How is the observed redshift caused by a recessional velocity?

[mysearch]

I know that the issue of cosmological redshift has been discussed in this forum before, e.g. https://www.physicsforums.com/showthread.php?t=368958", but I would appreciate any knowledgeable insights regarding the model outlined below. This model is only considering light in terms of a stream of photons, as it constrains the description to some fairly fundamental concepts, i.e. E=hf. As such, a photon has energy (E) proportional to its frequency (f), where (f) is assumed to be a relative measure linked to the tick of the clock in the emitter and receiver frames of reference, i.e. frequency would reflect any time dilation. There is also a basic assumption that the emitter and receiver are not positioned within any significant gravitational fields, plus the path of the photons is assumed not to pass through any significant gravity fields.

It is not clear whether gravitational effects still play a part in the photon’s trajectory over cosmological distances, if k=0 implies spatial flatness within the universe at large. The assumption is that when k=0, spacetime curvature is defined only in terms of the expansion of the universe with time, as linked to the Hubble parameter (H) and the FRW metric?

Based on these somewhat conceptual and tentative assumptions above, it was hoped to limit the discussion to the issues, and effects, surrounding the recessional velocity as defined by Hubble’s law, i.e. v=Hd. As such, the central question is:

How is the observed redshift caused by a recessional velocity?

Based on the initial assumptions above, it would seem that there are only 2 basic options. First, the recessional velocity is the ‘cause’ of some form of time dilation between the emitter and the receiver or the recessional velocity is simply an observed ‘effect’ of the expansion of space, which then causes a frequency-wavelength change in the stream of photons on-route. However, the idea of the photon’s wavelength ‘expanding’ in transit appears to raise some initial issues: i) there does not seem to be any accepted description of the structure of a photon, ii) space expansion is normally described has only affecting the relative position of ‘objects’ on the very largest scales of the universe. However, trying to analyse the effects of the recessional velocity in terms of relativistic time dilation also appears to raise problems – see model below. The following papers are both dated 2009 and are simply cited as examples of the scope of opinion on this issue:

http://arxiv.org/PS_cache/arxiv/pdf/0808/0808.1081v2.pdf"
http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.3280v2.pdf"

Basic Model:
The model outlined in the attached diagram does not seek to resolve the argument for and against the various mechanisms underlying cosmological redshift; rather it is simply an attempt to visualise and discuss some of the issues from my own learning perspective. This model is based on the data from a ‘cosmic calculator’ similar in scope to that described by Ned Wright – http://arxiv.org/PS_cache/astro-ph/pdf/0609/0609593v2.pdf" for some of the terminology and issues connected to this model. In this context, the attached diagram is only an attempt to summarise the data from the calculator based on a redshift z=1089, which is associated with the decoupling era, based on the basic LCDM model, some 370,000 years after the big bang. Not being an expert in these matters, the following interpretation may need correction, but the model is initially described as follows:

• Point (A) is an arbitrary point in spacetime. In the present era, it is receiving photons that are associated with the CMB radiation with a redshift of z=1089. As such, it is assumed that these photons have been in transit since decoupling, i.e. ~13.7 billion years.
  
  
• Point (B) represents the emitter of the photons, which could be in any direction from (A), but according to the calculator, point (B) would have been ‘located’ some 42 million lightyears from (A) when the photons, now arriving at (A), were first emitted.
  
  
• The diagram shows the value of (H) at the time the photons were first emitted, which is based on the initial separation of ~42 million lightyears and translates into a relative recession velocity that is 66 times the speed of light [c]. At this point in time, (B) is hidden behind the ‘horizon’ defined by the Hubble radius (R=c/H).
  
  
• After the photons are emitted from the initial position of (B), the recession velocity associated with the expanding universe causes (B) to recede to a distance of ~46 billion lightyears in the current era. However, due to the fall in (H) as a function of time, the actual recession velocity has fallen to v=3.3c despite the huge increase in distance; although it is still behind the Hubble radius.
  
  
• However, while it might be an inappropriate description, it seems that the photons make headway ‘swimming against the tide of expansion’ and emerge from behind the Hubble radius and eventually arrive at (A) after some 13.7 billion year.

So my next question is whether there are any fatal flaws in this description as a whole?

Other Issues:
At this stage, I will simply table the following issue, which occurred to me when considering the description of a cosmological redshift:

While the speed of light [c] may never be exceeded in any local frame, I cannot see how the apparent superluminal velocities involved can be transposed into any meaningful time dilation between (A) and (B)?

If (A) and (B) are both stationary with respect to the CMB frame and the physics of the expansion of space does not align to a kinetic explosion, i.e. it is more representative of an expansion of each unit volume of space; can the recession velocity be described as kinematic in nature, as per the Bunn & Hogg paper?

The twin paradox is normally resolved by determining the frame that has been subject to acceleration, which then indicates the direction of time dilation. In this conceptual case, both (A) and (B) have always been inertial frames, at least with respect to CMB, so would the tick of the clock be the same in (A) and (B)?

So is the answer assumed to be as per the Bunn and Hogg paper or does the expansion of spacetime actually have a ‘physical’ effect on the wavelength of the photons on-route to (A)?

Of course, there are other hypotheses, which sit outside accepted science and would possibly fall foul of the guidelines of this forum if discussed in any detail, but I would be interested in knowing if there has been peer review of this paper, which takes and altogether different view of this issue. Thanks

http://shpenkov.janmax.com/Cosmological_Redshift.pdf"

 

[Ich]

Hi mysearch,

the recessional velocity is the ‘cause’ of some form of time dilation between the emitter and the receiver or the recessional velocity is simply an observed ‘effect’ of the expansion of space

Expansion of space and recessional velocity are not causally connected, they are rather two words for the same thing. I think it's better to take a step back and peel off the layers of interpretation befor you ask questions about cause and effect.

The source of all is the observed redshift pattern. There is no further interpretation in this observation.
Then GR comes into play. GR says that the effects can be described as if they were of purely geometrical origin. It is a therefore a metric theory, and our first-step interpretation of redshift is a geometrical one. DaleSpam https://www.physicsforums.com/showthread.php?p=3165919#post3165919" in the other thread.
Now, the problem is that humans can't play with a four dimensional pseudo-Riemannian manifold intuitively, and that it's hard to calculate anything this way. So the next layer of interpretation is the use of coordinate systems. It is important to know that
1) coordinate systems are generally totally arbitrary. You can make them up just as you like. Of course, it's convenient if the coordinates are related to observations in a simple manner.
2) coordinate systems are the basis of our physical understanding. There is no natural global definition of distance, time, time dilation, recessional velocity, expansion of space, gravitational acceleration or potential etc, except in the context of a chosen coordinate system. All these things exist in our heads rather than nature (of course they correspond to natural phenomena in one way or another).

For example, if you want to make use of your Newtonian or post-Newtonian intuition, you have to use "Normal Coordinates". Pick any point as origin, and you can define such coordinates to arbitrary precision within a limited region. Then space is no longer homogeneous, you are at the bottom (or top) of a nearly parabolic gravitational potential, galaxies are actually moving away from you, and redshift is a combination of relativistic doppler effect (including time dilation) and gravitational redshift.
However, Normal Coordinates become generally more and more complex as you increase the region you want to describe. In the limit, they totally fail if you want to describe e.g. a whole closed universe.

That's why cosmologists often use a different coordinate systen, comoving coordinates, which are valid everywhere and where space is homogeneous. In these coordinates, you say that there is no gravitational potential, galaxies are "not moving", and redshift is produced by the expansion of space.
However, to use these coordinates, you have to throw out the intuition you learned in school and replace it with different mental pictures. This new intuition, along with the coordinate system, is utterly unsuited to describe local physics, like the expansion of Brooklyn or the dynamics of galaxy clusters.

You see, there is only one reality, but your understanding of it can be based on different descriptions, each best suited to understand certain aspects of reality und useless/misleading for some different aspects. Don't ask which description is "real" or causes the other. Instead, use as many different descriptions as possible to get a deeper understanding.

 

[Ich]

As to your papers and questions:

The Bunn&Hogg paper is good.

If (A) and (B) are both stationary with respect to the CMB frame and the physics of the expansion of space does not align to a kinetic explosion, i.e. it is more representative of an expansion of each unit volume of space; can the recession velocity be described as kinematic in nature, as per the Bunn & Hogg paper?

Yes. As I explained, different descriptions are not mutually exclusive. Use all there are.

I haven' read the Budko paper carefully. It seems his "exponential redshift" is nothing extraordinary worth a paper.

While the speed of light [c] may never be exceeded in any local frame, I cannot see how the apparent superluminal velocities involved can be transposed into any meaningful time dilation between (A) and (B)?

In this context, it is good to know that the "recession velocities" are actually, by their definition, "recession rapidities". In a "free floating" universe (the case a(t)~t in Budko's paper), the whole redshift is of kinematical origin. Redshift as a function of rapidity "R" is
$$1+z=e^R,$$
and R is proportional to comoving distance. That's the meaning of his eq. 21. It is not time dilation as he claims, though, it's redshift.

I didn't read the Shpenkov paper, it seems to be crackpottery. (EDIT: I skimmed it. It is crackpottery. But then, he's mostly talking about http://en.wikipedia.org/wiki/Redshirt_(character)" , not redshift, which changes the picture considerably.)

The twin paradox is normally resolved by determining the frame that has been subject to acceleration, which then indicates the direction of time dilation. In this conceptual case, both (A) and (B) have always been inertial frames, at least with respect to CMB, so would the tick of the clock be the same in (A) and (B)?

Yes - in cosmological coordinates. But there is time dilation in Normal Coordinates. There is no absolute truth to time dilation.
If you construct a twin paradox, like in a recollapsing universe, where both meet again at the big crunch, the net difference will of course be zero in both systems.

So is the answer assumed to be as per the Bunn and Hogg paper or does the expansion of spacetime actually have a ‘physical’ effect on the wavelength of the photons on-route to (A)?

Whatever you like more. The real - and probably not very useful - answer, according to theory, is one level deeper: It's all geometry.

 

[mysearch]

Sorry, if some of the following points are addressed in your 2nd post. I won't have time tonight to response further, but you to seem to have addressed some central issues. Thanks.

Ich,
I appreciate you taking the time to try to explain some of the issues, which appear to be a little abstract on first reading to somebody not fully versed in GR, but I guess this is my problem not yours. While I understand that a PF discussion is not really appropriate place for learning about all the complexity of GR, I would like to see whether it is possible to translate some of your statements in a way that might be more generally understood and possibly reconciled to the specific example given in post #1. Equally, others may have a different perspective. As such, I have raised some comments below primarily for clarification. Thanks

Expansion of space and recessional velocity are not causally connected, they are rather two words for the same thing. I think it's better to take a step back and peel off the layers of interpretation before you ask questions cause and effect. The source of all is the observed redshift pattern. There is no further interpretation in this observation.

In many ways, cause and effect is the way that most people are taught to think about the Newtonian world. As such, we start off anchored in Newtonian physics, which most of us then struggle to reconcile with GR. My confusion over the first sentence is possibly captured in my earlier question:

If (A) and (B) are both stationary with respect to the CMB frame and the physics of the expansion of space does not align to a kinetic explosion, i.e. it is more representative of an expansion of each unit volume of space; can the recession velocity be described as kinematic in nature, as per the Bunn & Hogg paper?

So when you say that expansion and velocity are not causally connected, but are attempting to describe the same thing, can we consolidate this apparent ambiguity in terms of the Hubble parameter (H). It would seem that you can use the units of (H) to either define metres/second per metre or redefine it as metres/metre per second. The first seems orientated to the description of a recession velocity, while the latter seems more orientated to the description of expansion. Is this valid from your perspective? If interpreted as a recessional velocity, should you be able to relate the tick of the clock in both (A) and (B) as used in my model? Do you agree with any of the positions taken in the referenced papers in post #1?

Then GR comes into play. GR says that the effects can be described as if they were of purely geometrical origin. It is a therefore a metric theory, and our first-step interpretation of redshift is a geometrical one.

When you say ‘metric theory’ are you making reference to a specific solution of Einstein field equations and might we focus on the FRW metric in a cosmology context? If so, can we simplify the form of the FRW metric, such that k=0, and make the assumption that ‘geometry’ of space has no spatial curvature; especially when viewed in terms of a homogeneous mass-energy density of the universe?

Now, the problem is that humans can't play with a four dimensional pseudo-Riemannian manifold intuitively, and that it's hard to calculate anything this way.

That is a relief. I was beginning to think that everybody but me was born with an intrinsic understanding of GR curved spacetime!

So the next layer of interpretation is the use of coordinate systems. It is important to know that 1) coordinate systems are generally totally arbitrary. You can make them up just as you like. Of course, it's convenient if the coordinates are related to observations in a simple manner. 2) coordinate systems are the basis of our physical understanding. There is no natural definition of distance, time, time dilation, recessional velocity, expansion of space, gravitational acceleration or potential etc. except in the context of a chosen coordinate system. All these things exist in our heads rather than nature (of course they correspond to natural phenomena in one way or another).

OK, as an arbitrary choice, can we use some orthogonal coordinate system, if k=0 allows us to assume that the spatial geometry is flat? If this is true, can we then describe spacetime curvature just in terms of expansion? For example, 2 photons apparently moving in parallel will over a very large period of time diverge due to the expansion of the universe? Your next point seems to suggest no.

For example, if you want to make use of your Newtonian or post-Newtonian intuition, you have to use "Normal Coordinates". Pick any point as origin, and you can define such coordinates to arbitrary precision within a limited region. Then space is no longer homogeneous, you are at the bottom (or top) of a nearly parabolic gravitational potential, galaxies are actually moving away from you, and redshift is a combination of relativistic doppler effect (including time dilation) and gravitational redshift.
However, Normal Coordinates become generally more and more complex as you increase the region you want to describe. In the limit, they totally fail if you want to describe e.g. a whole closed universe with their help.

To be honest, I don’t understand this description. Are you saying that the diagram attached to post #1 is not a valid description of the expansion of (A) and (B) in spacetime with respect to time and distance axes shown?

That's why cosmologists often use a different coordinate system, comoving coordinates, which are valid everywhere and where space is homogeneous. In these coordinates, you say that there is no gravitational potential, galaxies are "not moving", and redshift is produced by the expansion of space.

OK, in the context of my example, can we describe (A) and (B) to be stationary with respect to the comoving coordinates of CMB?

However, to use these coordinates, you have to throw out the intuition you learned in school and replace it with different mental pictures. This new intuition, along with the coordinate system, is utterly unsuited to describe local physics, like the expansion of Brooklyn or the dynamics of galaxy clusters.

Unfortunately, for me, school was a very, very long time ago, but I am still willing to learn, although I no longer feel obligated to accept things by rote. Apologises, if this makes my seemingly repetitive questions a bit of a pain in the neck for others. However, does this new mental picture redraw the model diagram in post #1?

You see, there is only one reality, but your understanding of it can be based on different descriptions, each best suited to understand certain aspects of reality und useless/misleading for some different aspects. Don't ask which description is "real" or causes the other. Instead, use as many different descriptions as possible to get a deeper understanding.

In many respect, my model was an attempt to get, at least, 1 picture based on the premise of the cosmic calculator, as defined by Ned Wright. This calculator is based on the LCDM model of an expanding universe, which would appear to account for curved spacetime. As stated, my central question was:

So my next question is whether there are any fatal flaws in this description as a whole?

The answer is?

 

[Ich]

Ok, let's start with the end:

So my next question is whether there are any fatal flaws in this description as a whole?

Your diagram seems to plot cosmological time versus "cosmological proper distance". This is not the same as "comoving distance", where A and B would indeed be stationary = vertical lines.
In your diagram, both B and the photon should start heavily inclined to the right (moving away with 66 or 65 c, respectively). The photon then follows the teardrop curve described in the Davis & Lineweaver paper. B keeps on moving outward, at first decelerating and then accelerating again.

Now, to the rest of your post:

In many ways, cause and effect is the way that most people are taught to think about the Newtonian world. As such, we start off anchored in Newtonian physics, which most of us then struggle to reconcile with GR.

it seems I left the wrong impression. Cause and effect are of course valid concepts in GR. It's just that "expansion of space" and "recessional velocity" do not cause each other, they are rather the same thing in disguise. Different coordinates, nothing more.

It would seem that you can use the units of (H) to either define metres/second per metre or redefine it as metres/metre per second. The first seems orientated to the description of a recession velocity, while the latter seems more orientated to the description of expansion. Is this valid from your perspective?

That's great. Marcus uses to say "the universe expands 1% in 140 million years". That's in fact the same as 71km/s/Mpc.

If interpreted as a recessional velocity, should you be able to relate the tick of the clock in both (A) and (B) as used in my model?

This is ultimately tricky.
In your diagram, you use cosmological time, which is defined as the proper time of each comoving observer since the big bang. So coordinate time is the same as proper time, and there is no time dilation. (Has to do with the definition of simultaneity).
If you'd use normal coordinates instead, with the Einstein synchronization convention, you'd have time dilation.
So, short answer: you can relate the ticks, but there are many different ways to do so.

Do you agree with any of the positions taken in the referenced papers in post #1?

Bunn & Hogg have a valid point. You'll encounter people that tell you that you mustn't see cosmological redshift as a Doppler shift, and the paper is a good refutation of that claim.
I ignore the other papers.
The paper I agree most with is http://arxiv.org/abs/0809.4573" .

When you say ‘metric theory’ are you making reference to a specific solution of Einstein field equations

No. That's what GR is, generally. I refer to DaleSpam's description.

and might we focus on the FRW metric in a cosmology context?

Again, a subtle point. Of course I speak of FRW metrics. But do not confine myself to Robertson-Walker coordinates. You can use normal Coordinates locally in every spacetime, including FRW.

If so, can we simplify the form of the FRW metric, such that k=0

If you insist. But believe me, the simplest and most enlightening example of a FRW spacetime is the Milne model with k=-1.

OK, as an arbitrary choice, can we use some orthogonal coordinate system, if k=0 allows us to assume that the spatial geometry is flat?

Coordinate systems extend to 4 dimensions, not only space. In a spatially flat, expanding universe, there is spacetime curvature, and you can't use orthogonal, normal coordinates globally.

If this is true, can we then describe spacetime curvature just in terms of expansion?

I don't understand. In a spatially flat universe, spacetime curvature is indeed proportional to the expansion, if that is what you mean. (Disclaimer: I don't think it's "proportional", but something to that effect.)

For example, 2 photons apparently moving in parallel will over a very large period of time diverge due to the expansion of the universe?

If you mean "parallel" in the context of "proper distance": No. If you mean "comoving distance": Maybe, I haven't done the calculations.

To be honest, I don’t understand this description.

That's a pity. I tried to describe the (near) universe in the simple terms of (post-)Newtonian physics. You can do that. But it's not what you depicted in your diagram (even if it were right).

However, does this new mental picture redraw the model diagram in post #1?

The "new mental picture" is IMHO quite exactly what you tried to plot.

 

[mysearch]

Response to Post #3:
If possible, I wanted to try to better understand some of the terminology you have introduced so that I might go away and reference the issues in more detail:

In this context, it is good to know that the "recession velocities" are actually, by their definition, "recession rapidities".

Did a Google search on ‘recession rapidities’. It only pointed to 1 instant of use, i.e. this thread. Does this term carry any important subtlety in its definition, which I should understand?

Yes - in cosmological coordinates. But there is time dilation in Normal Coordinates.

Again, I would be interested in understanding the meaning and implications attached to ‘normal’ and ‘cosmological’ coordinates. I found one reference to cosmological coordinates ‘http://www.chronon.org/articles/milne_cosmology.html"’ that also seems to be addressing some of the issues raised in this thread. I would be interested if people agreed with its basic tenets?

I am not sure whether the description of normal coordinate in the first 2 paragraphs ‘http://en.wikipedia.org/wiki/Normal_coordinates" ’ was an attempt to explain them or simply used as an excuse to make as many oblique references as possible. As a result, I am not too sure about their scope.

There is no absolute truth to time dilation. If you construct a twin paradox, like in a recollapsing universe, where both meet again at the big crunch, the net difference will of course be zero in both systems.

Sorry, are you implying that the idea of a relative time dilation cannot be applied to (A) and (B) or simply that it cannot be quantified? For example, if we had 2 hypothetical clocks in (A) and (B), you seem to be implying that they would both have the same time at the big crunch? So is there a suggestion that the tick of each clock is the same throughout or does the perception of a recessional (relative) velocity require the clock rates to depend on the frame of reference, i.e. A to B or B to A?

 

[mysearch]

Response to Post #5
Thanks for the additional response. Again, I am primarily raising comments in an attempt to better understand the basic terminology you are introducing, such that I might go away and reference the issues in more detail:

Your diagram seems to plot cosmological time versus "cosmological proper distance". This is not the same as "comoving distance", where A and B would indeed be stationary = vertical lines…………B keeps on moving outward, at first decelerating and then accelerating again.

Agreed. Given that (A) and (B) are said to be stationary wrt to CMB, the diagram does not relate to comoving coordinates. Equally, the diagram makes no attempt to correctly scale the (AB) events to the time and distance axes. The curved path of the photons and (B) was only intended to provide some general representation of expanding spacetime. However, in a spatially flat universe, it would seem that photons start out at (B), positioned 42MLYs from (A), and follow a radial path towards (A) through expanding space, while the original position of (B) follows a radial path away from (A) due to the continuing expansion of space. As such, there appears to be an inference that the photons travel a path that is 13.7BLYs in 13.7Byrs, if [c] can be assumed to be constant through out. However, the main inference of the diagram, based on the calculator data, was that the position and relative velocity of (A) and (B) can be ‘unambiguously?’ specified in spacetime at 370,000 and 13.7 billion years.

This is ultimately tricky. In your diagram, you use cosmological time, which is defined as the proper time of each comoving observer since the big bang. So coordinate time is the same as proper time, and there is no time dilation. If you'd use normal coordinates instead, with the Einstein synchronization convention, you'd have time dilation. So, short answer: you can relate the ticks, but there are many different ways to do so.

Does this link adequately define ‘http://en.wikipedia.org/wiki/Einstein_synchronisation" ? Yes, the synchronisation of time as described does seem tricky. However, the inference of the diagram is that both (A) and (B) both arrive at their respective positions in spacetime after 370,000 years. There is also the inference that (A) and (B) end up separated by 46BLYs after 13.7 years. As a hypothetical question, if (A) and (B) were described in terms of an identical evolution of biological life, would you expect life in (A) and (B) to be identical in their rates of evolution?

The paper I agree most with is http://arxiv.org/abs/0809.4573" .

Yes, this does seem to be a very good paper. However, I need to actually work through the implications of equation (2) through (11). The following statement from the paper also seems to be central to the issue I am trying to resolve:

“The redshift is thus the accumulation of a series of infinitesimal Doppler shifts as the photon passes from observer to observer, and this interpretation holds rigorously even for z ≫ 1. However, this is not the same as saying that the redshift tells us how fast the observed galaxy is receding.”

Just by way of reference, the following equation is essentially the core of the Ned Wright cosmic calculator. In light of the paper referenced above, which I need to study more carefully, is it correct to say that radiation, i.e. photon, energy is modeled in terms of expanding space and that this form of energy is subject to an additional (a) factor due to expanding space?

$$\Delta t = \frac {\Delta a}{aH_0 \sqrt{ (\Omega_M /a^3) + (\Omega_R /a^4) +(\Omega_\Lambda) } }$$

I will defer responding to the comments concerning the use of the FRW metric to another post, but primarily I was hoping that focusing on a specific solution of Einstein field equation would reduce the scope of GR maths, which appears to have developed to handle almost any assumption about spacetime geometry, even if there is no obvious parallel in the observed universe.

For example, if you want to make use of your Newtonian or post-Newtonian intuition, you have to use "Normal Coordinates". Pick any point as origin, and you can define such coordinates to arbitrary precision within a limited region. Then space is no longer homogeneous, you are at the bottom (or top) of a nearly parabolic gravitational potential, galaxies are actually moving away from you, and redshift is a combination of relativistic doppler effect (including time dilation) and gravitational redshift.
However, Normal Coordinates become generally more and more complex as you increase the region you want to describe. In the limit, they totally fail if you want to describe e.g. a whole closed universe with their help.

To be honest, I don’t understand this description. Are you saying that the diagram attached to post #1 is not a valid description of the expansion of (A) and (B) in spacetime with respect to time and distance axes shown?

That's a pity. I tried to describe the (near) universe in the simple terms of (post-)Newtonian physics. You can do that. But it's not what you depicted in your diagram (even if it were right).

Clearly, you are suggesting that it is important to understand your initial description. I did initially attempt to review the scope of ‘http://en.wikipedia.org/wiki/Normal_coordinates" using wikipedia as reference, but the implications of these coordinate are not yet clear to me. Anyway, I have appreciated your help in some of these issues. Thanks

 

[mysearch]

GR & the FRW Metric
I realize that the following comments are tangential to the original scope of this thread, but I would be interested in trying to understand what aspect of GR maths are considered essential to understand modern cosmology. I am assuming that many people who simply start out being interested cosmology have had to face up to the problem of the mathematical complexity of some aspects of GR. Einstein himself is quoted as saying “Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.” However, this post is not intended as a rant against a subject that I am attempting, albeit struggling, to fully understand, but rather as an attempt to see whether focusing on a specific solution of Einstein field equations allows certain topics to be prioritised. In the context of cosmology, the FRW metric is often cited as a GR solution of expanding spacetime based on the simplifying assumption of homogeneous universe, which is often presented in spherical coordinates as follows:

[1] $$ds^2 = - dt^2 + a^2(t) \left (\frac{dr^2}{1-kr^2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \right )$$

However, it would seem that we can immediately reduce the form of the equation above by only considering a radial path, which basically aligns to the model in post #1.

[2] $$ds^2 = - dt^2 + a^2(t) \left (\frac{dr^2}{1-kr^2}) \right )$$

Of course, if we assume k=0 and imply a zero spatial curvature, the form of the metric can be reduced even further:

[3] $$ds^2 = - dt^2 + a^2(t) dr^2$$

As far as I am aware, the definition of the expansion (a) in the equations above can be connected to the basic Friedman, Fluid and acceleration equations of cosmology. While the derivation of these equations are also said to linked to GR, it would seem that reasonable approximations can still be made using classical concepts of energy conservation underpinning the Friedmann equation with the laws of thermodynamic leading to the Fluid equation, while the acceleration equation comes from differentiating the Friedmann equation with respect to time and then substituting in the Fluid equation for the rate of change of density with time. Based on such simplifying assumptions, we might again introduce the equation that underpins the cosmic calculator, which then allows an approximate model of an expanding universe to be made.

[4] $$\Delta t = \frac {\Delta a}{aH_0 \sqrt{ (\Omega_M /a^3) + (\Omega_R /a^4) +(\Omega_\Lambda) } }$$

In contrast, the description according to GR seems to require a much longer apprenticeship in order to cover a broader range of mathematical concepts, which we might put under the umbrella of ‘differential geometry’. In this context, the basic ideas of vectors and matrices seem to quickly give way to tensors, contravariant and covariant coordinates, coordinate transforms, metric tensors plus the concept of parallel transport and the use of Christoffel symbols. However, it would seem that armed with these concepts, spacetime curvature can then be addressed without prejudging the geometric nature of curved spacetime via the introduction of the Riemann tensor, but which then seems to be immediately reduced from a rank (1,3) tensor requiring 256 components to the Ricci tensor, Weyl tensor and the Ricci scalar. It is said that the 10 independent components of Ricci tensor describes how the volume of spacetime changes in any given direction, while the Weyl tensor describes how it changes in shape. At this point, you might begin to have a basic grasp of the scope of Einstein (G) tensor, but are then left to understand the implications of the stress-energy tensor (T), before finally being introduced to the summary form of Einstein field equation.

However, if we return to the FRW metric in [1], [2] or [3] as a GR solution, the stress-energy tensor (T) appears to reduce to 4x4 matrix, where only the diagonal components are non-zero, i.e. [p,P,P,P], which then appears to align with the assumptions of the Fluid equation. Equally, while the metric tensor [g] is also a 4x4 matrix of spacetime, only the [g11] component plays an active role in [3] above.

However, while the simplifications outlined would appear to reduce the complexity enormously, it does not seem to reduce the scope for a broad range of opinions as seen in the formal papers referenced through this thread, which is also backed up by the scope of Ich comments. Anyway, I would appreciate any insights from those who have gone through this learning curve and successfully come out the other side. Thanks.

 

[Ich]

Hi mysearch,

a suggestion: we cover too many topics in this thread. The posts get too long, it takes too long to answer (which is a problem for me as I rarely have enough time), and we can barely scratch at the surface of every aspect, maybe losing the "big picture". Further, other peole won't read through all the posts and join the discussion. I'll try to anwer all points, but you may want to continue with one or two central topics that help you most.

Did a Google search on ‘recession rapidities’. It only pointed to 1 instant of use, i.e. this
thread. Does this term carry any important subtlety in its definition, which I should understand?

"Recession rapidity" is a non-standard term. "Rapidity" is standard, though. If you know that the alleged velocity is a rapidity, you can interpret Budko's result correctly.

I found one reference to cosmological coordinates ‘here’ that also seems to be addressing some of the issues raised in this thread.

Chronon (the link's author) is a member of PF. I personally tend to agree with his writings about cosmology. I fully endorse his point about the usefulness of "SR-like" coordinates, and his criticism of some well known papers on that ground.

Sorry, are you implying that the idea of a relative time dilation cannot be applied to (A) and (B) or simply that it cannot be quantified?

It can be applied, and it can be quantified (at least for observers not too far away from each
other). The dilation is zero in cosmological coordinates, and non-zero in normal coordinates. It is coordinate-dependent, just like energy or momentum.

For example, if we had 2 hypothetical clocks in (A) and (B), you seem to be implying that they would both have the same time at the big crunch?

Of course. In the FRW models, symmetry demands that every observer measures the same lifetime of the universe.

So is there a suggestion that the tick of each clock is the same throughout or does the perception of a recessional (relative) velocity require the clock rates to depend on the frame of reference, i.e. A to B or B to A?

Not "or". Both viewpoints can be correctly applied. Again: there is no absolute truth to time
dilation. It's a coordinate-dependent concept.

it would seem that photons start out at (B), positioned 42MLYs from (A), and follow a radial path towards (A)

at first, their path leads away from A

As such, there appears to be an inference that the photons travel a path that is 13.7BLYs in 13.7Byrs, if [c] can be assumed to be constant through out.

c can be assumed constant only in normal coordinates near the observer.

However, the main inference of the diagram, based on the calculator data, was that the position and relative velocity of (A) and (B) can be ‘unambiguously?’ specified in spacetime at 370,000 and 13.7 billion years.

No. Both position an relative velocity depend heavily on the coordinates you use.

Yes, the synchronisation of time as described does seem tricky.

The Einstein convention is relatively natural IMHO. The tricky thing is that cosmological coordinates don't follow this convention, and thus introduce a different idea of simultaneity.

As a hypothetical question, if (A) and (B) were described in terms of an identical evolution of biological life, would you expect life in (A) and (B) to be identical in their rates of evolution?

Time dilation affects everything. But it is dependent on your idea of simultaneity, which depends on the coordinates you use. In cosmological coordinates, there is no time dilation between comoving observers by definition. In "SR-like" coordinates, there is time dilation. There is no way to compare these "rates" unambiguously.

Yes, this does seem to be a very good paper.

It seems you refer to the Bunn & Hogg paper, while I linked the paper by Peacock. Bunn & Hogg is good, but Peacock agrees most with my point of view.

is it correct to say that radiation, i.e. photon, energy is modeled in terms of expanding space and that this form of energy is subject to an additional (a) factor due to expanding space?

Yes, this additional a-factor being redshift. More generally, the momentum of every particle is scaled by a^-1, which means no energy change for slow particles and an energy scaling of a^-1 for photons.

However, while the simplifications outlined would appear to reduce the complexity enormously, it does not seem to reduce the scope for a broad range of opinions as seen in the formal papers referenced through this thread, which is also backed up by the scope of Ich comments.

You will always have a broad range of possible interpretations, that lies in the nature of GR. So there will be a broad range of (sometimes conflicting) opinions.
If you want to understand the dynamics of the Friedmann equations, this is extremely simple (well, at least compared to other methods) in said "SR-like" coordinates. There, the term "rho + 3p" is the source of (otherwise Newtonian) gravity. In a homogeneous universe, you can then derive
$$\ddot a / a = -4/3 \pi G (\rho + 3 p)$$
Together with the continuity equation
$$d/dt (\rho a^3) = -p d/dt (a^3)$$
the evolution is completely described. In plain English: Think of a small sphere filled with an effective mass density (rho + 3p) in Newtonian gravity. The universe will behave exactly like this sphere, if you make it obey the continuity equation above.

 

[mysearch]

Hi mysearch,
a suggestion: we cover too many topics in this thread. The posts get too long, it takes too long to answer, and we can barely scratch at the surface of every aspect, maybe losing the "big picture". Further, other people won't read through all the posts and join the discussion. I'll try to answer all points, but you may want to continue with one or two central topics that help you most.

Fair comment. You have already helped point the way in a number of key areas and therefore I had already recognised the need to sit down and study some of the terminology in more detail. To be honest, some of the detail I put into my threads is primarily as an archive for my own use at a later date. Again, I appreciated the help.