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The cosmological redshift

  1. Jan 31, 2010 #1
    I’ve made several efforts to provide a simple description of how the cosmological redshift works. Each has been closer to the mark than its predecessor, and now I think I’ve got it nailed. The premise underlying the redshift is simply that a photon must travel at a local velocity of c through each and every infinitesimal local inertial frame along its worldline from the distant receding emitter to the stationary observer.

    Accumulated Doppler shift

    Each local frame can be thought of as containing its own fundamental observer moving exactly with the local Hubble flow. Since each such local frame has a different Hubble velocity HD relative to the ultimate observer, the photon in effect must change its coordinate velocity (relative to the ultimate observer) at every local frame crossing. Each such local frame crossing results in an infinitesimal Doppler shift (relative to the ultimate observer’s frame), and the accumulation of those Doppler shifts over the entire worldline yields the total cosmological redshift. As Peacock says at p. 87 of his textbook:

    "One way of looking at this issue is to take the rigid point of view that 1+z tells us nothing more than how much the universe has expanded since the emission of the photons we now receive. Perhaps more illuminating, however, is to realize that, although the redshift cannot be thought of as a global Doppler shift, it is correct to think of the effect as an accumulation of infinitesimal Doppler shifts caused by photons passing between fundamental observers separated by small distances..."

    One way to understand this accumulating Doppler shift is to think of it as an accumulation of physical stretching of the proper distance between successive wave crests of traveling light. The first wave crest always has an inward velocity (toward the ‘stationary’ observer) which is faster than the second wave crest’s velocity, because the first wave crest's proper distance from the observer is smaller than the second wave crest's. Therefore the Hubble velocity HD (relative to both the emitter and the origin) that the first wave crest must match at a local velocity of c is always is contemporaneously greater (less negative) than the HD the second wave crest is required to match.

    It's like a line of evenly-spaced joggers following each other at a constant peculiar velocity over a series of moving sidewalks which are moving in the opposite direction the joggers are running -- with each successive sidewalk having a lower 'negative' velocity than the prior one. The line of joggers will progressively stretch apart. This is true even if the speed of every such sidewalk is simultaneously reduced over time by the same proportion. Note that the outcome does not depend on any paradigm of ‘space itself’ stretching (or a stretching hypersphere.) Even if ‘space itself’ does not expand, two successive wave crests will progressively separate due simply to the different contemporaneous local Hubble velocities HD each of them is required to match with a local velocity of c. It’s like a long train with stretchable couplers between cars, where the engine is running on a part of the track with a faster speed limit than where the caboose is. Note that the radial distance between photons in a discrete pulse of light increases in the same proportion as their wavelength, and for the same reason.

    Another way to think of this Doppler shift is as an accumulation of losses of locally-measured momentum by the photon as it is required to adjust its local velocity c to match progressively higher (less negative) Hubble velocities HD as it approaches the observer. Peacock and Peebles both comment that it is appropriate to think of the cosmological redshift as a progressive loss of momentum by the photon.

    The accumulated Doppler shift equation

    The equation for the accumulated Doppler shifts is the multiplicative series (1+H1dt)(1+H2dt)...(1+Htdt). This equation is equivalent to Peacock’s equation 3.67. In units of c=1, the light travel time dt is an easier to use substitute for the light travel distance dr, since dt=dr. Note that Ht = da/(atdt), and (1+da/a1)(1+da/a2)…(1_da/at) = 1/at, because z = z/at-1. So the accumulated Doppler shift yields exactly the same result as the standard cosmological redshift formula.

    This equation for the Doppler shift does not include any element of SR time dilation. This is because no time dilation occurs as between fundamental comovers in the FRW metric. To illustrate this point, consider the empty Milne model. In that case, SR Minkowski coordinates are converted to FRW coordinates by applying a Lorentz length expansion gamma factor 1/[tex]\gamma[/tex] (see Peacock p. 88), which offsets the SR time dilation gamma factor [tex]\gamma[/tex] and yields a constant cosmological time shared by all fundamental comovers. So (1+Htdt) can be thought of as classical Doppler shift with a stationary observer, (1+v/c), or if that offends relativistic sensibilities, it can be thought of as a relativistic Doppler shift with the time dilation element eliminated by the transformation to FRW coordinates.

    The role of gravity in the cosmological redshift

    In FRW coordinates, gravity plays only an indirect role in the cosmological redshift: it acts to reduce the Hubble rate Ht over time. As the Hubble rate progressively diminishes, the rate of accumulation of incremental Doppler shift diminishes in the same proportion, because the difference between the Hubble velocities HD that successive wave crests must contemporaneously match (each at a local velocity of c) decreases.

    This temporal change in the Hubble rate is already incorporated in the accumulated Doppler shift equation stated above, so no additional correction for gravity is required or allowed. It is popular to think of gravity as applying a blueshift factor to photons approaching an observer, but applying that as a separate factor would result in an incorrect calculation. Returning to the idea of the distance between wave crests increasing due to their differential local Hubble velocities HD, it is obvious that there is no place for reduced stretching (i.e. blueshift) between the wave crests due to gravity. Artificially inserting such a gravitational stretching reduction factor would make it impossible for both successive wave crests to maintain their local velocities at c (unless an additional offsetting Doppler shift is also inserted). Or if one thinks of the redshift as a loss of local momentum, that momentum loss is already fully accounted for by the accumulated Doppler shift alone, without any separate gravitational factor.

    The tethered galaxy exercise shows that an inward gravitational force vector exists, which causes the untethered galaxy to move inward (if Lambda=0) toward the observer. However, while this is true for non-relativistic particles, it is not true in the case of relativistic photons. The velocity of a photon cannot be accelerated inward because its local velocity is constrained to remain at c regardless of any acceleration force. And as described above, the effect of gravity is already fully incorporated into the accumulated Doppler shift formula as a reduction in the Hubble rate over time. In other words, gravity causes exactly the same increase in the photon's inward momentum as it causes in the inward (less outward) momentum of the fundamental observers the photon passes along its worldline. So gravity causes no 'additional' blueshift with respect to those freefalling fundamental observers, in FRW coordinates. (As Bunn & Hogg have written, gravity can be viewed as creating its own blueshift if certain non-FRW coordinates are used).

    It helps to keep in mind that gravitational time dilation occurs only as a result of a clock rate differential between an emitter and observer, and not literally as a result of the photon ‘gaining energy’ from the gravitational ‘pull’. Since the clocks of all FRW fundamental observers run at the same rate, there is no opportunity for gravitational time dilation for a photon moving between them.
    Last edited: Jan 31, 2010
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  3. Jan 31, 2010 #2


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    Correct, that is why gravitaional redshift is a non-factor in cosmological redshift. The Doppler-like effect of expansion is the only known explanation.
  4. Feb 1, 2010 #3


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    That depends upon what coordinates you use, though. There are many different ways of talking about this. What nutgeb posted is just one way of looking at the physical phenomenon that's going on here. Another perfectly-equivalent way of looking at the situation is to use comoving coordinates where there is no relative velocity, and all of the redshift is gravitational (from the expansion of space).

    That said, nutgeb, there's a few small errors in the equations you wrote down.

    You started from this equation:
    [tex]1 + z = \left(1 + H_1 dt\right)\left(1 + H_2 dt\right) ... \left(1 + H_t dt\right)[/tex]

    Then you said that Ht = da/(atdt), but this doesn't make any sense to me. Instead, we can simply write:

    [tex]H dt = \frac{1}{a}\frac{da}{dt} dt = \frac{da}{a}[/tex]

    (this is just using the chain rule)

    Now, plug this back into your original equation:

    [tex]1 + z = \left(1 + \frac{da}{a_1}\right)\left(1 + \frac{da}{a_2}\right) ... \left(1 + \frac{da}{a_t}\right)[/tex]

    We can solve this equation pretty easily if we consider that [tex]a_n = a + n da[/tex], so we have:

    [tex]1 + z = \left(1 + \frac{da}{a_1}\right)\left(1 + \frac{da}{a_1 + da}\right) ... \left(1 + \frac{da}{a_t}\right)[/tex]

    If we then add the fractions by finding a common denominator:

    [tex]1 + z = \left(\frac{a_1 + da}{a_1}\right)\left(\frac{a_1 + 2da}{a_1 + da}\right)\left(\frac{a_1 + 3da}{a_1 + 2da}\right) ... \left(\frac{a_t + da}{a_t}\right)[/tex]

    With this we can easily see that the numerator of the first term perfectly cancels the denominator of the second. Then the numerator of the second cancels the denominator of the third, and so on. In the end, we are left with:

    [tex]1 + z = \frac{a_t + da}{a_1} = \frac{a_t}{a_1}[/tex]

    ...which just means that one plus the total redshift is the ratio of the scale factors between the emitter and observer. So I think this is correct, if [tex]a_t[/tex] is the scale factor now (usually defined to be 1), and [tex]a_1[/tex] is the scale factor at the source.
  5. Feb 1, 2010 #4
    Chalnoth, your proof that the series (1+Htdt) calculates exactly the same total redshift as the standard scale factor formula is certainly clearer than what I wrote on that point, so I am inclined to adopt it.
  6. Feb 2, 2010 #5


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    I think you'll need some more tries.

    For example,
    is definitely not what Peacock had in mind. It isn't compatible with SR either, in the end you're just rewording the description in cosmological coordinates (proper distance).

    Your section "This equation for the Doppler shift does not include any element of SR time dilation..." simply doesn't make sense. You're mixing frames at will. Choose either one and stick to it, then you'll see that in one case there is time dilatation, in the other not. That's how it is, and no amount of hand waving will change it.
    And in the case of infinitesimal doppler shifts, it is irrelevant from the beginning, as only first order terms are relevant. But we discussed this again and again. I won't answer to this kind of thread in the future, but urge you one last time to thoroughly investigate the two coordinate frames in the Milne model.
  7. Feb 2, 2010 #6
    Ich, you and I have debated the point about time dilation in FRW coordinates enough times that it's clear neither of us will change the other's mind.

    "...consider the empty Milne model. In that case, SR Minkowski coordinates are converted to FRW coordinates by applying a Lorentz length expansion gamma factor 1/[tex]\gamma[/tex] (see Peacock p. 88), which offsets the SR time dilation gamma factor [tex]\gamma[/tex] and yields a constant cosmological time shared by all fundamental comovers."

    And as I said my equation is equivalent to Peacock's equation 3.67, which like mine contains no element of SR time dilation.

    For what it's worth, I agree with you that the second order terms in SR time dilation are immaterial in a high-quality FRW cosmological redshift calculation, because they approach zero as the local frame segments approach zero length. However that hardly makes the distinction irrelevent. The fact that this error doesn't change the calculation may help explain why it seems to have gone unnoticed in the literature. And yes I understand you don't think it's an error.
    Last edited: Feb 2, 2010
  8. Feb 3, 2010 #7


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    "converted by applying a Lorentz length expansion gamma factor". That's the prototype of hand waving.

    Just answer the following questions correctly, and if you still disagree then, I won't bother you anymore.

    In the empty universe, fundamental observer A is defined to be at the coordinate origin. Fundamental observer B is at FRW comoving distance 1/H (Hubble distance).
    Now, in Minkowski coordinates, and only Minkowskli coordinates (no more mentioning of FRW):
    1. What is their relative speed?
    2. What redshift measures A for light originating from B?
    3. What formula connects redshift and speed, a) classical doppler without time dilatation, or b) SR doppler with time dilatation?
    4. So, in Minkowski coordinates, is there time dilatation between A and B?
  9. Feb 4, 2010 #8
    Ich, I'm not sure I completely follow what your questions are getting at, but I'll try to answer them.

    1. Relative speed of A and B in Minkowski coordinates: According to Peacock p.88, v/c = tanh [tex]\omega[/tex], where v/c is the Minkowski velocity parameter and [tex]\omega[/tex] is the FRW velocity parameter. Since [tex]\omega[/tex] = 1 at the Hubble radius, tanh 1 = 0.7616, and that is the Minkowski relative velocity.

    2. Measured redshift: Since we are using Minkowski coordinates, redshift is calculated using the SR formula

    [tex] z+1 = \sqrt{\frac{1 + v/c}{1- v/c}},[/tex]

    [tex] z+1 = \sqrt{\frac{1.7616}{0.2384}} = 2.7183[/tex]

    3. In Minkowski coordinates, redshift is calculated by applying the SR formula to the SR version of the parameters.

    4. In Minkowski coordinates, yes there is time dilation between A and B.

    If the above is converted back to FRW coordinates, [tex]\omega[/tex] = 1, there is no time dilation between A and B, space has negative curvature instead of being flat, and the redshift is calculated by applying the GR formula to the GR version of the parameters.
    Last edited: Feb 4, 2010
  10. Feb 5, 2010 #9


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    I just wanted to make sure that you're not longer claiming that the Bunn&Hogg procedure is wrong. And that you see that, as they are using local inertial frames (=local Minkowski frames), they're formally talking about accumulation of SR doppler shifts, including time dilatation. Of course it doesn't matter for small enough distances, as the result is the same (I understand that you no longer doubt that?).
  11. Feb 5, 2010 #10
    Ich, as I said a couple of posts ago, I agree with you that for small distances, or for an accumulation of very small steps of dt, the error introduced by using the SR Doppler shift formula is immaterial.

    However, I believe that Bunn & Hogg are technically incorrect in applying the SR Doppler shift formula to an accumulation of local Doppler shifts. If they were defining a shift that occurs with respect to an inertially moving galaxy entirely within a single large (or 'global') inertial frame, then the SR formula would be correct. But instead they are referring (in the same way my redshift equation does) to Doppler shifts that occur at local frame crossings as between fundamental comovers, who are always in non-inertial velocity relationships with respect to each other. Moreover, their references to 'comoving coordinates' demonstrate that they are accumulating Doppler shifts over 'global' FRW coordinates, not 'global' Minkowski coordinates. Therefore their use of the SR formula is an error. They should use the GR formula instead.

    But their paper is excellent on its main thesis and in other respects.
  12. Feb 8, 2010 #11


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    That's all inventions of yours. There's no global coordinate system used.
    Bunn & Hogg are indeed very precise in describing their procedure:
    I highlighted some parts that show explicitly what I am talking about all the time: They add velocity and redshift from one local inertial frame to the next, using the usual SR formulas that apply here. There is no error introduced (other than neclecting gravitation in the local frame, a second order error), and there is no reference to global coordinate systems.
    This procedure is certainly valid, and I don't see how your constant reference to global coordinate system contributes anything except confusion.
    Try to follow their argument, not the arguments you invented; you won't find a flaw.
  13. Feb 8, 2010 #12
    Maybe you're right that they aren't specifically invoking FRW coordinates in their 'world-tube'. Rather they build their 'world-tube' out of an aggregation of fragmented local Minkowski frames. However, doing so introduces an element of error in any case involving gravity, which they freely admit.
    We agree that they are measuring Doppler shift as between one local frame to another. However, the local frames do NOT have inertial velocity relationships if the cosmic gravity is nonzero.
    B&H admit there is error in their approach when spacetime is curved. Even if it's second order, it remains an error. We already agreed that the error was second order.

    "This region of spacetime can be considered as flat Minkowski spacetime, up to errors of order [tex]\epsilon[/tex]2."

    "The errors in this method are of the same order as the departures from flatness in the spacetime in a neighborhood containing both me and you; as long as I’m willing to put up with that very small level of inaccuracy, I can interpret that coordinate velocity as your actual velocity relative to me.”

    B&H can gloss over the admitted inaccuracy introduced by their use of the SR formula because they know that the error can be made to be vanishingly small in calculations. All well and good. But if their approach were exactly correct (as my version of Peacock's is) they could avoid having to continually remind their readers that their approach does in fact include an error.
    Last edited: Feb 8, 2010
  14. Feb 9, 2010 #13


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    I see your answer has greatly improved, that's good.
    Now we've come to the point:
    You always insisted that they introduce an error by using SR doppler shift instead of classical doppler shift. Deviation from flatness has nothing to do with that.

    In your last (edited) post, you neither claim this kind of error, nor that there is no time dilatation between nearby comoving observers when viewed in a local inertial frame.
    May I take this as a sign that we finally agree on both points?
  15. Feb 9, 2010 #14
    Ich, no I don't think we agree yet, and I want to reconsider the point about whether a global FRW metric is implied in B&H's approach. I'll come back with more analysis after I return in a week or so.
  16. Feb 10, 2010 #15


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    Sure, take your time. Try not to read between the lines, there's only paper.
    Freefalling (=comoving) observers and parallel transport of velocity are all coordinate-independent concepts, while the operational definition I quoted makes explicitly use of SR coordinates.
  17. Feb 27, 2010 #16
    Here’s the conceptual problem I see with Bunn & Hogg's use of the SR Doppler shift in their 'world tube’ analysis.

    We’ve discussed the fact that the Minkowski and FRW velocity parameters become quite different as distance increases. In the example involving the Hubble distance, the Minkowski velocity parameter is .7616 while the FRW velocity parameter is 1. The difference in the parameters arises from the fact that in FRW coordinates, relative to Minkowski coordinates, the time coordinate [tex]\ dT = dt/ \lambda [/tex], and proper distance coordinate [tex] dD = dX/ \lambda[/tex].

    The two different velocity parameters cannot be used interchangeably. So how can the B&H ‘world tube’ model use the standard FRW Hubble velocity parameter HD without any first converting it to the Minkowski version? (Conceptually the B&H analysis must start with the FRW HD parameter because the Minkowski version of the recession velocity parameter does not scale exactly in proportion to distance.) Their justification is that the resulting error becomes small as the size of the inertial frames is decreased. But it is conceptually incorrect to use the FRW velocity parameter in the special-relativistic formula, even if the error is small.

    Also, in the B&H approach, the total light travel distance is the sum of all of the dX measurements from all of the separate local inertial frames. This aggregate light travel distance is not the Lorentz-contracted red line shown in http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH"(lower) Minkowski diagram. Instead it is the longer red curve shown in his (upper) FRW diagram. In other words the B&H analysis implicitly uses an aggregation of distance fragments measured in FRW coordinates, not Minkowski coordinates. The correct light travel time must then be calculated by combining the longer FRW light travel distance and faster FRW recession velocity parameters, not the shorter and slower Minkowski versions.

    Since the B&H approach implicitly aggregates to FRW parameters, it is conceptually incorrect to apply the SR time dilation factor in calculating redshift.
    What I am saying is that the classical Doppler shift aggregation formula (or if you prefer, the SR shift with time dilation factored out) that Peacock and I use is always exactly correct, as long as the Hubble velocity value HD used for each segment is the true geometric mean value for that segment. On the other hand, the SR aggregation formula can only asymptotically approach a correct figure at an infinitely small segment size. So any error introduced by using approximated data is by definition always slightly larger when the incorrect SR formula is used. That error approaches zero if an (unachievably) perfect calculation is performed.
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  18. Mar 1, 2010 #17


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    They don't use the "FRW velocity parameter".
    It's neither. It is t(observation)-t(emission), where t denotes cosmological time. AKA Light travel time.
    I can't make sense of this statement.
    First, light travel time is not exactly a FRW parameter - but that's a subsequent error.
    Then, this is physics, not philosophy. Whatever their calculation "implicitly aggregates" to, they are explicitly using local inertial coordinates. In such coordinates, it is not a question whether the "SR time dilation factor" may or may not be "applied". They're calculating in SR coordinates, so there is time dilation. That's not a matter of choice or holistic conceptual considerations.
    What you're doing here is to compare calculations in the two different approaches. That's not the point.
    The question is: in Bunn&Hogg's approach, whether you like it or not, is there time dilation or not?
    That's plain wrong. Again, the question is: in Bunn&Hogg's approach, is there more or less error when the doppler effect is calculated relativistically? (Better deal with gravitation also; if you include all second order terms, you get Peacock's #16 in his "diatribe".)
  19. Mar 1, 2010 #18


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    Would it be valid to use Peacock's #16:


    as a series of infinitesimal Doppler shifts: [itex]1+z=(1+z_1)(1+z_2)...(1+z_n)[/itex],



    The reason I'm asking is that nutgeb and myself have struggled on another forum with the fact that when the gravitational factor is left out in such a series, the Newtonian Doppler shift equation produces a result closer to the correct (1+z=1/a) value than what the special relativistic Doppler equation produces. This may perhaps solve that puzzle (and maybe even settle this thread's argument :)

    Edit: I might note that we have used an incremental Hubble velocity over small distance increments, which is not exactly the same as 'SR-velocity'. I'm not quite sure how to determine the recession velocity ([itex]v_i[/itex]) of a galaxy in the locally Lorentz coordinates of a comoving observer.
    Last edited: Mar 1, 2010
  20. Mar 1, 2010 #19
    [Here is the re-edited version. I also responded to one point Ich added in his subsequent post while my editing was in progress.]
    I agree that B&H are measuring local light travel time, which equals local light travel distance in units of c=1.

    I also agree that the red lines in the Ned Wright diagrams don't represent light travel distance per se, rather they illustrate a photon's spacetime worldine along the light cone.

    However, in Ned's upper diagram, the total elapsed light travel time for any given photon's worldine in non-time dilated FRW coordinates (i.e., the coordinates on the time and distance axis) clearly must be greater for the same photon mapped in length contracted and time dilated Minkowski coordinates in the lower diagram. (Just as the hyperbolic lines in the lower diagram are longer than horizontal and vertical lines to the axis).

    Consequently, the elapsed light travel distance for the photon, when measured by local observers along the path and accumulated to a total, must equal the longer FRW light travel distance, not the shorter SR-contracted Minkowski distance, because in each 'ideal' local frame the Minkowski length contraction and time dilation of the photon's path are eliminated.
    If your point is that FRW coordinates can be mapped onto a Minkowski chart, I certainly agree, but that fact adds nothing to the topic we're discussing. The lower diagram uses Minkowski time and distance coordinates on each axis. That chart shows that in Minkowski coordinates, both time dilation and length contraction exist. Conversely, the hyperbolic lines on the lower diagram show how Minkowski coordinates can be extrapolated to FRW coordinates by eliminating the time dilation and length contraction. The upper diagram is simply a ‘normalized’ version of the hyperbolic lines in the lower diagram, such that the coordinates on each axis map directly to FRW coordinates. The important point here is that in FRW coordinates there is no time dilation or length contraction between comovers.
    The answer has nothing to do with whether I like it, or with philosophy. The amount of redshift actually observed does not depend on which set of charts or coordinates one chooses to apply. But time dilation itself is not an observation; it is an underlying cause we attribute to certain observations. And attributed causes most definitely can be coordinate-specific.

    B&H’s analysis adopts the premise of an ‘ideal’ local inertial frame in which spacetime curvature (gravity) has been entirely eliminated. They recognize that the presence of any spacetime curvature renders their analysis a close approximation, not an exactly correct answer. (Not to digress, but in contrast to the B&H approach, the formula Peacock and I use is equally exact both with and without spacetime curvature.)

    In any ideal inertial frame, regardless of whether it is ‘local’ or ‘global’ in extent, clearly the SR time dilation formula applied to Minkowski coordinates will calculate exactly the correct observed redshift, and not merely an approximation. SR time dilation definitely contributes to the redshift of a photon passing between two observers whose own clocks are SR time dilated relative to each other in the applicable coordinate system.

    But in the same ’ideal’ local or global inertial frame, the standard cosmological redshift formula (proportional expansion of the scale factor since emission) applied to FRW coordinates also will calculate exactly the correct observed redshift. By the nature of FRW coordinates, no time dilation exists as between fundamental comovers. That is true at both the global and local frame levels (assuming that FRW coordinates are rigidly adhered to even at the local frame level, and not ‘approximated over to Minkowski coordinates’.) The logic is irrefutable that if no time dilation exists as between the original emitter and the ultimate observer, then it cannot somehow ‘creep in’ as between any two intermediate comoving observers along the worldline, regardless of how close together those intermediate observers are. SR time dilation definitely does not contribute to the redshift of a photon passing between two observers whose own clocks are not SR time dilated relative to each other in the applicable coordinate system.

    Thus, if and only if SR time dilation exists between comovers at a global level in a particular coordinate system, it also exists at the level of an ‘ideal’ local frame in the same coordinate system.

    B&H’s world tube analysis adds together local Minkowski worldline segments, but these do not actually sum to the correct length of the global Minkowski worldline between the original emitter and the ultimate observer. The reason is that, by the nature of the Lorentz transformation, all (or really asymptotically approaching all) of the SR time dilation and length contraction are eliminated from the local segment measurements. This should be obvious: no comoving Milne observer sees the distances to galaxies near his own location to be contracted; only distances to galaxies far away from his location are contracted. When each such Milne observer contributes to the group calculation his own measurement of only the local part of the worldline as he observers it, each such local measurement is non-contracted, and therefore the global sum of all such local measurements cannot be length contracted.

    This logic shows that the sum of local Minkowski measurements is not the Minkowski length-contracted global light travel distance; in fact it equals (or really asymptotically approaches) the FRW global light travel distance.

    Therefore it is wrong to use the SR formula to calculate each local contribution to the global redshift. Instead local calculations should be made using a formula compatible with FRW coordinates – a formula without time dilation.

    My purpose here is not to critisize B&H's overall approach to explaining the redshift, which I think is excellent, just because they employ a commonly-used approximation. The error arising from incorrect use of the SR formula can be made vanishingly small by using infinitely small local segments. My intent is just to point out that an even simpler formula is readily available that avoids this vanishly small error. With accurate segment Hubble velocity HD data, the approach Peacock and I use can yield an exactly correct calculation.
    I said very clearly in my last post that when the Doppler shift is calculated with the SR formula in this context using approximated date, there is always more error than when it is calculated without time dilation, unless the calculation could actually be performed with infinitely small segment sizes (which of course is impractical in real life), in which case the error using the SR formula will asymptotically approach the same exactly correct answer that the GR (non-SR) formula calculates.
    Inserting an additional '2nd order element' for gravity in Peacock's and my accumulated Doppler shift method would yield a flatly wrong answer in FRW coordinates. (As B&H point out, a different non-FRW coordinate system can be applied in which gravitational time dilation does become an explicit element.) I discussed why in my OP. Since the equation I gave already calculates exactly the correct answer, it can't be improved upon by inserting an additional factor.
    Last edited: Mar 1, 2010
  21. Mar 1, 2010 #20


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    I don't know if that's the reason. We're pretty close to exponential expansion, where "repulsive gravitation" adds second order redshift. I don't remember the details right now (have to check), but maybe this could be mimicked by an incorrect treatment of time dilation.
    Hey, that's a spoiler. I just wanted to give nutgeb some hints to edit his post, and this clue would have had a very central position.

    Of course, that's not the relative velocity in an inertial frame. nutgeb knows the correct formula (v=tanh(H*d)), that's why I was a bit irritated that he came up with the same argument.

    OK, let's wait until tomorrow. Here's a second hint for the editing: Go to the lower diagram at Ned Wright's site, and mark "simultaneous" intervals on different worldlines there. If you use the hyperbolic simultaneity, all intervals have the same "length" - hence no time dilation. If you use horizontal lines (Einstein convention), the intervals have different length - hence time dilation. Only one physical situation, but still both dilation and no dilation. That's how it is.
  22. Mar 2, 2010 #21


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    I think it can also be mimicked by an incorrect definition of relative velocity. This is part of the reason why I stay clear from cosmo-redshift as a Doppler shift - too many complications. The good old "expanding space" (1+z=1/a) treatment avoids most of those, provided it is done correctly.
  23. Mar 2, 2010 #22
    Jorrie I don't think it's very interesting to debate which one of the two rival paradigms for expansion -- 'expanding space' or 'kinematic' is more correct. The two paradigms have been refined over time such that they yield entirely identical calculations and have indistinguishable behaviors. It is somewhat interesting to describe in detail how the 'expanding space' version of the math works, as you have done elsewhere.

    But the 'expanding space' paradigm brings with it a very limited set of tools to address questions such as whether SR and gravitational dilation/contraction effects occur in certain scenarios, and which versions of parameters should be applied where. The 'kinematic' model brings along a much greater toolset that enables us to map out scenarios and mathematically test alternative approaches in sophisticated way. If more tools means more complexity, I'll accept the tradeoff.

    I'll summarize some of the points I made about Minkowski and FRW coordinates in my most recent (edited) post and elsewhere in this thread:

    1. Radial distances stated in FRW coordinates are just radial distances stated in Minkowski coordinates, divided by [tex]\lambda[/tex]. The same is true of elapsed times in these two coordinate systems. In other words, in this context FRW coordinates are Minkowski coordinates with the SR time dilation and Lorentz contraction eliminated.

    2. Therefore the clocks of all fundamental comovers in FRW coordinates run at the same rate, regardless of whether the emitter and observer are infinitesimally close together or arbitrarily far apart. And in FRW coordinates, there are no Lorentz contractions of distances between galaxies far from the observer, which would disrupt the homogeneity of the 'FRW dust' distribution and the direct proportionality of the Hubble recession velocity to distance.

    3. Time dilation does not contribute to the redshift of a photon passing between two fundamental comovers if their clocks run at the same rate in the selected coordinate system. This is true with respect to both SR and gravitational time dilation.

    4. Local SR dilated light travel times or SR contracted light travel distances observed in every adjacent local Minkowki frame/segment along the photon's worldline do not sum to the 'global' SR time dilated or Lorentz contracted light travel time or distance. Instead their sum asymptotically approaches the 'global' FRW light travel time or distance (as the frame/segment size goes to zero).

    5. Local FRW non-dilated light travel times or non-contracted distances observed in every adjacent local FRW frame/segment along the photon's worldline sum exactly to the 'global' FRW light travel time or distance.

    I'll also mention that if one intends to denigrate certain parameters inherent to FRW coordinate systems by characterizing them as 'Newtonian' or 'non-relativistic', that reflects a misconception. FRW parameters are more accurately described as 'general relativistic', because that's what they are. There simply is no place for special relativistic effects in calculating parameters in FRW coordinates which can't be localized to a single comoving location. An obvious example of this is the GR Hubble recession velocity HD, which is never subjected to Lorentz transformations regardless of distance, and scales smoothly above c, in FRW coordinates.

    Of course in proper cases a 'special relativistic' parameter that can be localized to a single comoving frame can be used in the same calculation with a 'general relativistic' parameter. For example, the total redshift of a distant object with peculiar velocity is calculated by multiplying the SR Doppler shift of the peculiar velocity component (localized to the object's comoving frame) and the GR cosmological redshift attributable to that same comoving frame's Hubble recession velocity component.
    Last edited: Mar 2, 2010
  24. Mar 2, 2010 #23


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    I think it is accepted today that one should apply the 'paradigm' that yields the simplest (correct) solution. I think the jury is still out as to whether your version of the 'Doppler solution' is correct, but I'll leave that to the advisors...

    I'm not competent to evaluate all 5 of the points you made, but just want to clarify this one:
    I know of no possible coordinate system in curved (GR) spacetime where the clocks of radially separated comoving (free-falling) observers 'run at the same rate'. If you bring in gravitational time dilation, you are forced to consider things according to Peacock's diatribe eq. 16:


    which includes both velocity- and gravitational time dilation. You also have to use the correct relative velocity for the chosen scenario, v/c = tanh(H*d), as Ich has pointed out.

    Anyway, this may be irrelevant, because that's not what you have used. IMO, your 'no-time dilation' solution is not a Doppler shift solution, but rather a round-about way to arrive at the 'expanding space' solution, as Chalnoth has explicitly shown in https://www.physicsforums.com/showpost.php?p=2558042&postcount=3".

    What you calculate is effectively:

    1 + z = \left(1 + \frac{da}{a_1}\right)\left(1 + \frac{da}{a_2}\right) ... \left(1 + \frac{da}{a_t}\right) = \frac{a_t}{a_1}

    This looks more like multiplying infinitesimal expansions than multiplying infinitesimal redshifts.

    Nevertheless, the equality [itex]1 + H dt = 1 + da/a [/itex] is quite interesting and does demystify the 'expanding space' influence on wavelengths somewhat. Maybe you (and Chalnoth) have shown us a new way of explaining the subject. :)
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  25. Mar 3, 2010 #24


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    I think I have spotted an important issue here. Peacock's diatribe equation #16 can be written as: (more fully and more conveniently for this purpose)

    1+z=\frac{1+v/c}{\sqrt{1-v^2/c^2}}\, \sqrt{1-2GM/(rc^2)} = \frac{1+v/c}{\sqrt{1-v^2/c^2}}\, \sqrt{1-\Omega_m H_0^2r^2/(2c^2)}

    The velocity time dilation and the gravitational time dilation factors will cancel out if [itex]
    v^2 = \Omega_m H_0^2r^2/2[/itex]. This is the condition for escape velocity, which leaves #16 as simply: [itex]1+z=1+v/c[/itex]. This is essentially what you are using - the low-speed, Newtonian Doppler shift for flat space.

    The issue is: does a cancellation of gravitational and velocity effects at escape velocity justify leaving them both out in general? I think not.
    Last edited: Mar 4, 2010
  26. Mar 3, 2010 #25
    'Ocam's razor' is a useful guideline, but not the only one. A paradigm also must be able to provide principled predictions about behaviors that are not otherwise explained. The 'expanding space' paradigm can be made to mimic behaviors already predicted by the 'kinematic' paradigm, but I am not aware of any unique predictions it has generated. In any event, I'm not going to spend much time debating which paradigm is more correct.
    Jorrie, I know you are very competent to offer analysis of these individual points if you wanted to make the effort. I'll (perhaps baselessly) speculate that you don't because you're fond of the 'expanding space' paradigm and don't want to lend credence to the other camp. Do you at least consider yourself competent to comment on the simple math in my point #1? (which is sourced directly from Peacock p.88.)
    Jorrie, I have to say I'm startled by this statement. One point that the physics literature seems to agree upon without controversy is that that the clocks of fundamental comovers run at the same rate in FRW coordinates. E.g.:

    Peebles textbook p. 59: "This means each observer sees that the clocks of all the neighboring observers are synchronized with the observer's own clock. The cosmological principle says this construction is always possible, for isotropy allows synchronization of neighboring clocks, and homogeneity carries the synchronization through all space."

    Peacock textbook p. 67: "COSMOLOGICAL TIME: The first point to note is that something suspiciously like a universal time exists in an isotropic universe. ... We can define a global time coordinate t, which is the time measured by the clocks of these observers - i.e. t is the proper time measured by the observer at rest with respect to the local matter distribution. The coordinate is useful globally rather than locally because the clocks can be synchronized by the exchange of light signals between observers, who agree to set their clocks to a stand time when e.g. the universal homogeneous density reaches some given value."

    Harrison textbook p. 139: "Homogeneity of the universe also means that all clocks in the universe, apart from local irregularities, agree in their intervals of time. Suppose our imaginary explorer rushes around the universe adjusting clocks everywhere to show a common time. On subsequent tours she finds the clocks all running in synchronism and showing the same time. This universal time is known as cosmic time."

    Davis and Lineweaver p.80: "Moreover, if you consider the proper time of fundamental (comoving) observers, T′, as your constant time surface, dT′ = 0, then the Milne universe is homogeneous. That is, fundamental observers all measure the same density at the same proper time. This is exactly the choice made in FRW coordinates. The time coordinate, t, is chosen to be the proper time of comoving observers. When this choice is made the universe is homogeneous along a surface of constant t ..."
    First, Peacock's 'Diatribe' is neither a textbook nor a peer-reviewed paper, so it isn't a good source for an equation on this Forum. But it is widely referenced.

    Second, you seem to be missing the point of his eq 16: The velocity shift is a redshift, and the gravitational shift is a blueshift, so the two factors tend to negate each other. Indeed, his equation can be valid only if the SR time dilation and gravitational time dilation are exactly equal with opposite signs, which results in the emitter and observer sharing the same cosmological time. I haven't thought about this subject for a while but I recall from previous work that one should apply the Schwarzschild metric to this scenario so that both SR and gravitational effects can be applied discretely. When the external Schwarzschild metric is applied, at escape velocity the SR Lorentz contraction exactly offsets the gravitational spatial curvature, resulting in locally flat space. When calculating the net time dilation between the emitter and the origin, the internal Schwarzschild metric should be applied, and the SR and gravitational time dilation do not fully offset each other, resulting in net 'global' time dilation. I haven't tried the math to compare the Schwarzschild redshift result with the standard FRW cosmological redshift formula. Maybe you can do that and tell us what you find.

    I'd also point out that Peacock's eq 16 obviously doesn't work in FRW coordinates in an empty Milne universe, where there is no gravitational shift to offset an SR time dilation factor. In this respect, Davis & Lineweaver comment (p.65): "Peacock (1999) claims that using the special relativistic Doppler formula to calculate recession velocity from large cosmological redshifts, although generally incorrect, is appropriate in the case of an empty universe. We maintain it is not appropriate, even in the empty FRW universe." They go on to say that Peacocks' claim is correct, however, for an empty universe in Minkowski coordinates. I think their distinction between the two coordinate systems on this point is exactly right. Peacock's formula works fine at zero gravity in Minkowski coordinates, although of course in that case the eq 16 reduces to just the standard 'global' SR time dilation formula, Peacock's eq 15.

    I'll also repeat that I never said gravity plays no role in the cosmological redshift. What I said is that in FRW coordinates, the contribution of gravity is already embedded indirectly in the accumulated Doppler shift equation, so inserting a separate gravitational 'adjustment' would calculate the wrong answer. This is easy to verify with your spreadsheet. Please re-read the Gravity section of my OP in this thread. My perspective on this point is consistent with the Bunn & Hogg paper, which explains that gravity is not a separate factor in FRW coordinates, but it can be if different non-FRW coordinates are used.
    What? I brought that equation to the discussion, not Ich, (in my post Feb5-10, 03:56 AM) and you can see there that I said: " Since [tex]\omega[/tex] = 1 at the Hubble radius..." In FRW coordinates at the Hubble radius, by the definition of that term, H*D = 1 relative to an observer at the origin.

    Anyway, we seem to agree this is the correct formula for converting between FRW and Minkowski velocity parameters.
    Yes, but you are missing half of the point. The equation does two things at once: First, it is in the form of the classical Doppler shift equation for a moving emitter and stationary observer, (1+v/c). Since dt = dr in units of c=1, Htdt = Htdr, and Htdr is the Hubble velocity H*D measured across the length of an individual local segment of the worldline. So it is explicitly portraying an accumulation of discrete segment Doppler shifts. Second, you are correct that the equation also implicitly implements the 'expanding space' paradigm, which it must equal in order to be valid. By doing both things at once, the equation proves irrefutably that in FRW coordinates, the accumulation of classical Doppler shifts exactly equals the standard cosmological redshift formula.
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