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Misconceptions re: Cosmological Redshift

  1. Jul 8, 2009 #1
    Apparently my recent thread on the cosmological redshift assumed more general insight into this subject than is the case. So the purpose of this thread is to help dispel some popular misconceptions about the cosmological redshift and hopefully spur a vigorous discussion. This thread is NOT about comparing the 'expanding space' and 'kinematic' paradigms, nor about comparing Newtonian gravity to GR.

    The starting point of course is that the cosmological redshift is proportional to the change in the cosmic scale factor a(t) between the time of emission and the time of reception:

    [tex] \frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}} = (1 + z) [/tex]

    Keep in mind that in addition to increasing the wavelength of photons by this factor, the cosmological redshift also increases the proper distance separation between photons by the same factor.

    Misconception #1

    The first widely disseminated misconception is that 'expanding space' -- which really means an 'expanding hypersphere' of geometry -- acts like a stretching rubber sheet -- as a force -- causing photons and photon wave crests to physically separate. This idea assumes the 'expanding hypersphere' paradigm for the expansion of the universe, applying the FRW metric.

    However, it is well established in peer-reviewed analyses in recent years that if Lambda=0, an 'expanding hypersphere' metric does not act as a force or rubber sheet. An expanding hypersphere metric is incapable of motivating a physical separation in proper distance between photons, wave crests, or any other objects. The expanding hypersphere will cause coasting particles to continue separating only if they began with a proper velocity away from each other in the initial conditions. The hypersphere's action is like a flywheel's momentum (think 1st derivative of distance) rather than like an accelerator (2nd derivative of distance). This analysis is most intuitively confirmed by the https://www.physicsforums.com/showthread.php?t=315550&highlight=misconceptions", about which much has been much written.

    Consider the scenario where the 'observer' is at the coordinate origin, and non-relativistic particles are fired from the distant 'emitter' at constant time intervals dt. Over a short period of time, the Hubble rate Ht will remain essentially constant. In a simplified scenario without gravitational effects, at the instant after each particle is fired, its proper velocity will be zero relative to the previously fired particle, dv=0. With this as the starting condition, the 'expanding hypersphere' will never have any effect on the proper distance between the particles, dD = Htdt = constant.

    Next make the scenario more realistic by adding gravitational effects. From the observer's perspective, there is a sphere of cosmic matter centered at the observer with its radius at the proper distance of any given 'incoming' particle. Birkhoff's Theorem allows us to disregard all matter outside of that sphere. Since the cosmic matter density is deemed to be homogeneous, a larger sphere will exert a relatively greater gravitational acceleration at its surface than will a smaller sphere. Considering a string of equally spaced particles moving radially toward the observer, at all times the lead particles will be subject to less gravitational acceleration than the tail particles. Therefore the velocity of the tail particles will increase relative to the lead particles over time, and the particle string will compress longitudinally. So, quite contrary to what the 'rubber sheet' metaphor would suggest, the proper distance separation between particles actually will decrease rather than increase, despite the effect of the 'expanding hypersphere' on the background geometry.

    You may say but wait, the lead particle experienced a period of gravitational acceleration before the second particle was fired, so doesn't that give it a head start? As it turns out, the answer is no, because the Hubble recession rate does not remain constant. The same sphere of cosmic matter causes the Hubble velocity (H * D) near the emitter to decelerate at essentially the same rate as the lead photon accelerates. The net effect is that at the time it is fired, the second particle has the same proper velocity relative to the observer as the first particle does. (Actually, after firing, the second particle has a small net proper velocity toward the first particle, since the gravitational sphere acting on the first particle was slightly smaller than the sphere acting on the emitter's Hubble velocity. This just further compresses the photon string.)

    You may also wonder why the sphere of cosmic matter centered on the emitter doesn't cause the traveling particles to experience gravitational acceleration away from the observer. In a sense this does occur, but at the same time, the emitter experiences even greater acceleration toward the observer than the particles do (because the matter sphere is larger). These effects offset each other, and the particles' acceleration toward the emitter has no net effect whatsoever on their acceleration or velocity toward the observer. This may seem confusing, but it's helpful to remember that, assuming perfect homogeneity, the cosmic gravity causes everything to accelerate toward everything else, and never causes anything to accelerate away from anything else.

    The entire discussion above has been about non-relativistic particles. Exactly the same effect applies to relativistic photons, with the same consequences. In addition, the gravitational acceleration causes a gravitational blueshift in the photons' wavelength.

    However, photons have one attribute that non-relativistic particles do not: photons always must travel through every local frame at exactly c. This unique attribute causes photons to experience coordinate velocity acceleration toward the observer over their worldlines, as they ascend the Hubble velocity gradient. This coordinate acceleration results in a loss of comoving peculiar momentum, and additional redshifting, as viewed in the observer's frame. You can read about that topic in my thread about the mechanics of the cosmological redshift.

    Misconception #2

    Another widely disseminated misconception is that the cosmological redshift can be calculated by integrating (multiplying together) the SR redshifts that occur over a very large number of infinitesimal local frames along the photon's worldline. Presumably, although this usually isn't rendered explicit, each local SR redshift is calculated using the change in recession velocity (relative to the observer) at each local frame-crossing.

    Anyone with a cosmic calculator and a spreadsheet can convince themselves that this math does not generate an SR redshift anywhere close to the expansion of the scale factor. In my own relatively crude spreadsheet with 70 integration intervals from z=127, the calculated SR redshift is over 400,000 times larger than the correct answer, 128. This should not be surprising, because the recession velocity is well above > c for the majority of the worldline, and the geometric mean of the recession velocity is > c. As velocity approaches c, the SR redshift becomes infinite.

    There is another reason to discard this approach. As mentioned in my other post, the time portion of the FRW metric is obviously linear, so no time dilation at all can occur between fundamental comovers regardless of distance. They all share a common cosmological proper time. Since SR time dilation is an inherent component of the SR redshift, it is impossible for the SR redshift formula to apply between two local frames if no time dilation is permitted between those frames.

    . . . . . . . . . .

    Hopefully these two misconceptions can be dropped from our cosmology dialog, but let me know if you agree or disagree!
     
    Last edited by a moderator: Apr 24, 2017
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  3. Jul 8, 2009 #2

    sylas

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    It is worth noting that "loss of momentum", or perhaps better, difference in momentum as observed locally by co-moving observers, applies for all particles, whether moving at c, or relativistically, or at much lower velocities.
     
  4. Jul 8, 2009 #3
    Good point Sylas. The loss of comoving peculiar momentum at 1/a is a key factor in the mechanism of the cosmological redshift. No misconception there! But this thread so far has discussed changes in proper distance and velocity, and I don't want people to get that mixed up with peculiar velocity, that is, velocity relative to the local Hubble flow. Two different but related things.
     
    Last edited: Jul 8, 2009
  5. Jul 9, 2009 #4

    Ich

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    Mostly agree on the first part, vehemently disagree on the second.

    Extending to relativistic velocities: the "cosmological proper distance" between particles will suffer Lorentz decontraction eventually growing to a fixed proportion [tex]\gamma(v)[/tex].
    In the limit v->c the decontraction has no upper limit, the distance growing infinitely with time. In an empty universe, this coordinate effect accounts completely for the cosmological redshift.
    Better switch viewpoints completely: The emitter is unaccelerated at the sphere's cents, and the observer accelerates toward it. That overcompensates the gravitational redshift of the photons, giving the same blueshift as in the other frame.
    That's not a misconception, it is simply true. It's easily proven (see Peacock), and I've calculated it myself for my (successful, btw) attempt to create "Minkowski-like" coordinates in cosmology.
    Now that would be stupid. The only thing that counts is of course the recession velocity relative to the previous frame. Fix your spreadsheet according to that, and you'll see how it works. You'll even get a velocity assigned to the emitter that corresponds to the observed redshift as a relativistic doppler effect. Including time dilatation, even if every single shift could be treated classically.
     
  6. Jul 9, 2009 #5
    I guess, my problem with all this started with the initial equation quoted above. It is equating the redshift explicitly to the scale factor. When doing an activity to correct luminosity distance to an observer corrected distance, BOTH (1+z) and the scale factor are applied. They are not the same factor. Scale factor is a geometric one that is present even in the static case. The redshift (1+z) correction is one due apparent recession -- expanding universe.

    Entangling the two seems to be the problem. One corrects for apparent velocity the other for distance -- not quite the same thing. And will vary with the presumed geometry -- also why two correction terms.

    The increase in separation is real and measurable.
     
    Last edited: Jul 9, 2009
  7. Jul 9, 2009 #6

    sylas

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    Sure, the increase is real and measurable. But what is your measure? There are many different ways to define distance.

    The formula you quote is CORRECT. It tells you the cosmological redshift. It's not a "definition", but a result.

    Scale factor is not an alternative to recession velocity. You can get recession velocities (using what ever co-ordinates you like) in terms of redshifts or in terms of scale factor difference. The increase in scale factor corresponds to things really separating.

    Remember, the rate of expansion is not a velocity. The recession velocity of a distant galaxy is proportional to how far away it is, and also to the rate of expansion. The growth of scale factor IS the expansion.

    You mention luminosity. That is sometimes used as a measure of distance also. So is angular size. In sufficiently small scales, all these measures are identical. On the large scales of the really and truly expanding universe, the relationships of different distance measures is more complicated, but none of them is more "real" than the others. They are just different co-ordinate systems on the same real expanding universe.

    Cheers -- sylas
     
  8. Jul 9, 2009 #7
    I see we are actually agreeing. It is the model determined 'scale factor' that I was thinking of as a 'real' separation -- with redshift for correction of luminosity distances.

    My concern was that the relation would be taken as a definition meaning a 'universally' accepted statement met by all models.
     
  9. Jul 9, 2009 #8

    sylas

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    Um... that equation IS a universally accepted statement met by all models. It's also something that doesn't depend on co-ordinate systems... except that "a" itself can be used as a time like co-ordinate, because it is a function of proper time in the FRW models. But regardless, the scale factor is a well defined quantity, independent of co-ordinate choices for quantifying separations, and it is used in all the various models.

    I think we need to be clear on terminology.

    For simplicity, let's use the usual convention that the universe is homogenous... everywhere the same at a given time, and the same in every direction, from any point. That's not quite true, of course; the universe has inhomogeneities like galaxies and voids and clusters; but on very large scales, this is a pretty good approximation. This approximation works pretty well, much like we can often consider the Earth as a sphere for many descriptions.

    Expansion

    Now... expansion is actually very simple in principle. It just means everything is moving apart from everything else, on average. Imagine a bubble of gas, with uniform density, which is expanding. Imagine it always has a uniform density at any time, except that as time passes and the bubble expands, the density drops. That is exactly what the expansion of the universe is doing... except that it has no boundary... the gas goes on indefinitely, without any end or boundary. When you describe a gas like this on very very large scales, you do need GR to describe some of what goes on; but it all reduces over small scales to precisely this simple expansion of a bubble of gas.

    The average distance between gas molecules is increasing... and the further apart two molecules are in the bubble, the greater the rate at which the distance between then is increasing. In fact, the rate of separation is proportional to the distance between the molecules (messed up a bit by the noise of random motions over and above the general expansion trend).

    That is precisely the Hubble law... recession velocity is proportional to distance; but there are also some local motions of galaxies on top of that.

    Scale factor

    The scale factor is simply a dimensionless number; the factor by which separation distances have increased by comparison with some arbitrary chosen reference... typically we define the scale factor to be "1" at the present.

    Hence... when the scale factor was "0.5", everything was about twice as close together as it is now. When the scale factor becomes "2", everything will be about twice as far apart as it is now. Except that when I say "everything", I really mean on very large scales. Locally, things don't just move in straight lines, they are bound together in gravitational orbits and so on... hence our galaxy is not expanding. Expansion of the universe is not a "force" trying to make our galaxy expand, either. It's just the motion of the universe, or of things in the universe. So our galaxy doesn't expand and there's no reason for it to expand... but distant galaxies receding from us continue to do so... and that is expansion.

    Models

    The different models for the universe are concerned with what happens to the scale factor over time. To say that the universe is expanding is to say that the scale factor is increasing. What a model tells you is how fast it is expanding and ... more importantly ... how the rate of expansion changes.

    Gravity is the crucial force here... gravity tends to pull matter together, so gravity works to slow expansion down. The rate at which expansion is slowed depends on the density of the universe. This is "deceleration" of expansion. On really really big scales, like a universe, you do need GR to describe gravity well... but it's the same as Newtonian gravity on smaller scales. The tendency of gravity to slow the expansion of the universe is precisely the same as the tendency of gravity to compress a cloud of gas on smaller scales.

    But descriptions using general relativity allow scope for an additional factor to come into play... a strange kind of additional kick of energy given to particles just by the space between them. This is Einstein's "cosmological constant", or "dark energy". Put simply, it means there's another factor at work trying to push things further apart, faster and faster. Different models have different amounts of matter to give gravitational attraction, and different amounts of dark energy; but basically, a model for a homogenous universe comes down to sorting out how the scale factor changes with time.

    Recession velocity

    Scale factor is a property of the universe at a given time. Recession velocity is a property of a particle (galaxy) in the universe, with respect to us. the velocity of a particular particle is proportional to how rapidly the scale factor is increasing, and ALSO to how far away that particle is.

    Cheers -- sylas
     
  10. Jul 9, 2009 #9
    No. Scale factor represents the geometry of the model. Its results are model dependent.

    The redshift correction (1+z) is the DEFINITION of the measurement.

    Equating the two and demanding that all models conform to this is misleading.

    Some scale factors need not be functions of time -- at least not in the way implied. This approach is predefining the acceptable models.

    The redshift correction for luminosity distance can be separate from and additional to the geometric scale factor one.

    I gather that in the models you are familiar with and addressing that the (1+z) redshift is incorporated into the scaling factor. Is that correct?
     
  11. Jul 9, 2009 #10

    sylas

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    OK... I don't want to give offense here, but in all honesty, it seems to me that you are in the category here of "not even wrong". I tried to give a simple explanation above, because I think you are dreadfully mixed up about this. You are using words in the quoted extract in non standard ways, or else are just wrong.

    I strongly recommend a basic level introduction on this material. If you feel your own understanding is sufficiently good that you don't need that any longer, then we are at an impasse.

    z, or 1+z, is just a way of quantifying the redshift. The scale factor is just a way of quantifying how much the universe has expanded at a point in time for a homogenous isotropic model. And the equation you quoted is, as I told you before, a basic result for how these two quantities are related. The redshift of a photon from a distant galaxy that is co-moving with the expansion of the universe will have precisely that relation you have quoted, in all the models which can be characterized with a scale factor.

    More general models that allow for the small inhomogeneities of the universe don't admit description simply in terms of a scale factor, but on large scales they are well approximated by the FRW models... and they ALL satisfy the relation you have quoted. Photons from a real galaxy will have small contributions to the redshift from peculiar motions of the galaxy, but the equation holds as a kind of average, and certainly holds for co-moving galaxies.

    Or are you are thinking of fringe models that are sometimes proposed as variants of the basic Big Bang cosmology? I honestly have no idea what you are thinking here. I have no idea what kinds of models you think DON'T satisfy that relation... but I don't think that has anything to do with my lack of understanding of basic modern cosmology.

    But by all means, if you can think of any model that does not satisfy that relation, it would be very helpful to give a citation.

    Cheers -- sylas
     
    Last edited: Jul 9, 2009
  12. Jul 9, 2009 #11
    Sorry. I'm not trying to cause a problem.

    Its just seemed to me that the way the equation was written implied that the redshift -- which is a measurement -- could only be addressed with a model conforming to the given scale factor relation.

    It was my poor attempt at trying to keep what we know for sure -- the measurements -- separate from the theory.

    I'm not currently a student -- and at my age will likely never be again -- so I do not have access to libraries and university materials. This means I MUST use the internet as my primary source.

    When it comes to cosmology, there are many many theories out there. The outsider has no good way of knowing when he starts to read something wither it is 'mainstream', 'alternative' or just 'nuts'. I've learned to ask pointed questions to find out.

    (Note, if need be I can create such a model -- but that is NOT the purpose of this forum as I understand it.)
     
  13. Jul 9, 2009 #12
    It reads:
    [tex]\frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}} = (1 + z) [/tex]

    It should say, normal scale factors:
    [tex]\frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}}[/tex]

    Where by measurement:
    [tex]\frac{ \lambda_{o} }{\lambda_{e} } = (1 + z) [/tex]

    The order and placement of the ratios was misleading -- to me.

    I also question whether the scale factor ratio is upside-down? I do not know. But I have seen in other writings where the scale factor should be multiplied by the dimension to give the corrected value.
    Meaning, isn't the relation:

    [tex]a_{0} \lambda_{o} = a_{e} \lambda_{e} [/tex]
     
  14. Jul 9, 2009 #13
    Suit yourself Rymer, I was just summarizing the simple mathematical equivalence, and you should feel free to re-order and expand on it however is most sensible to you.
    No. The usual convention, for convenience, is to set [tex] a_{0} = 1 [/tex]. So if the scale factor was 1/2 that at z=1, [tex] a_{0} / a_{e} = 2 [/tex], correctly showing that the wavelength has doubled. Your equation should be:

    [tex]a_{0} \lambda_{e} = a_{e} \lambda_{0} [/tex]
     
  15. Jul 9, 2009 #14

    sylas

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    No... it is correct as given originally.

    The subscript o means observed, and e means emitted. If we see a photon which is redshifted, that means we observe a longer wavelength than observed by the emitter. As the universe has expanded since it was emitted, and the wavelength has increased, you need the ratio of wavelenghts equal as equal to the ratio of scale factors.

    The redshift value z is by definition that ratio of wavelengths, minus 1.

    It is a consequence of relativity that co-moving observers will see a photon with a wavelength that is proportional to the scale factor at the time of observation. Whether you think of this in terms of recession velocities, or expanding space, or transforms between different frames, is really matter of what explanation works for you. The underlying physics ... general relativity ... is the foundation and all explanations are attempts to give a simple intuition for this underlying foundation.

    In my opinion, nutgeb is barking up the wrong tree in the quest for a "mechanism" for the redshift. It's a consequence of the geometry of space time and the perspective of different observers, rather than a consequence of something "changing" the photon. The frequency of a photon depends on the observer. Observation of a photon by different co-moving observers widely separated from each other is no more and no less a "mechanism" than observation by local observers moving relative to each other, giving the conventional Doppler shift.

    There are indeed other models that have been proposed that are not based on the expanding universe, or "Big Bang". Steady state, tired light, electric universe, etc etc. Some might have been credible alternative notions back a few decades ago, but by now they are long since falsified dead ends; and generally speaking they don't even use the concept of a scale factor. There are still a few supporters of such models, but they are by now pretty much the equivalent of creationists in scientific standing.

    My own favourite introduction to cosmology is Ned Wright's Cosmology Tutorial. It's the one to use if you don't mind a bit of algebra, and it will help to know special relativity. You don't need all the detail of general relativity and tensors. Check it out... it's been a great help to me over the years.

    Cheers -- sylas
     
  16. Jul 9, 2009 #15
    Ich, I'm at a loss how to respond to your repeated insistence on introducing SR effects as between comovers in the non-empty FRW metric. We debated the point endlessly in the past, and it's simply wrong. I do not recall reading any textbook or peer reviewed source which suggests that either proper distances or photon worldlines become Lorentz-contracted as between comovers in the non-empty FRW metric. You can't just treat a matter-filled model as if it were empty, a point that all the books and articles make. There is no place mathematically in the FRW metric for SR time or space dilation/contraction effects between comovers.
    I have Peacock's textbook, and he doesn't prove it there. (Note that his web posting "Diatribe on Expanding Space" is not peer-reviewed.) Equation 3.67 in his textbook is actually very similar to the one I proposed in my thread on the Mechanics of the Cosmological Redshift, although he doesn't address all the elements of the redshift or how they work, and he misleads readers by simplistically stating that de Broglie wavelengths scale with the expansion, which sounds like he is portraying the 'expanding hypersphere' as a stretching rubber sheet. (Perhaps he didn't intend that particular interpretation). His equation, like mine, calculates a classical Doppler-like effect with no SR gamma factor:

    [tex]\frac{dz}{1 + z} = \frac{H_{z}dl_{z}}{c} [/tex]

    His wording uses the terminology "Doppler shift", not "relativistic Doppler shift" or "Lorentz transformation." Peacock goes on to say (p72): It is common but misleading to convert a large redshift to a recession velocity using the SR formula ... Any such temptation should be avoided - it is only valid in a universe with zero matter density." And at p87: "A common but incorrect approach is to use the SR Doppler formula... This would be appropriate in the case of a model with [tex]\Omega=0[/tex]..., but is wrong in general... The reason the redshift cannot be interpreted this way is that a non-zero mass density must cause gravitational redshifts; even this interpretation is hard to apply rigorously when redshifts of order unity are attained." [Note: I've tried adding in gravitational redshift, and the math doesn't work at all.] At the end of the paragraph he does mention using a Lorentz transformation, but only as an aside to show that relativistic and non-relativistic particles both experience a loss of momentum.
    I don't know what approach you used, but I challenge you to demonstrate that integrating the SR redshift formula yields a result that is as accurate as integrating the classical Doppler shift formula. My spreadsheet calculation showed an error 4 times larger using the SR formula.
    That language is inappropriate for this forum.
    Your suggestion is exactly what my cosmological redshift equation does. It's when I run it that way (the same as you suggested) that I get 4 times the error for SR as for classical Doppler.
     
  17. Jul 9, 2009 #16
    You're entitled to your opinion Sylas, and I'm glad to hear it. But if you are suggesting that the cosmological redshift is a unique, sui generis effect of physics, which is not constructed out of components of existing physics, then your hypothesis is far more radical than mine. Before you propose a theory like that, shouldn't you rule out the logic of the conventional physics I suggested, and the math I proposed? My suggestion happens to be quite logical and the math works really well. It is also consistent with the math used in Peacock's textbook.

    Also, when I said (in the other thread) that cosmological redshift happens all along the photon's worldline, rather than discretely at the observer, that's because there is no known physics to describe WHY a photon emitted from an emitter with a certain recession velocity would be observed in the (stationary) observer's frame at a redshift equal to the expansion of the universe. In other words, there is no end-to-end formula that avoids the need to integrate infinitesimal effects all along the worldline. If you think there is one, please describe it.

    The purpose of this thread is not to debate the mechanics I proposed in the other thread, but rather to see if we agree that the two misconceptions I described in this thread are really wrongheaded. If, contrary to my conclusion, you think either of them is correct, then it would help if you could say so, and be specific as to why. I sense that you are harboring an affection for one or both of them. If so, let's get it out on the table.
     
    Last edited: Jul 9, 2009
  18. Jul 9, 2009 #17

    sylas

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    Sorry, nutgeb; I am certainly not proposing anything different from conventional explanations. It sounds like you find conventional explanations inadequate for some reason. Meaning no offense, but I find your posts very hard to follow. Maybe it is me, maybe it is you, but I honestly haven't figured you out.

    I'm very dubious about your claims above indeed, but am not sufficiently expert in my own right, nor sufficiently confident I can even follow what you are trying to say; and there appear to be others willing to take up the discussion and I prefer to leave it to them.

    So. For the record, then. I don't really understand you, and perhaps that my problem; not yours. In any case, I am certainly not convinced, and content to continue referring to other sources than your posts for my own understanding. I find them clearer and more consistent and with the benefit of wide mainstream acceptance to give me confidence. My current understanding is that general relativity works just fine for calculating the world lines of photons, and for calculating their wavelength with respect to any local observer. In the case of the FRW solutions, that calculation has as a consequence that co-moving observers observe a wavelength that is proportional to scale factor.

    The only way I can make any sense of your remark is that you want to know the mechanism for general relativity. It is, after all, basically a theory of geometry rather than a theory of mechanisms. If that is the case, you will need to wait until people have sorted out the unification of relativity and quantum mechanics; with some "grand unified theory". In the meantime, I am content with the conventional general relativity as a description of how matter and space and time and energy interact, for the purpose of world lines for photons; and the relation with scale factor for FRW expansion falls out from that without any great additional mystery.

    Cheers -- sylas
     
  19. Jul 10, 2009 #18
    Sylas, good for you for being dubious. That's a good way to react.

    If you have specific issues to raise on this topic, please feel free. I'm still interested in your opinion about the two misconceptions.
     
  20. Jul 10, 2009 #19

    George Jones

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    I, too, have great difficulty following nutgeb's posts.
     
  21. Jul 10, 2009 #20

    Ich

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    To help your memory, here's the context of my statement:


    The "cosmological proper distance" of two photons in your scenario obviously increases. That's redshift.
    So does the distance of two relativistic particles, but not the distance of two nonrelativistic particles.
    The respective formula is:
    [tex]\frac{D_{observer}}{D_{emitter}}=\frac{\sqrt{1-(Hx)^2}}{1-Hx\,v}[/tex]
    How do you explain that?

    And if your spreadsheet doesn't work with small SR doppler shifts, but has no problems with small classical doppler shifts: go fix it. They are the same.
    If instead you merely report a larger error (the factor 400000 after integrating past infinity has disappeared, I gather): make smaller steps.
     
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