of F' .
    At P2 (x'=-30 t'=30 ), (x=-10.,t=10)
    F' would calculate that the measurement (L) in F would be decreased.

    At P1 (x'=3.334 ,t'=3.334) , (x=10 ,t=10)
    F' would calculate that the measurement (L) in F would be increased.

    If you assume that has any meaning then[/I:
    ]P1 (x=10) L cannot equal P2(x=-10) L

    Different measurements at the same distance in a frame would seem to be inconsistent with the 1st Postulate.


    Maybe you could explain a little about the indeterminancy?
    Do you mean on a quantum level? Or just as a statistical margin of error?
    I am thinking that you could make receptor array as large as you like.
    I just read about people using the Canon latest CMOC cameras to determine photon counts , weighted values for frequency [which the receptors also determine], with claimed reasonable margins of error.
    SO a (100 sq. m) matrix should make it fairly workable.

    I checked out Unruh not applicable , black body radiation through acceleration.

    Thanks
     
  15. Jul 6, 2010 #14
    austin0-----Remember the source is stationary in both frames

    Only if you think source means a mechanism. I was refering to the source of the sphere.

    Unattached to any frame. The location of the initial observation of the phenomenon.


    I purposely didn't locate a mechanism in the description ;the beginning of the light burst was simply a colocated event with the coincidence of the origens.

    Would it make any real difference if it was the result of a mechanism located on a different frame passing orthogonally ??
     
  16. Jul 6, 2010 #15

    Ich

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    Yes.
    I'm not sure exactly how your setup is supposed to look like. I took it to be a "homogeneous" light sphere, and a light sphere looks homogeneous only in the source frame.
     
  17. Jul 6, 2010 #16

    Thanks for the relevant math. YOu gentlemen have convinced me I am going to have to
    work it in.

    Just a first take;
    at v=.8c it appears to be a reciprocal factor of 3 and .33

    the luminance flux falloff at the distances in my setup [ x'=-30 and x' = 3.334]

    are: 302 =900 ---------- 3.3342 = [tex]\approx[/tex]10

    Hugely different ratios. In my problem its the ratios that are important as there is no quantitative comparison merely greater ,equal or lesser.
    But I will keep working with it.
    Also there is the CMOS info I posted above , which seems like it might remove the doppler factor because it makes possible weighted photon counts Devidng the enrgy by wavelength and deriving photon counts. It is purely phonton count that is the basis of apparent paradox.
    I am sorry I cant think of a few words to explain it maybe some will come to me. But if you haven't read it , it may be a little early to conclude DaleSpam was right?????
    Thanks
     
  18. Jul 6, 2010 #17

    Dale

    Staff: Mentor

    Yes, I mean on the quantum level. The best example I know of is a coherent state:
    http://en.wikipedia.org/wiki/Coherent_state

    In a coherent state the number of photons has an uncertainty relationship with the phase of the state. And detecting a photon leaves the state unchanged. Essentially, any state which is not an eigenstate of the Hamiltonian will not have a definite number of photons.

    Yes, I agree that the Unruh effect is not applicable, it was just intended as an example of the fact that the number of photons is frame variant. I don't know if there is a similar effect which is applicable for a pair of inertial frames.

    :biggrin: It is never too early to conclude that DaleSpam was right!!!
     
  19. Jul 6, 2010 #18

    Dale

    Staff: Mentor

    Could you clarify. Are you interested in how the luminance or luminosity transforms? They are different quantities.
     
  20. Jul 6, 2010 #19
    Once again you came up with a very interestin refernce. Thanks.
    It appears to me to be more related to coherence in an emitting source as resonant phase buidups or reinforcements. I dont quite see how it would apply to a singular ideally short interval burst ? Or measuring the same through complex detectors and filters. Once again it doesnt seem to apply??
    The initial conditions could be changed to install such detectors instead of CMOS but I also dont really see how that would significantly alter the picture.
    Also it seemed to indicate that the higher the emitting energy the less statistical deviation was expected but I could have read that wrong.
    Do you think that this level of quantum flux would really be significant given large detectors , the problem does not call for any kind of quantitative accuracy but rather large disproportionate ratios??

    Its never too early to conclude DaleSpan has some sense of humor!!!
     
  21. Jul 6, 2010 #20

    Dale

    Staff: Mentor

    Well, if you are really interested in number of photons then you will need to write out the wavefunction for your ideally short interval burst. Then you will need to find the eigenstates of the Hamiltonian. If your wavefunction is not an eigenstate (which I suspect it will not be) then the number of photons is not definite even in a single frame.

    My thought is that this would be some serious overkill for what you really want to know which is how the luminance (or luminosity, please clarify) transforms. I would take a more starthaus-like approach and look at continuous fields rather than discrete photons. The transformation laws for the fields are well known.
     
  22. Jul 6, 2010 #21
    This can't be since the two frames are moving wrt each other.


    What is this supposed to mean? What do you mean by "both frames agree"? Agree on what?
     
  23. Jul 6, 2010 #22
    In actuality I misused the term luminosity. I think luiminance or luminance flux is more correct.
    In any case I believe it falls off as a function D2 or with a sphere -(r2).
    AM I wrong in this?? In any case the whole subject of light measurement as normally applied is both complex and somewhat ambiguous about terms from what I have read.
    But really; is it relevant?
    In this case there is no need to relate it to any factor of the source intensity or normalize it in any was. WE know we are concerned simply with photon count relative to receptor area. Purely based on spherical surface area and expansion it would seem to follow the above function. No?

    As far as transforms; are you talking about the energy function of relative velocity that espen posted?

    Once again you can apply that but it must be applied reciprocally to both frames and:
    a) It would have no real meaning because it is just another relative perspective
    whereas the events are frame frame independant [between these two frames only, obviously]
    b) If it was possible to resolve the problem with unilateral application this would in effect mean an actually preferrred frame. Which is really the basis of the paradox.

    In all respect I suspect that you all ; read the post, assumed it was another twins or barn and ladder with a simple resolution and haven't really thought to much about understanding the basis of it.

    I will tell you this. Last year I posted a version with the pinhole scenario. JesseM replied , mentioning having noticed the same enigma in the light spheres but didn't elaborate or propose any details, explanation or resolution.
    That wasn't neccessary as I immediately not only saw how it would apply but that the sphere was a much better context. Less directly obvious and more complex but offering a more complete exposition.
    SO it may in the end ,turn out to be explainable but I really dont think it is going to be simple or obvious. But then I can vaguely remember being wrong once before.
     
  24. Jul 6, 2010 #23
    The wave function is unknown because the wave frequency is unknown, you cant start with the source information. This is a onetime emission out of the cosmos.

    The transformation laws by default simply return relative information. This problem is not based on determining actual values for the colocated measurements. This would be equivalent to calculating clock desynchronization.

    This is a logical proposition. You assume agreement ,or difference , greater or lesser and look at the outcome of these possibilities.
    There aren't that many.

    But you seem to be suggesting that the photon detection, photometric measurement that takes place in labs all over the world is flawed because they didnt calculate the wave function of the source.
    Of course there is going to be a degree of quantum uncertainty,
    A small statistical error in instrument imperfection. But that doesn't negate there validity for their purpose. And much of it is directly based on measureing number of photons.

    In this situation we can assume the light from the source will be equivalent in both frames and equal uncertainty and instrument error.
     
  25. Jul 6, 2010 #24
    Only if you think source means a mechanism. I was refering to the source of the sphere.

    Unattached to any frame. The location of the initial observation of the phenomenon.

    COnsidered by both frames to be a staionary location in their frame [the origen]


    I purposely didn't locate a mechanism in the description ;the beginning of the light burst was simply a colocated event with the coincidence of the origens.

    Would it make any real difference if it was the result of a mechanism located on a different frame passing orthogonally ??


    Agree on the colocation and simutaneous measurement at those coordinates.
    At both P1 and P2 .
     
  26. Jul 6, 2010 #25
    If you're referring to the center of the sphere as defined as the point equidistant between simultaneous detections of light in either direction, then yes, the center of the sphere is stationary in each frame. But notice that the origin of F' is not equidistant between the sensors you specified in your post 1, so they don't represent the simultaneous "edges" of a light sphere centered on the origin of F'. Those sensors will detect either edge of an expanding light sphere (centered at origin of F') at different times and distances from its center.

    The center of a light sphere that is detected simultaneously by those sensors in F' (x'=3.334 and x'=-30) would be located at x'=-13.334 in F'.
     
    Last edited by a moderator: Jul 6, 2010
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