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Why expanding space ?by Peter Watkins
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#55
Mar3109, 05:29 PM

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#56
Apr109, 03:31 AM

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Flat speed now: 0.999969 cosmological speed now: 5.545 Flat distance now: 13.69958 GLY cosmological distance now: 76 GLY If you're going to stop the expansion now (cosmological time), the flat distance (now the only sensible one) would be 1754 GLY. 


#57
Apr109, 07:21 AM

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Specificly, here's the procedure: I start with the FRW metric ds² = dt²  a(t)²dr². Depending on the details of the spacetime and the transformation I use, the other two space dimensions deviate from flat space in second order. that doesn't bother me, I'm after first order effects only. Now, at a specific epoch t0, I can linearize the funktion a(t) by setting a(t)=const. * (tt0'), where [tex](t_0t_0')=1/H_0=a/ \dot a[/tex] and the constant ensures that a(t0)=1. Now that a(t) is linear, I can get rid of it by the same transformations that bring the empty FRW coordinates to Minkowski coordinates, i.e. [tex]t_{FRW} = \sqrt{t_{mink}^2  x^2}[/tex] [tex]r = 1/H_0 \tanh^{1}(x/t_{mink})[/tex] In these standard coordinates, comoving observers have the claimed velocities. That works because these velocities are proportional to [tex]\dot a[/tex] and independent of [tex]\ddot a[/tex]. They are not a curvature effect. 


#58
Apr109, 02:04 PM

P: 87

and that [tex]\dot a[/tex] is always independent of curvature effects. That assumption is quite wrong. The point is that the affine connection is curved in general. This means that the Riemann tensor is nonzero, but also that there are curvature effects via nonzero connection coefficients. For a flat connection, the nonzero connection coefficients can be eliminated via a suitable coordinate transformation. That cannot be done (globally) in a curved manifold. To illustrate this point; for a flat FRW model the timedependent connection coefficients are proportional to [tex]\dot a[/tex]. The only difference between the line elements of Minkowski spacetime on the one hand and of the flat FRW model on the other (using standard coordinates), is the presence of a timedependent scale factor. Yet the latter line element yields nonzero connection coefficients proportional to [tex]\dot a[/tex]. The coordinate systems used are the same, so the nonzero connection coefficients cannot be blamed on a coordinate effect in flat spacetime. The only reasonable explanation is that the nonzero connection coefficients (and thus [tex]\dot a[/tex]) comes from curvature. This means that your assumption that [tex]\dot a[/tex] is independent of curvature effects is incorrect. If you do not agree with this, we should agree to disagree. I see that I wrote somewhere that the interpretation of spectral shifts in a nonempty, open FRW model can always be interpreted as motion in flat spacetime for small times/distances. In light of my subsequent posts this view would be wrong  for a nonempty open FRW model the spectral shift should be interpreted as a mix of curvature effects and velocity in flat spacetime. Also some comments on the initialvalue problem for open FRW models were a bit misleading. Otherwise, what I have written in this discussion should be reasonably correct (except some minor nitpicks). Anyway, since it is now quite clear at what points we disagree, we should round off this discussion. By the way, it's All April Fool's Day today. Do you consider yourself fooled? 


#59
Apr109, 03:27 PM

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I suspect you are mixing up definitions of H. In conventional GR based cosmology, H can be defined as (da/dt)/a where a is the scale factor (wrt to present) and t is a proper time coordinate. H~1/t is the nice simple solution you get for an empty universe using GR. In SR, you can't define H in terms of a scale factor, because SR doesn't have expansion of space. However, you can assume that particles with recession velocity v are at a distance proportional to v. That is, we all started out close together and have been moving apart at constant velocities. The particle with redshift z has recession velocity [(1+z)^21]/[(1+z)^2+1].c, and it is at distance given by recession velocity by some constant time T by its recession velocity, on the assumption that everything started out from close together; that is, v is proportional to distance. H can be defined as the relationship between distance and v. Now of course, under this assumption, the value of "H" for an observer at different times is proportional to 1/t. We can't test that, because we can't take observations billions of years apart in time. However  and THIS is the key point you seem to be missing  H is defined here as a common feature of all observations, no matter how distant they may be. In GR the function H is a function of proper time, and so when you look into deep space you are seeing things when H was larger. Given information about time between events in deep space (SN data, for example) you can put strong constraints on the development of the scale factor over time. That is, there is a function from z to age, and from age to the scale factor, and from that to a value for H which was in play at the time the photon left whatever we are observing. But in the SR model, H is a description of the observation, and it is identical for every particle we observe. When we look at distant particles, we are looking back in time, but the H is the same for all those particles. THAT's what is meant by constant H, I am pretty sure. Davis and Lineweaver is excellent as an educational tutorial, helping to clear up all kinds of common popular misconceptions. It's perfectly normal to think they've done something wrong; and this is precisely because they tackle popular and entrenched misconceptions. If you think that they have made a mistake, you are probably in a good position to be learning something. Cheers  Sylas 


#60
Apr209, 06:52 AM

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But you should know also that, in suitable coordinates, spacetime can be, locally and to first order, approximated by flat minkowski spacetime with zero connection coefficients (to be sure: first order). You simply have to find the correct local tranformation, and then show that lines of constant r have the appropriate velocity in these coordinates. That's what I have done, maybe you should try also. And what does "flat FRW model" mean? The empty one? One with flat space? Anyway, it was fun. cheers Ich 


#61
Apr209, 07:27 AM

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Maybe you want to read what Ned Wright has to say, or you want to convince yourself. Start with FRW coordinates (a(T)=T, T0=age of the universe) [tex]ds^2=dT^2T^2dr^2[/tex] and apply the transformations [tex] T = \sqrt{t^2  x^2} [/tex] [tex] r = T_0 \tanh^{1}(x/t) [/tex] You'll get [tex]ds^2=dt^2dx^2[/tex] and you can perform the necessary calculations (redshift, luminositiy distance, angular size distance...) purely in SR. Pease understand that I'm not trying to sell a pet theory of mine. Davis&Lineweavers' analysis contradicts textbook wisdom, you can convince yourself if you're familiar with th idea of a metric, you can read what other authorities in the field have to say. Or you can take the fact that even Old Smuggler, who disagrees generally with everything I say, agrees with me as evidence with the status of a mathematical proof. Really, I'm not doing original research here, that chapter is simply wrong. 


#62
Apr209, 07:43 AM

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P: 1,750

The SR model described in Davis and Lineweaver is the model obtained by taking redshift as due to motions in a simple nonexpanding space, and calculated as Doppler shift. That's DIFFERENT from the FRW solution with an empty universe. There's no error in the Davis and Lineweaver paper on this point, because they are quite clear on what they mean by SR model. It's not just taking an FRW solution and applying SR. It's taking redshift as being a Doppler shift in nonexpanding space. The luminosity distance with z arising from Doppler shifts for particles receding with at uniform velocity from a common origin event is different from that in the empty FRW model. Cheers  Sylas 


#63
Apr209, 08:02 AM

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I hope I can contribute here. I think you (sylas and ich) are both basically right.
The empty FRW universe is indeed only 'ruled out' at 3 sigma, but as sylas suggests this is not the model D&L mean by saying 'SR model', they are referring to a particular assumption, valid at low redshift, that gives a bogus result at high redshift. The point that leads to disagreement is actually a bit subtle. In post #61 ich makes a conformal tranformation between the empty FRW metric and a Minkowski like metric. This is all well and good, however this is only valid radially. If you put the angular terms back into the first line you will see that your transformation does not return a fully conformally Minkowski metric. This means that you cannot use this to determine either the angular diameter or luminosities distances. You need to do a more complex fully conformal transformation in order to do this. Some technical details of this can be found here. I*think* that the error in the SR model the D&L discuss is that if you work through the details, you can see that that way we define distance in the SR model violates simultaneity, which is why it is okay for small distances but gets worse and worse the further you go. So yes, a *correct* SR model is identical to an empty FRW universe and to work out the relationship between the FRW coordinates and the coordinates of this model you need to do the fully conformal transfomation, but D&L are talking about a model that, due to the misidentification of the meaning of coordinates, is only a low redshift approximation. I hope that helps! 


#64
Apr209, 08:07 AM

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Now this is your claim, please back it up with calculations. You are probably in a good position to be learning something. 


#65
Apr209, 08:26 AM

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I really think the points of agreement are much more than those of disagreement here, stemming from maybe some loose terminology. I think we can all agree that the 23 sigma model from D&L is not a 'correct' SR model. The disagreement appears to be just how incorrect it is, yes? 


#66
Apr209, 08:28 AM

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What I'm doing here is an exact coordinate transformation. The angular directions (hyperbolic to flat space) transform correctly, no need to bend the laws of physics. We're talking about a flat spacetime in both cases. 


#67
Apr209, 08:36 AM

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Alright, I don't want to introduce additional disagreement. As you say, minkowski space and an empty FRW metric are both flat spacetimes (they have a vanishing Ricci scalar). You can transform between these two coordinate systems, without being forced to be vaild only to a given order, via a fully conformal transformation.
I get what you are saying, any coordinate transformation is exact, so if your original spacetime is flat the transformed one is as well. Just pointing out that the one you suggest doesn't work, on it's own to relate FRW comoving coordinates to their Minkowski counterparts. Clearly you agree with this point, it just wasn't clear to me what you were demonstrating with it original, but now I see. 


#68
Apr209, 08:36 AM

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#69
Apr209, 02:59 PM

P: 87

For the benefit of readers who may possibly fall for the misunderstandings Ich seems to be promoting,
I will contribute with one last post in this discussion. may be included into [tex] \dot a [/tex](t_0) (as is the case, in general). This simple misunderstanding may be appropriately called "the Bunn/Hogg fallacy", and you have endorsed it. sections. The nonzero connection coefficients are proportional to [tex]\dot a [/tex], as usual. But here, since we cannot perform any relevant coordinate transformation in order to change the connection coefficients (the coordinates already have the standard form), the correct flat spacetime approximation is to neglect [tex]\dot a [/tex] altogether. On the other hand, for a nonempty, open FRW model where the line element is expressed in comoving coordinates, a coordinate transformation to standard coordinates will change the connection coefficients, but not get rid of them altogether. What is left should be due to curvature and must be neglected in the correct flat spacetime approximation. It is only for the empty FRW model a coordinate transformation from comoving to standard coordinates can completely get rid of all the connection coefficients. On the other hand, approximating a(t) as a Taylor series to first order the way you do, is effectively to include all the crucial effects of the connection coefficients (expressed in comoving coordinates) at the time t_0, since the relevant connection coefficients expressed in comoving coordinates are always proportional to [tex] \dot a [/tex]. After making a local transformation to standard coordinates, the resulting nonzero velocity field is then just an expression of the fact that the connection coefficients (expressed in comoving coordinates) at the time t_0, are proportional to [tex] \dot a [/tex](t_0). You have absolutely no guarantee that these connection coefficients do not include some effects of curvature so that this procedure yields the correct flat spacetime approximation for the issue we were discussing. In fact, it fails. the coordinate system covers the relevant part of the manifold. That is basic differential geometry. You should try to learn it some time. That concludes all I have to say in this discussion. You are on your own now. Good luck. 


#70
Apr209, 05:07 PM

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P: 1,750

In any case, I'll go away and try my own analysis, and report back. Cheers  Sylas 


#71
Apr309, 02:05 AM

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I'm not an expert in GR; I can solve the differential equations for scale factor and energy density which are used in the FRW models; but I can't derive the equations themselves. In any case, I didn't need any of that, because the issue is simply the SR model. The SR model corresponds to a realistic situation that could, in principle, be set up and tested right now, and SR is the appropriate way to analyze it. Take a large collection of particles, and at a point in time, have them all start moving at constant velocity from a common point. (An explosion in space.) After elapsed time t, an observer on one of the particles makes observations of all the others. Consider a signal received by one exploding particle from another, and compare with the signal from another equivalent particle at the same distance, but with no velocity difference. The signal received from the moving particle is weaker by a characteristic amount. The factors to consider are
But that is precisely the relation for all the FRW models, empty or otherwise. Davis and Lineweaver, in their section 4.2, used a factor of (1+z) for the socalled SR model, which can only be seen as an error. There are still differences in comparing z with the apparent magnitude across the different FRW solutions, but the ratio of angular distance and luminosity distance is the same for everything. Using Ned's formulae for the empty universe, I get the angular distance as follows: [tex]D_A = \frac{c}{H_0}(1(1+z)^{2})/2[/tex] Using Lorentz transformations for the SR model I have described here, and using H_{0} as the inverse of time since the explosion, which makes sense, I get the same thing. Hence the SR model gives the same relation between z and luminosity distance as the empty FRW solution. Thanks very much. I have learned something indeed. Cheers  Sylas 


#72
Apr309, 07:49 AM

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In the next paragraph, you seem to concede this point, but then write: I showed you how to get the first order result, and you've done nothing to show where, explicitly, the procedure fails in you view. I appreciate your general, wellmeaning, and repeatedly uttered advice that I better learn basic principles of mathematics and physics, and I will certainly continue to do so with the help of this forum, but this discussion seems to lead nowhere. You didnt'd really believe that you'd have the last word, did you? 


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