The Mystery of Superluminal Recession Velocities in Cosmology

[jonmtkisco]

Hi Wallace,
I bought Hartle's textbook some months ago on your recommendation. I'm looking at Section 18.2 The Cosmological Redshift. As I read it, it seems to describe cosmological redshift only in terms of the size increase of the scale factor. I don't see any specific mention of an additional effect caused by the gravity gradient of the cosmic fluid's decreasing density over time. For example:

"In an expanding universe where a(t) grows with t, the ratio a(t_e) / a(t₀) will be less than 1 and the received frequency w₀ less than the emitted one w_e. That is the cosmological redshift. As the universe expands, the frequency of the photon decreases, and its wavelength increases linearly with the scale factor a(t)."

Section 9.2 describes gravitational redshift in terms of the Schwarzschild metric, but not in terms of the FLRW metric.

Can you please point me to Hartle's description of how the temporal gravity gradient of an FLRW cosmic fluid affects the cosmological redshift?

Jon

 

[Wallace]

Okay, looking back I somewhat misread your comment. I was merely stating that it is clearly the change in time of the metric that causes redshfit in the FLRW line element. Since the metric encodes the effects of gravity (the metric IS gravity if it is a valid solution to the field equations) then in effect the metric already encodes the physical effects you mention. I don't think it is as direct as you state, at least not a term in the equation you can point to, but the effect is in there in the form of a(t).

 

[yuiop]

Hi all,

I have given the subject some further thought and research and concede the point by Wallace and others that causes of redshift are also causes of time dilation which ever model you choose. The various models only differ in the assumed recession velocity of the galaxies or supernovae at the time of emmision and in distances at the time of emmision calculated from flux and luminosity measurements.

The red tear drop curves in the attached diagram are the light paths in co-moving model with gravity assuming (t/to)^(2/3 )with Ωo = 1. They are plotted using the equation (derived by George in the Relativity forum) of x = n(t^(1-1/n)-t) with a value of 3 for n giving x=3(t^(2/3)-t). This is an exact fit for the curve given by Ned Wright here http://www.astro.ucla.edu/~wright/cosmo200.gif

The blue tear drop curves in the attached diagram are the light paths in a low density model with negligable density which is basically the the match for the curve given by Ned Wright here http://www.astro.ucla.edu/~wright/omega0.gif and is plotted using a value of n=1000 in the x = n(t^(1-1/n)-t) equation. There is not much difference between the curves for n=1000 and n approaching infinity.

The interesting part is that both models satisfy the requirement that (z+1) = (to/te) and both satisfy the (z+1) = time dilation factor. In the diagram for z=1.7 the time of emmission te is represented by tA and is numericaly equal to 0.3707 and when multiplied by (z+1) =2.7 this gives a value of 1 which is where t0 is situated on the diagram. At time tA the velocity of the supernovae at the time of emmision in the low density model is about 0.99c while in the mass dominated model (t/to)^(2/3) the velocity at the time of emmision is 0.85c. Note that in neither model is the velocity equal to z.

The really interesting part of the diagram is that if you look at the middle blue and red teardrop curves terminating at epoch t1, the curves cross over at about z=0.5. The distances at the time of emmision in the low density model are further away (and darker) below z=0.5 and nearer (and brighter) at redshifts above z=0.5 than would be expected for the matter dominated model. This is basically the observation that is at the root of the conclusion that the rate of expansion is accelerating. The low density model seems to produce the same brightness anomally relative to the assumed (t/to)^(2/3) model without requiring the rate of expansion to be accelerating.

To fill in some more of the details of the diagram, the coloured curved lines going out to the right (labelled vs) are the trajectories of galaxies in the expanding models. The straight black lines on the right is the geometry of a model without gravity, relativity or expanding space where light travels in straight lines and is not representative of anything physical but gives something to compare the physical models with. The lines in the green section on the left is the equivalent special relativity model of the low density model at z=1.7

 

[Wallace]

Your on the right track, except that an empty Universe does not look the same as an accelerating one. Everytime the Supernovae data is analysed this is examined and it simply doesn't fit the data points. It might look similar to an accelerating model, but not similar enough to fit the data nearly so well.

 

[jonmtkisco]

Hi Wallace,

Since the metric encodes the effects of gravity (the metric IS gravity if it is a valid solution to the field equations) then in effect the metric already encodes the physical effects you mention. I don't think it is as direct as you state, at least not a term in the equation you can point to, but the effect is in there in the form of a(t).

To be fair, the FLRW metric for a flat matter-only universe encodes only that it is physically stretching at the escape velocity of its contents. So in that very basic sense, gravity is encoded in a(t).

But the off-the-wall idea here is that there is an additional component of gravitational redshift beyond that attributable directly to the stretching of the scale factor. A factor which reflects the experience of a photon moving through regions that are not only stretched but also are characterized by a progressively diminishing gravity well in the cosmic fluid.

I don't see any reference to that in Hartle or Peebles, or in pages 367-68 of Hobson, Efstathiou & Lasenby (2005).

Jon

 

[Wallace]

So you are proposing that there is an additional effect due to gravity that is not described by General Relativity? In that case then I agree, it is an off the wall idea and not surprisingly doesn't appear in Hobson, Peebles etc.

 

[jonmtkisco]

Hi Wallace,
The hypothetical effect I'm asking about is based on GR and as far as I can tell would be completely consistent with it. It's nothing more than identifying a (possibly overlooked?) situation where plain old gravitational redshift could occur, and applying standard GR and the FLRW metric to calculate the answer.

I don't see how it's any more radical than, say, the way the Lewis & Francis paper on Radar Ranging innovatively applies Gauss' Law to explain the "overshoot" of a returning radar signal. The authors don't attribute that idea to another author.

If you believe that the hypothetical application of GR I'm asking about is fundamentally inconsistent with GR, I'd appreciate if you could explain why.

Jon

 

[Wallace]

Because the metric already encodes everything that GR has to say about gravity! Redshift (and this is talking about the effect due to the homogeneous universe only) is determined by the difference in scale factor at the cosmic time of emission and reception. All effects of gravity the GR know about have already been included in this, you can't add any other effects and still be consistent with GR. This has not been 'overlooked', it is already included, you can't add it in twice.

The paper you mention makes a Newtonian analogy to explain why the effects of GR are maybe not as strange as might first be thought in the situation being examined. By thinking about the qualitative behavior you would get by using Newtonian gravity, understanding the GR result becomes easier. They certainly are not suggesting that you actually use Gauss's law to add additional effects in that have been 'overlooked'. The numbers are all crunched simply from the metric and the geodesic equation, Guass's law is invoked only as a guide to understanding, not used to calculate the numbers since although the qualitative answer would be the same (i,e. whether the rocker over or undershoots or makes a symmetric journey), the specific numbers would be wrong (i.e the Newtonian 'Gauss Law' results differs from the GR) if the journey went to a reasonable cosmological distance. To see an example of this see Barnes et al 'Joining the Hubble Flow' where the GR and Newtonian curves for one tethered galaxy model are shown. The behavior has the same form but the specifics are different.

If you want to think about cosmological redshift being produced in the manner you describe then that's fine, I think it's a reasonable mental picture. My original comment made the point that your description was a reasonable description of what the GR equations tell you. This effect however is already included in those equations and can't be double counted by adding it in again.

 

[jonmtkisco]

Hi Wallace,
Hmmm. OK, thanks for the explanation.

It just seems to me that if you consider an SR recession of two rockets away from each other in an almost-empty FLRW universe, and the rockets use their motors to maintain their recession speed (relative to each other) as a function of time such that it exactly duplicates the (subluminal) relative motion of two comoving particles in an FLRW Omega(m)=1 universe, you'd get the same Doppler redshift over time as you would get between the two particles in the FLRW Omega(m)=1 universe at each respective proper distance. Maybe that's not true.

Jon

 

[Wallace]

I'm pretty sure that's not true, but it would be interesting to see what the results would be. In the end though, to know for sure there's only one way, and that is to do the calculation. I think you're understanding this stuff pretty well, but as I've said before, to get to the next level you really need to start playing with the equations. Analogies and concepts will only get you so far and will lead you astray if you try and push them too far, which I think you are doing.

Try and re-create some of the results in the papers about this stuff, most of them go through from the basics and if you have Hartle that should give you all you need that isn't covered. I'd be interested to know the results from the thought experiment you mention above, but without cranking the handle we're just punching smoke to try and work out the answer just from 'mental pictures' alone. It's always better to know the result from an exact calculation, and then try and understand in simpler terms what the results are telling you. That's pretty much the format of most of the work on this. If you want to push something to an area not explicitly covered already then unfortunately I don't think there are any shortcuts to getting there.

 

[yuiop]

...
It just seems to me that if you consider an SR recession of two rockets away from each other in an almost-empty FLRW universe, and the rockets use their motors to maintain their recession speed (relative to each other) as a function of time such that it exactly duplicates the (subluminal) relative motion of two comoving particles in an FLRW Omega(m)=1 universe, you'd get the same Doppler redshift over time as you would get between the two particles in the FLRW Omega(m)=1 universe at each respective proper distance. Maybe that's not true.

Jon

I'm pretty sure that's not true, but it would be interesting to see what the results would be. In the end though, to know for sure there's only one way, and that is to do the calculation. I think you're understanding this stuff pretty well, but as I've said before, to get to the next level you really need to start playing with the equations. Analogies and concepts will only get you so far and will lead you astray if you try and push them too far, which I think you are doing.

Try and re-create some of the results in the papers about this stuff, most of them go through from the basics and if you have Hartle that should give you all you need that isn't covered. I'd be interested to know the results from the thought experiment you mention above, but without cranking the handle we're just punching smoke to try and work out the answer just from 'mental pictures' alone. It's always better to know the result from an exact calculation, and then try and understand in simpler terms what the results are telling you. That's pretty much the format of most of the work on this. If you want to push something to an area not explicitly covered already then unfortunately I don't think there are any shortcuts to getting there.

I found this interesting question:
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Optional Problem 10: “Minkowski space in disguise” (hard!): Show by a clever choice of coordinates that the FRW metric with ΩΛ =Ωm =Ωγ =0, Ωk = 1 (this is the special case with a(t)= t, k = −1, corresponding to an empty and maximally open universe) is simply the Minkowski metric in disguise.
------------------------------------------------------------------------------

in this problem set http://ocw.mit.edu/NR/rdonlyres/4973019C-C55D-43D0-81AF-6FDA7D4444C7/0/ps8.pdf that seems to be related to the issue you are discussing. Anyone able to answer that question?

 

[jonmtkisco]

Hi Wallace,

Since the metric encodes the effects of gravity (the metric IS gravity if it is a valid solution to the field equations) then in effect the metric already encodes the physical effects you mention. I don't think it is as direct as you state, at least not a term in the equation you can point to, but the effect is in there in the form of a(t).

I'd like to get a more specific understanding of what you said here.

Is, or is not, the normal GR cosmological redshift calculation, proportional to a(t), divisible into two distinct, directly additive components: (1) the gravitational redshift proportional to the temporal change in cosmic density from emission event to reception event, and (2) SR Doppler redshift proportional to recession velocity as between the emitter and receiver? This question assumes that the gravitational redshift component is calculated using only the standard form of the GR gravitational redshift equation in the Schwarzschild metric, and the Doppler redshift component is calculated using only the standard SR Doppler redshift equation.

Jon

 

[yuiop]

Hi Wallace,

I'd like to get a more specific understanding of what you said here.

Is, or is not, the normal GR cosmological redshift calculation, proportional to a(t), divisible into two distinct, directly additive components: (1) the gravitational redshift proportional to the temporal change in cosmic density from emission event to reception event, and (2) SR Doppler redshift proportional to recession velocity as between the emitter and receiver? This question assumes that the gravitational redshift component is calculated using only the standard form of the GR gravitational redshift equation in the Schwarzschild metric, and the Doppler redshift component is calculated using only the standard SR Doppler redshift equation.

Jon

Hi Jon,
Hope you don't mind me jumping in and adding my 2 cents worth. As I understand it the redshift is due to the "expansion of space" stretching the wavelength of the photon during its journey and a component that is due to the motion of the emitter away from the receiver during the interval that the photon is emitted which is equivalent to classic doppler red shift without the SR time dilation component. There is no SR time dilation because the source is stationary with respect to the local space. However, you can swap coordinate systems and treat the space as not expanding so there is no stretching of the wavelength during its travels and then SR time dilation does apply because in this coordinate system (conformal) the source is moving relative to the local space (which remains static) and the doppler shift is calculated as per SR doppler shift. The equivalence of the SR conformal system with the co-movingsystem is illustrated by Ned Wright here http://www.astro.ucla.edu/~wright/cosmo_02.htm The end result is the same, but both of the systems Ned Wright is using do not have the the effect of gravity included, which has to factored in in a non empty universe.

 

[Wallace]

Hi Wallace,

I'd like to get a more specific understanding of what you said here.

Is, or is not, the normal GR cosmological redshift calculation, proportional to a(t), divisible into two distinct, directly additive components: (1) the gravitational redshift proportional to the temporal change in cosmic density from emission event to reception event, and (2) SR Doppler redshift proportional to recession velocity as between the emitter and receiver? This question assumes that the gravitational redshift component is calculated using only the standard form of the GR gravitational redshift equation in the Schwarzschild metric, and the Doppler redshift component is calculated using only the standard SR Doppler redshift equation.

Jon

No, as Kev referred to, if you use the FRW co-ordinates then a(t) gives you the redshift. That's it. You can slice up how a(t) came to be what it is if you like, but the effects you refer to are already in there. In these co-ordinates co-moving particles are at rest, so there is no Doppler shift calculation. As we know, this physical effect of Doppler shift has already been included by the choice of co-ordinates.

This is one of the many reasons why FRW co-ordinate are so useful. You just have to remember not to take phrases like 'the expansion of space' literally!

By the way, I'm not sure that you recover the same redshift as FRW if you use an SR doppler shift with an added gravitational redshift using a Schwarzschild metric? Maybe you do, I haven't seen this demonstrated so neatly though. I know there have been arguments that in principle redshift can be divided up this way but again, while this is a reasonable qualitative argument, I don't know that you get exactly the same answer. GR is non linear so you get to trouble trying to linearly add things if you're not careful. Have you seen anyone show this relationship explicitly? I'd be interested to see if this had been looked at in detail.

 

[jonmtkisco]

Hi Wallace,
Yes as I went back and reread some papers I realized that I must have originally picked up the idea from Alan Whiting's 4/04 http://arxiv.org/abs/astro-ph/0404095" . He takes a couple of different approaches to dividing the cosmic redshift into its component parts.

He applies a sort of Newtonian Shell Theorem approach to calculating the gravitational redshift component, and combining it with SR Doppler redshift he recovers the cosmic redshift formula in a backhanded way but says it is a special case for a flat FRW universe with Lambda=0 only.
His Shell Theorem approach seems to measure the difference between the total matter density now and zero matter density now.

As far as I can tell he does NOT approach it specifically from the perspective of the temporal/spatial density gradient from Event E to Event O, which it seems to me is more likely to be fruitful.

Jon

 

[Wallace]

Whitings formula is, as I say, only approximate. It gives you back a similar behavior but it quantitatively different from the GR approach. As I said earlier, this was shown in Barnes et al "Joining the Hubble Flow". It works on the level of conceptual understanding.

You asked a specific question, whether SR doppler shift + a gravitational redshift calculated with the Schwarzschild metric gives you the same result as FRW. I don't think this is the case, but Whitings Newtonian result will certainly differ from both FRW and this approach you describe.

I'm not saying Whiting was 'wrong', his paper was intended to demonstrate a point of understanding. He probably acknowledges somewhere in there that his formulas would surely break down at some point. You need to separate quantitative calculations from lines of argument intended to give a greater intuiative qualitative understanding.

 

[jonmtkisco]

He Wallace,

... Whitings Newtonian result will certainly differ from both FRW and this approach you describe.

Well I already suggested that Whiting's result doesn't seem fruitful... He compares total density NOW to zero density NOW. That has nothing directly to do with the past worldline of a photon emitted long ago. I made it clear that I was not saying his result was "correct." Why would it be?

What I suggested is that my approach to attacking the problem makes more sense than his. Therefore by definition it yields a different result.

Previously you scolded me for not combining quantitative calculations with intuitive lines of argument; now you scold me (without basis) for not keeping them separate.

I think MY idea might be original and it definitely is interesting enough to warrant further analysis.

Jon

 

[Wallace]

He Wallace,

Well I already suggested that Whiting's result doesn't seem fruitful... He compares total density NOW to zero density NOW. That has nothing directly to do with the past worldline of a photon emitted long ago. I made it clear that I was not saying his result was "correct." Why would it be?

Whitings 'results' are perfectly fruitful in terms of achieving what he sets out to do. He his not trying to re-derive cosmological redshift.

What I suggested is that my approach to attacking the problem makes more sense than his. Therefore by definition it yields a different result.

What is the problem?? We know perfectly well how to calculate cosmological redshift. Whiting is not trying to re-solve this problem. I'm not sure why you would even compare your suggestion to Whitings paper, which is clearly purely pedagogical.

Previously you scolded me for not combining quantitative calculations with intuitive lines of argument; now you scold me (without basis) for not keeping them separate.

What I'm stressing is that while they both need to work together to best approach a problem, in the end you need to know the difference. If you want to compare to data you need to use a proper calculation. If you want to talk about how best to teach cosmology to students you need to go beyond just saying "work out the maths". This is why you see the kind of co-ordinate transformations etc that are use in a lot of the literature on this stuff. The FRW metric is always the easiest to use to get an numerical answer, but can be a little tricky and confusing conceptually. Con-formal co-ordiantes or Newtonian analogies etc can be a useful way of re-arranging the maths in order that the equations take a more intuitive form, i.e. 'this term represents this, that term represents that'. Of course, no one (included Chodorowksi etc) is actually suggesting that these representations should actually be used by professional cosmologist when assessing data. Clearly the FRW metric is most suitable for that. It is a conceptual exercise.

I think MY idea might be original and it definitely is interesting enough to warrant further analysis.

Jon

YOUR idea is, as I said originally, a rough qualitative description of how redshift is calculated. I'm not quite sure what is particularly original about it? Indeed, what 'further analysis' do you think can be done on this idea? If you use it to motivate a different mathematical method for calculating redshift then you already know the answer that you must end up with. Possibly you can re-calculate the answer through a set of equations that doesn't come straight from the FRW metric but instead is the mathematically description of your words. If this set of equations can be shown to be equivalent to FRW then great, you've come up with a new co-ordinate transformation which might well be an interesting way of thinking about the issue. If you get a different result then the equations might be an approximation, or possibly simply wrong. Certainly it sounds as though you could describe your proposal mathematically such that you do get the right answer.

No one is going to do this analysis for you however, and if you really want to claim you have surpassed Whiting paper then you going to have to demonstrate that with something a lot more concrete than what you have. Although as I say, I think you are wildly misinterpreting the aims of conclusions of that paper.

Lets get this straight. Are you suggesting a new way of calculating redshift such that previous methods are wrong and yours will be correct? Are you instead suggesting a new way of thinking about how redshift occurs which would lead to a new form of equations that give you the correct answer? Are you suggesting a new qualitative description that isn't intended to get a correct numerical answer? None of the above? I think if you can make this clearer it would greatly aid the discussion.

 

[jonmtkisco]

Hi Wallace,

Whitings 'results' are perfectly fruitful in terms of achieving what he sets out to do. He his not trying to re-derive cosmological redshift.

Scold, scold, scold.

First you scold me for supposedly swallowing (not!) that Whiting's result is a correct derivation of cosmic redshift. Now you scold me because, although he did not succeed, you don't think I recognize that he never really tried. In lieu of more scolding, can't we just agree that my proposal suggests a way to attempt this and his proposal doesn't?

Possibly you can re-calculate the answer through a set of equations that doesn't come straight from the FRW metric but instead is the mathematically description of your words. If this set of equations can be shown to be equivalent to FRW then great, you've come up with a new co-ordinate transformation. If you get a different result then the equations might be an approximation, or possibly simply wrong. Certainly it sounds as though you could describe your proposal mathematically such that you do get the right answer.

Hurray! I might be right! Well if you don't want to be the first to find out, that's your prerogative. If you're interested in the subject matter, you might enjoy doing the calculations. It's obvious you are better at it than I am. If you're not interested, I'll struggle with it myself. If there's any merit to the idea, eventually someone will include it in a published paper. If the idea sucks, then it will end up on the scrap heap along with a lot of good company.

Jon

 

[Wallace]

Jon, I am not 'scolding' you. If you don't want feedback then why post your ideas in the first place? I love how you trawl through my whole post ignoring the meat of it to highlight one sentence where you feel that you have had some kind of victory. As if this was some kind of online debating competition. I don't get it. You will make much more progress much more quickly if you are willing to take on board advice and criticism.

 

[jonmtkisco]

Hi Wallace,
So the meat of your post was that my idea is wrong? Or was the meat of it that no idea is really worth discussing if it it isn't accompanied by a complete and accurate set of equations?

I have no interest in scoring debating points. My interest is solely in gaining a deeper understanding of the subject matter. I have greatly appreciated your substantive answers, but often they have been diminished by excessive scolding.

Alan Whiting over 4 years ago suggested at least the possibility that cosmic redshift might be cleanly separable into gravitational and Doppler components. I for one am interested in pursuing the substance of that inquiry. If you share that interest, you haven't made it very clear.

Jon

 

[Wallace]

Hi Wallace,
So the meat of your post was that my idea is wrong? Or was the meat of it that no idea is really worth discussing if it it isn't accompanied by a complete and accurate set of equations?

If you want to discuss whether you can correctly split redshifts into different components then of course this needs to be accompanied by the equations describing this split. I'm not sure why that can be construed as being a controversial statement? It either is or isn't possible. From memory I think this was actually done by Gron and Elgaroy in a paper a few years ago. Can't remember the exact reference but I think it's in there somewhere. I could be mis-remembering though.

Alan Whiting over 4 years ago suggested at least the possibility that cosmic redshift might be cleanly separable into gravitational and Doppler components. I for one am interested in pursuing the substance of that inquiry. If you share that interest, you haven't made it very clear.

Again, I think you are misinterpreting Whitings paper. He was not suggesting that the equations he derived correctly split the redshift in this way. It was only conceptual, to demonstrate the illusionary nature of 'expansion of space'. This redshift split was not really the essence of Whitings paper.

I am always interested in sensibly formulated discussions of cosmology, I'm just trying to unpick what exactly you are aiming at here, since it isn't clear. I know it's clear to you, because you wrote it, but surely asking for clarification when it isn't clear is not 'scolding'. If something isn't clear it is never entirely the fault of the author or the reader!

So anyway it sounds like you are trying to formulate a way to correctly calculate cosmological redshift with an equation that neatly splits into a motion and a gravitational part. This is an interesting question, but Whiting is not the place to start, since his Newtonian approach is guaranteed not to do this (since that is not what is was intended to do). You'd need to start from GR somewhere. As I say I think there is a Gron and Elgaroy paper that does this, maybe check that out (they have a few on this topic I think so it might take some searching).

Edit:

I found it! Turns out they only had the one paper on this stuff, not several. Anyway the paper is astro-ph/0603162 , see section IV where they do precisely what you had in mind and end up with a formula

z = z_doppler + z_grav

where the two z's are written in terms of cosmological parameters. Their derivation is valid for dust dominated Universes only, but should be able to be generalized I would think.

 

[jonmtkisco]

Hi Wallace,
Thanks for finding the Gron and Elgaroy paper. I'm a little surprised at how "conservative" their perspective is. They want to attribute cosmological redshift to the stretching of wavelength due to the expansion of space, despite pointing out the problems that have been identified with that physical description. Oh well, I'm more interested in their specific results than in how they characterize them.

G&E attribute the z = z_gravity + z_doppler formulation to Bondi (1947). They say this is an approximation which is valid only at small fractions of the Hubble radius. There must be some way to generalize the result to larger radii as you say, but apparently no one has done it yet.

I had thought that the gravitational redshift component of the equation would be "positive", in the sense that it would contribute redshift rather than blueshift. Because the past worldline of a photon from a distant source passes through regions of smoothly decreasing density over time, which, in G&E's terminology, would suggest that the photon has climbed "uphill" out of a gravitational well, i.e. redshifted. However, G&E uses the Shell Theorem/Gauss' Law approach to describe the photon's worldline as "falling downhill" due to the gravitational acceleration towards the observer at the coordinate origin, i.e. blueshifted.

I've attached a copy of a chart from Davis & Lineweaver's 10/03 paper illustrating the relative effects of SR Doppler redshift, GR cosmological redshift, and Classical Doppler redshift. It show that, relative to any given "actual" recession velocity, the SR Doppler equation calculates the highest amount of redshift. The GR cosmological redshift equation for an Omega(m)=1, Lambda=0 is blueshifted relative to the SR Doppler equation. The "empty" universe model is further blueshifted from that. And the Classical Doppler equation is the most blueshifted of the three.

It seems that the degree of redshifting in each respective scenario reflects the degree of time dilation in that scenario. Time dilation = 0 for the Classical Doppler scenario which is least redshifted.

It seems contrary to G&E's analysis that the Omega_M=1, Lambda=0 scenario is more redshifted than the empty universe scenario. In other words, adding matter to the universe causes more redshift, indicating that in the "matter" scenario the photon's past worldline has indeed climbed "uphill" rather than falling "downhill." That outcome is more consistent with my description: climbing out of a gravitational well as the temporal/spatial density decreases.

Unless this is just another one of those situations where the weird Milne model confuses the result.

Jon

 

[Wallace]

G&E attribute the z = z_gravity + z_doppler formulation to Bondi (1947). They say this is an approximation which is valid only at small fractions of the Hubble radius. There must be some way to generalize the result to larger radii as you say, but apparently no one has done it yet.

Hmm, I didn't read the fine print. I've never read the back half of that paper in much detail, just vaguely remembered that redshift split thing. This formula in that case may well be equivalent to Whitings, since the derivation of it is very Newtonian in spirit. Maybe it can be generalised to a full GR formula, but then any maybe it just doesn't work. As I said earlier, GR is non-linear so you can't always add things in the way that might seem intuitive. That's quite a sweeping statement though, it may well be possible in this case.

I had thought that the gravitational redshift component of the equation would be "positive", in the sense that it would contribute redshift rather than blueshift. Because the past worldline of a photon from a distant source passes through regions of smoothly decreasing density over time, which, in G&E's terminology, would suggest that the photon has climbed "uphill" out of a gravitational well, i.e. redshifted. However, G&E uses the Shell Theorem/Gauss' Law approach to describe the photon's worldline as "falling downhill" due to the gravitational acceleration towards the observer at the coordinate origin, i.e. blueshifted.

Hmm I would have expected the gravitational component should be 'positive' or blue-shifting. Consider a static space with a varying density. The redshift of a photon depends on the difference in potential between two places which will be an integral over the gravitational force due to the density. There is no direct relationship between the density at two places and the redshift in a static space, so I don't know that this would be expected in a temporal sense either? Obviously in familiar potential wells like Galaxies the density does drop with radius so the 'mover to lower density get redshifted' rule of thumb works, but if you imagine a static potential well that for some reason got more dense with radius the relationship wouldn't hold. What would stay the same is that the photon gets redshifted by climbing out of the well.

I've attached a copy of a chart from Davis & Lineweaver's 10/03 paper illustrating the relative effects of SR Doppler redshift, GR cosmological redshift, and Classical Doppler redshift. It show that, relative to any given "actual" recession velocity, the SR Doppler equation calculates the highest amount of redshift. The GR cosmological redshift equation for an Omega(m)=1, Lambda=0 is blueshifted relative to the SR Doppler equation. The "empty" universe model is further blueshifted from that. And the Classical Doppler equation is the most blueshifted of the three.

It seems that the degree of redshifting in each respective scenario reflects the degree of time dilation in that scenario. Time dilation = 0 for the Classical Doppler scenario which is least redshifted.

It seems contrary to G&E's analysis that the Omega_M=1, Lambda=0 scenario is more redshifted than the empty universe scenario. In other words, adding matter to the universe causes more redshift, indicating that in the "matter" scenario the photon's past worldline has indeed climbed "uphill" rather than falling "downhill." That outcome is more consistent with my description: climbing out of a gravitational well as the temporal/spatial density decreases.

Unless this is just another one of those situations where the weird Milne model confuses the result.

Jon

Something is up. I doubt either G&E or D&L made such a serious error as to give the conflicting results you are suggesting. One or other would have to be very very wrong for this to hold. I'm not sure of the context of the D&L plot, can you post the link to the paper? Without the actual caption it isn't clear what they are getting at with that plot.

 

[yuiop]

Your on the right track, except that an empty Universe does not look the same as an accelerating one. Everytime the Supernovae data is analysed this is examined and it simply doesn't fit the data points. It might look similar to an accelerating model, but not similar enough to fit the data nearly so well.

Hi again,

I have taken on board your comment that the empty model is not a good fit to the observed data (although strangely it is a better fit than the mass dominated model). I have given this some more thought and think I now have a better grasp on the matter.

Please look at the attached digrams which illustrate co-moving distances in the FRW metric and compare them with Special Relativistic subluminal distances similar to the Milne model.

In the first diagram is a hypothetical universe where the mass content of the galaxies is almost zero and provides a good simple starting point. The left part of the diagram shows the classic "tear drop" curve of light as it travels towards us against the Hubble flow. The path of three receding galaxies are shown (solid diagonal red, blue and green lines) with redshift of 2, 1 and 0.5 respectively. The dashed lines show the corresponding nominal velocities of z equal to the redshift value but this is not an actual physical velocity. The "corrected velocities" are still superluminal in this model for redshift greater than z=1.7. On the right is the same three galaxies plotted on a coordinate system that has been stretched out as described by Ned Wright here http://www.astro.ucla.edu/~wright/cosmo_03.htm by " dividing the spatial coordinate by a(t)". The left part of the diagram clearly shows the lack of gravity in the model as the galaxies follow parallel paths that remain at constant distance from each other. What is surprising about the right diagram is that despite the lack of gravity, light from distant galaxies appears to slowing down as it aproaches us, initially supeluminal and finally c locally. It is surprising because this co-moving model has no gravity! The curved light path is exactly what you would expect to see if you were deep in a greavitational well. This shows that comoving coordinates introduce a fictitious gravitational field even when there is no mass. I suspect that in trying to compensate for the fictitious gravity that is an artifact of the comoving coordinates that the cosmological constant representing repulsive gravity has to be introduced to make the comoving model math the observation that the universe is essentially flat on large scales. Anyway, the distance AD is the "co-moving distance" that is quoted in cosmological calculators and is preferred to the corresponding distance AB as it closer to distances measured as angular or luminosity distances after suitable corrections of 1/(1+z) and (1+z)^2 respectively. Because there is no gravity in this particular model the comoving distance is equal to EF and FG but that is just coincidence because the galaxies are following straight paths rather than curved paths.

In the second diagram, the left part represents subluminal motion and distances in Special Relativistic coordinates compared to the superluminal comoving distances on the right. It is easy to see that the SR distances A'B' C'D' E'F' are all considerably shorter than the comving distances AB, CD, EF in the comoving model and this is one reason that the comoving model is preferred over the SR model because the comoving distances are closer to what is observered. However, the SR model should not be dismissed so lightly. The third diagram shows the SR coordinates with the effect of rapid inflation in the early epoch of the universe added in. The inflation is represented by the flat truncated base of the universal light cone. Conventional time as measured by the decay of radioactive atoms and the formation of large scale structures starts at time O and not at time O', because matter is not in the form of atoms during the inflation period. The dashed circle is just there to illustrate that the distance of a galaxy receding at redshift z=1 has been set to the same value by a suitable choice of inflation radius. By including inflation the distances of galaxies with z<1 is greater (E'F'>EF) and less luminous than the empty model and the distances of galaxies with z>1 is less (A'B<AB) and more luminous than in the empty model (without inflation). This is the effect that is currently explained by an initially collapsing universe that starts to expand at an accelerating rate in more recent times. Although gravity has not been introduced into the model yet this hints that when inflation is taken into account the universe can be modeled largely using SR with some modification for gravity and GR, in a way that matches observation without requiring superluminal motion, expanding space or dark energy accelerating the expansion.

 

[jonmtkisco]

The Davis & Lineweaver http://arxiv.org/abs/astro-ph/0310808" is "Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe" 11/03.

Jon

 

[Wallace]

Hi again,

I have taken on board your comment that the empty model is not a good fit to the observed data (although strangely it is a better fit than the mass dominated model). I have given this some more thought and think I now have a better grasp on the matter.

Why is this strange? The empty universe fits better than matter dominated because in a universe with matter and dark energy the attractive and repulsive parts roughly cancel each other for a period making the Universe looks closer to the empty model. A matter only Universe only decelerates, so is a worse fit than the empty Universe.

What is surprising about the right diagram is that despite the lack of gravity, light from distant galaxies appears to slowing down as it aproaches us, initially supeluminal and finally c locally. It is surprising because this co-moving model has no gravity! The curved light path is exactly what you would expect to see if you were deep in a greavitational well. This shows that comoving coordinates introduce a fictitious gravitational field even when there is no mass. I suspect that in trying to compensate for the fictitious gravity that is an artifact of the comoving coordinates that the cosmological constant representing repulsive gravity has to be introduced to make the comoving model math the observation that the universe is essentially flat on large scales.

In a sense the FRW co-ordinates do introduce an apparent gravity, in that in GR gravity is curvature. In very rough terms the co-ordinate transformations that get you from a flat Minkowski metric to the equivalent FRW metric of the empty Universe curve the spatial and temporal parts individually, such that the overall curvature remains zero. As I say, I am speaking very loosely here, but is essence that is what is done. This means if you take surfaces of constant FRW time, as you have done, you find that the surface is curved, hence there is an apparent gravity.

But, and this is a very important but, none of this matters when assessing the correctness of the model. Regardless of which co-ordinate system you use you will find the same underlying physical model. There is no way in which this 'apparent' gravity can be mistaken for real gravity and spoof results. Co-ordinate transformations do not change observable quantities. This is how your diagrams can mislead you. What you need to do is work out when distance (e.g. luminosity distance) vs redshift curves you get given different physics. Using either FRW co-ordinates or Minkowski co-ordinates you will get the same curves, because these curves relate to things you would observe, and these are not dependent on co-ordinates.

It is very easy in GR to mistake properties of co-ordinates with observations, but it is an important distinction.

this is one reason that the comoving model is preferred over the SR model because the comoving distances are closer to what is observered. However, the SR model should not be dismissed so lightly.

There is no such thing as 'the co-moving model'. There are co-moving co-ordinates, but the model is General Relativity. For an empty universe, the co-moving co-ordinates description of the GR model is exactly equivalent to SR. For a non-empty universe there are 'SR like' co-ordinates (as defined in recent papers by Chodorowski and Lewis et al) as well as the usual co-moving co-ordinates. But these are most definitely not different models, again when related to observables they give exactly the same results. They are just different co-ordinate descriptions of the same models.

I don't really follow you diagrams, but as I say if you want to discriminate between models you need to work out the model predict for what we can observe, which is almost always something as a function of redshift. The actual distances to a given redshift are not measured but can be derived once the model is set via the something vs redshift data. As it stands I think you plots are interesting, but can't be used in the way you are trying to, since they are in terms of arbitrary unmeasurable quantities. To select the best model it is crucial that the models be related to what can be observed.

 

[jonmtkisco]

Hi Wallace,

I doubt either G&E or D&L made such a serious error as to give the conflicting results you are suggesting.

Unless you come up with a better explanation, my guess is that it's just a typo in Davis & Lineweaver Fig 3. I think the labels on the (0,0) and (1,0) lines were inadvertently switched. The chart makes good sense with those two labels reversed.

Hmm I would have expected the gravitational component should be 'positive' or blue-shifting.

OK, I think your reasoning and prediction are persuasive, so I will set aside my idea about the declining cosmic density acting as a gravity well. Blueshift it is.

I want to try out a simpler way to think about gravitational redshift. The standard explanation makes it sound as if gravity acts directly on the energy of a photon, either decreasing the energy (redshift) if it is "climbing out of a gravity well," or increasing the energy (blueshift) if its "falling into a gravity well." To me, it seems easier if one does NOT think of gravity acting on the energy level. Instead, the only effect of gravity is to cause time dilation or contraction.

Thus, if a photon travels from the surface of a massive planet to an observer at a distant point in empty space, the local clock on the planet runs slower relative to the space clock. The observer's faster pace of life (faster clock) causes him to perceive the photon as having a lower frequency (fewer oscillations per clock tick) than would be calculated using the planet clock. The photon has not "changed" at all, it is just perceived differently depending on which clock is used to measure its frequency. This description is consistent with the concept that gravitational redshift does not "occur enroute", instead it "occurs as a single discrete event" at whatever point its frequency is measured. It also is consistent with the fact that in a matter-only universe the Integrated Sachs-Wolfe effect does not affect the redshift of photon which passes through a succession of overdense and underdense regions along its path. The gravitational redshift is calculated solely by the relative clock differential at the two end points. E.g., by comparing the clock at the observer with the clock at the emitter using the SR time dilation equation.

With this narrative in hand, maybe we can separate cosmological redshift cleanly into a velocity component and a clock component as follows.

1. Velocity Component. The equation for this is:

\frac{ \left( V_{emit} + V_{rec} \right) }{2}

where V_{emit} is the proper recession velocity between the emitter and receiver at the time of emission, and V_{rec} is the proper recession velocity between them at the time of reception. The Velocity Component follows the Classical Doppler effect equation v=cz.

2. Clock Component. The equation for this is:

\frac{ \left( \Delta T_{emit} + \Delta T_{rec} \right) }{2}

where \Delta T_{emit} is the relative clock rate difference due to the proper velocity between the emitter and receiver at the time of emission (V_{emit}), and \Delta T_{rec} is the relative clock rate difference due to the proper recession velocity between them at the time of reception (V_{rec}). The Time Component follows the SR time dilation formula.

These equations account for the gravitational decrease in the receiver's proper recession velocity (relative to the emitter) which occurs between the time the photon is emitted and the time it is received. The equations divide by two in order to exclude the gravitational decrease in the emitter's proper recession velocity occurring place after the photon is emitted.

I've attached a diagram showing the proper distance and elapsed time over which (V_{rec}) is calculated at the time of reception.

Jon

 

[Wallace]

Hi Wallace,

Unless you come up with a better explanation, my guess is that it's just a typo in Davis & Lineweaver Fig 3. I think the labels on the (0,0) and (1,0) lines were inadvertently switched. The chart makes good sense with those two labels reversed.

I think your right, I've been scratching my head over that plot and couldn't work out what they were on about. If it's just a typo then I agree it makes sense once the labels are switched.

I want to try out a simpler way to think about gravitational redshift. The standard explanation makes it sound as if gravity acts directly on the energy of a photon, either decreasing the energy (redshift) if it is "climbing out of a gravity well," or increasing the energy (blueshift) if its "falling into a gravity well." To me, it seems easier if one does NOT think of gravity acting on the energy level. Instead, the only effect of gravity is to cause time dilation or contraction.

This is consistent with the standard explanation, since redshift is a measure of the time dilation between the observed and emitted frames. Redshift if you like is the way we can measure time dilation.

Thus, if a photon travels from the surface of a massive planet to an observer at a distant point in empty space, the local clock on the planet runs slower relative to the space clock. The observer's faster pace of life (faster clock) causes him to perceive the photon as having a lower frequency (fewer oscillations per clock tick) than would be calculated using the planet clock. The photon has not "changed" at all, it is just perceived differently depending on which clock is used to measure its frequency. This description is consistent with the concept that gravitational redshift does not "occur enroute", instead it "occurs as a single discrete event" at whatever point its frequency is measured. It also is consistent with the fact that in a matter-only universe the Integrated Sachs-Wolfe effect does not affect the redshift of photon which passes through a succession of overdense and underdense regions along its path. The gravitational redshift is calculated solely by the relative clock differential at the two end points. E.g., by comparing the clock at the observer with the clock at the emitter using the SR time dilation equation.

Yep, redshift and time dilation are always equivalent.

With this narrative in hand, maybe we can separate cosmological redshift cleanly into a velocity component and a clock component as follows.

1. Velocity Component. The equation for this is:

\frac{ \left( V_{emit} + V_{rec} \right) }{2}

where V_{emit} is the proper recession velocity between the emitter and receiver at the time of emission, and V_{rec} is the proper recession velocity between them at the time of reception. The Velocity Component follows the Classical Doppler effect equation v=cz.

I don't quite follow this? How can the recession velocity 'at time of reception' of the distant galaxy be a factor? Once the photon leaves the emitting galaxy it doesn't know or care about what that galaxy does. Unless I'm missing the point of that expression? It would help if you could make it clearer what that expression refers to, i.e. what exactly is it that equals that expression?

2. Clock Component. The equation for this is:

\frac{ \left( \Delta T_{emit} + \Delta T_{rec} \right) }{2}

where \Delta T_{emit} is the relative clock rate difference due to the proper velocity between the emitter and receiver at the time of emission (V_{emit}), and \Delta T_{rec} is the relative clock rate difference due to the proper recession velocity between them at the time of reception (V_{rec}). The Time Component follows the SR time dilation formula.

These equations account for the gravitational decrease in the receiver's proper recession velocity (relative to the emitter) which occurs between the time the photon is emitted and the time it is received. The equations divide by two in order to exclude the gravitational decrease in the emitter's proper recession velocity occurring place after the photon is emitted.

I've attached a diagram showing the proper distance and elapsed time over which (V_{rec}) is calculated at the time of reception.

Jon

Hmm, I'm still confused. Maybe if you demonstrate an example of using this to calculate a redshift it would help?

 

[yuiop]

...

Unless you come up with a better explanation, my guess is that it's just a typo in Davis & Lineweaver Fig 3. I think the labels on the (0,0) and (1,0) lines were inadvertently switched. The chart makes good sense with those two labels reversed.

I think the Davis & lineweaver diagram is correct without the switch (No typo). The (0,0) model has no gravity and no antigravity while the (1,0) model has significant gravity due to normal matter and no antigravity. The reason the velocity for a given redshift is lower in the (1,0) model is due to the slow down of the recessional velocity by gravity which eventually leads to a collapse of the universe. The greater redshift in the (1,0) model relative to the (0,0) model for a given recessional velocity is due to gravitational redshift which is present in the former but not in the latter.

...
Thus, if a photon travels from the surface of a massive planet to an observer at a distant point in empty space, the local clock on the planet runs slower relative to the space clock. The observer's faster pace of life (faster clock) causes him to perceive the photon as having a lower frequency (fewer oscillations per clock tick) than would be calculated using the planet clock. The photon has not "changed" at all, it is just perceived differently depending on which clock is used to measure its frequency. This description is consistent with the concept that gravitational redshift does not "occur enroute", instead it "occurs as a single discrete event" at whatever point its frequency is measured. It also is consistent with the fact that in a matter-only universe the Integrated Sachs-Wolfe effect does not affect the redshift of photon which passes through a succession of overdense and underdense regions along its path. The gravitational redshift is calculated solely by the relative clock differential at the two end points. E.g., by comparing the clock at the observer with the clock at the emitter using the SR time dilation equation.

Your basic concept that the redshift of a photon emitted in an early epoch of the universe is partly due to the reduction in the energy density of the universe during its travel time is good one and I think it is also correct to say it is equivalent to the redshift of a photon climbing out of a gravity well. However, it is synonomous to the concept of the redshift being due to the "stretching" of the photon wavelength by the expansion of space. In other words you can calculate the increase in wavelength using either concept/model and get the same results but it is important to realize they are essentially the same thing and not to apply the both concepts at the same time and double count. Mathematically they are the same, just the interpretation of "what is really happening" is different.

That is one component of the redshift. The other component in comoving coordinates is classic doppler redshift due to recessional velocity relative to the observer at the time of emission. There is no SR time dilation component in comoving coordinates. In the SR model there is a component due to SR time dilation calculated using the subluminal SR velocity with an additional component due to classic doppler redshift due to the SR recession velocity and a component due to gravity but there is no component due to "expansion of space" in SR coordinates. The spacetime in SR coordinates is static and does not admit a cosmological constant or acceleration of expansion by an antigravity effect. I tried to show geometrically in an earlier post that lack of an anti gravity effect in the SR model can be compensated by assuming a (large) component due to rapid inflation right at the start. When I get more time I will try to show what I am talking about in mathematical terms rather than visual geometrical terms. Hopefully I can produce some equations that give the same numerical results as the cosmological calculators and people might find that more convincing. You should be able to do the same the equations you are trying to produce for the components of cosmological redshift. That is the acid test ;)