Superluminal velocities & redshift

[nutgeb]

We all know that nothing can travel faster than the speed of light in a local frame. We also know that distant galaxies in our observable universe have recession 'velocities' relative to us which are far in excess of the speed of light, c, despite the fact that the velocity in their local frames never exceeds c. This apparent contradiction is most often resolved by adopting the paradigm that distant galaxies are not in a local reference frame and are not actually moving away from us; instead the 'expanding space' between us causes them to become progressively more distant. As I've mentioned in other threads, in the 'expanding space paradigm' empty (Lambda=0) vacuum doesn't act like some kind of self-reproducing expansion force that pushes objects apart. Instead, the hypersurface substrate of geometry simply expands over time, resulting in increasing separation between 'stationary' comoving objects.

However, there is another well known situation in which a particle (including a photon) has a coordinate velocity faster than c, and the 'expanding space' paradigm is sometimes but not usually invoked to explain it.

That situation is believed to occur inside the event horizon of a black hole. Professors Taylor & Wheeler give a very accessible description of this situation in their textbook 'Exploring Black Holes'. If you don't have that book, some of the Taylor & Wheeler description is paraphrased in the Wikipedia Article on http://en.wikipedia.org/wiki/Gullstrand-Painlev%C3%A9_coordinates" . Taylor & Wheeler make a three-part argument in favor of faster than light travel:

1. A particle gravitationally accelerated by the BH from rest at an infinite distance plunges radially through the horizon at a coordinate velocity of exactly c, as calculated by a faraway observer (which is the BH's escape velocity at that radius). Since the in-falling particle continues to accelerate gravitationally after it passes the horizon and moves closer to the center, its coordinate velocity must exceed c.

2. They plot the equation:

$$\frac{ dr } { dt_{rain }} = - \left( \frac{2M}{r} \right) ^{1/2}$$

where dr is the Schwarzschild r-coordinate and dt_rain is their name for the proper time lapse in the frame of a particle plunging radially at escape velocity. Their plot shows that the coordinate velocity increases without limit above c as the particle approaches the center.

3. They show that the elapsed proper time of the in-falling particle from the horizon to the center represents an average speed greater than c.

On the other hand, they emphasize that in its own local rest frame, the in-falling particle never observes light traveling faster than c, and measures in-falling light to pass by it at exactly c. (Note that light inside the event horizon can never travel outward.)

You might think that there isn't much time for an in-falling particle to conduct measurements inside the horizon before the particle is torn apart by growing tidal forces. But the larger the mass of a BH, the more travel time and distance there is inside the horizon before tidal forces become significant. The authors use the example of a "20-year BH" of 10¹¹ solar masses, for which 20 years is required in an in-falling particle's proper time for it to pass from the horizon to the center, even at superluminal coordinate velocities.

The authors also show that a flash of light emitted radially inward from the in-falling particle inside the horizon will have a coordinate velocity faster than the particle's, and therefore even more in excess of c. And they show that a flash radially outward from the in-falling particle will approach the center with a coordinate velocity that can be less than c.

. . . . . . . . . .

So naturally I'm thinking that the 'kinematic' paradigm for homogeneous, expanding, matter-only, FRW space can be analogized to a BH situation. If so, then superluminal recession velocities might also be explainable without resorting to the 'expanding space' paradigm (not that I have a problem with that paradigm), and without introducing any new physics. (Of course we still have to assume that the superluminal recession velocities of the original cosmic expansion resulted from some exotic physics (Big Bang, inflation, whatever) in the initial conditions.) No one knows for sure what happens inside a BH event horizon, or what combination of effects is possible.

In an FRW model, the Hubble Sphere is defined as the distance at which the coordinate velocities of galaxies become superluminal. Apparently recession velocity always exceeds c outside the Hubble Sphere, but never exceeds c inside it.

The Hubble Sphere in a spatially flat FRW model also happens to be the smallest cosmic sphere containing exactly the total mass of matter (matter parameter) which exceeds the BH threshold, in the sense that 2M > r. That's easy to see intuitively, because any radius in a spatially flat FRW model changes at exactly the Newtonian escape velocity of its mass parameter; and the escape velocity at a BH horizon is exactly c. So we see that a spatially flat, homogeneous cosmos always has BH characteristics outside the Hubble Sphere, and never has BH characteristics inside the Hubble Sphere. This is an interesting correlation. It seems entirely logical that galaxies beyond our Hubble Sphere must have coordinate recession velocities > c; otherwise they would all be inevitably collapse into a true BH.

I’ll refer to these Hubble Spheres as cosmic BH’s, even though they have some unique characteristics (such as the fact that they can expand, and the fact that they do not have BH characteristics inside the Hubble Sphere).

[to be continued in another post]

 

[JesseM]

However, there is another well known situation in which a particle (including a photon) has a coordinate velocity faster than c, but the 'expanding space' paradigm is not typically invoked to explain it.

It doesn't have a velocity faster than c in locally inertial coordinate systems, and if you're talking about some non-inertial coordinate system like Schwarzschild coordinates, there is nothing special about black holes in this regard; the speed-of-light limit only applies to inertial frames, so even in flat SR spacetime you can have coordinate speeds greater than c if you use a non-inertial coordinate system, like Rindler coordinates.

So naturally I'm thinking that the 'kinematic' paradigm for homogeneous, expanding, matter-only, FRW space can be analogized to a BH situation. If so, then superluminal recession velocities might also be explainable without resorting to the 'expanding space' paradigm (not that I have a problem with that paradigm), and without introducing any new physics.

I think you may be taking the "expanding space" idea too literally, as I understand it it's just a sort of layman's conceptual explanation, not something very rigorous (see this paper which was discussed in this thread, as well as this this paper, for some objections by physicists to this way of conceptualizing what's happening in cosmology), and in any case there is no reason in relativity to expect objects to have coordinate velocities under c in a non-inertial coordinate system.

The Hubble Sphere also happens to be the smallest cosmic sphere containing exactly the total mass of matter (matter parameter) which exceeds the BH threshold, in the sense that 2M > r.

Exactly? Do you have a source for that claim? Pretty sure there's no theoretical reason in GR for this to be true, and if not I doubt empirical estimates of the mass in the Hubble sphere have shown this.

 

[nutgeb]

Kinematic acceleration of in-falling cosmological photons

Now let’s imagine a flash of light emitted by a galaxy beyond our Hubble Sphere, and eventually received by us. Every intervening galaxy along the travel path can sample the photons and observe them passing by locally exactly at c. In the 'kinematic' paradigm, each intervening galaxy has a coordinate velocity radially away from us. Some of those galaxies are outside of our Hubble Sphere, so the photons passing them locally at c are actually receding away from us, in some cases at coordinate velocities in excess of c. In that sense the photons are propagating backward, away from us. (In the very early universe, these galaxies had much, much higher coordinate recession velocities, and the photons passing by them were propagating backward away from us at much, much higher coordinate velocities.)

During their trip, the photons' coordinate velocity toward us is always less than c (at first it was negative), and the photons attain a coordinate velocity of c only just when they arrive here. Using one of the online cosmology calculators, and considering the coordinate origin to be at our location, one can readily calculate that the coordinate velocity of these photons toward us accelerates smoothly over their entire travel path. At first the photons' coordinate distance increases rapidly, then more slowly; then as they cross our Hubble Sphere (presently at z ~ 1.2 in the flat Lambda=0 model) it begins decreasing, then more rapidly with time.

The early part of the photons’ journey can be analogized to the “outward” flash emitted inside a BH horizon, which propagates backward toward the center, except that in our scenario, the photons start out with a coordinate velocity greater than c. The photons begin outside our Hubble Sphere, so they are inside the 'horizon' of the cosmic BH centered on us. In the 'kinematic' paradigm, the 'inside the horizon' gravitational pull of our cosmic BH accelerates the photons from a coordinate velocity that is a negative multiple of c to a coordinate velocity of zero exactly as the photons cross our Hubble Sphere.

A gravitational pull of 'inside the horizon' BH strength seems required in order to eventually stop the recession of the photons that originally were propagating backwards away from us. Yet once the photons cross our Hubble Sphere, a gravitational pull of less than 'inside the horizon' BH strength is sufficient to further accelerate them from a coordinate velocity of zero to c just as they reach us.

[to be continued in another post]

 

[nutgeb]

The cosmological redshift

In the previous post we looked at how the cosmic gravitation field accelerates the in-falling photons’ proper coordinate velocity. Next let’s focus on what happens to the photons’ peculiar velocity, that is, their velocity relative to the Hubble flow in each local frame they pass through.

It is well established that the peculiar velocity of a non-relativistic particle decays in inverse proportion to the growing cosmic scale factor, or 1/a. A comoving observer interprets the decay in a non-relativistic particle’s peculiar velocity as a decrease in the particle’s momentum.

For a particle which is ‘outbound’ from the coordinate origin, this peculiar velocity decay reflects simply that the particle finds itself overtaking successive galaxies that have increasingly large Hubble velocities, since Hubble velocity is proportional to proper distance from the origin. For a particle which is ‘inbound’ toward the coordinate origin, the peculiar velocity decay is also 1/a, but the picture is less intuitive. An inbound particle observes the local Hubble velocities it passes through to be decreasing, all the way to exactly zero, as it approaches the origin. But since the particle’s peculiar velocity is negative (toward the origin), and the Hubble velocity is positive (away from the origin), the decreasing Hubble velocity in fact represents a decreasing peculiar velocity. For example, the photon’s negative proper coordinate velocity and the positive background Hubble flow initially combine for a peculiar velocity greater than the particle’s proper coordinate velocity; but by the time the particle arrives at the origin, its peculiar velocity has decreased such that it exactly equals its proper coordinate velocity.

Of course the peculiar velocity of relativistic photons does not decay as the cosmic scale factor increases. Photons must retain a peculiar velocity of exactly c in every local frame they pass through. Tamara Davis comments on this behavior in her http://arxiv.org/abs/astro-ph/0402278v1" at the page numbered 51:

“It may seem strange that momentum decaying as 1/a means the peculiar velocities of massive objects decay until the objects are comoving, and yet the peculiar velocities of photons always stay at c. It seems that photons are getting some velocity boost that massive particles miss out on.”

The photons effectively are experiencing a velocity boost: their velocity is boosted each time they pass from one infinitesimal local frame to the next which has a different Hubble velocity. But in this situation there is no external source of incremental energy (such as gravity) driving the velocity boost. Therefore, energy conservation requires the photon’s momentum, as experienced by a comoving observer, to decrease in the same proportion as its velocity increases. Thus in an expanding universe a photon’s comoving momentum decays at 1/a, the same momentum decay as a non-relativistic particle. The decay in the photon’s momentum at 1/a means that the photon’s wavelength is observed by a comover to be redshifted in exact proportion to the expansion of the universe since the photons were emitted.

Note that the photon’s comoving momentum does not decay from the perspective of the emitter’s frame. Thus the redshift results from the emission occurring in a local frame with a very high Hubble velocity, with the photon then passing through local frames with progressively lower Hubble velocities, and finally ending up in the observer’s frame which has zero Hubble velocity.

The ‘kinematic’ gravitational acceleration of the photon which I described in my last post contributes nothing to the comoving redshift. In that situation incremental energy was imparted to the photons by gravity, which increased their comoving momentum accordingly. Note also that under the FRW metric, SR time dilation does not contribute to the cosmological redshift, because there is no net dilation of proper time as between comovers. Therefore the only source of the FRW cosmological redshift seems to be the classical Doppler shift. I think the following simple equation describes the cosmological redshift:

$$z = \int_{t_{e}}^{t_{o}} 1 + \frac{dH_{t}dr_{t}}{cdt} = \frac{a_{o}}{a_{e}}$$

where r is the photon's proper distance from the origin and the subscripts e and o respectively signify the time of emission and observation.

. . . . . . . . . .

Besides the increased wavelength of individual photons, the other key attribute of the cosmological redshift is that the separation between traveling photons increases in proportion to the increasing scale factor. This appears to a comoving observer as a pseudo time dilation. See e.g. Appendix A of the http://arxiv.org/abs/0804.3595v1" ‘Time dilation in type Ia supernova spectra at high redshift’ by Blondin, Davis et al in Astrophysics J 2008. (I use the term ‘pseudo’ because the authors do not claim that clocks run slower at the emitter than at the observer. They mean more specifically that the time required to receive a full photon train is increased because of the spatial elongation of the train.)

It is logical to interpret the successive boosts of the photons’ velocity as the cause of their increasing physical separation. The velocity boosts occur as a function of location (passing by a galaxy with a particular local Hubble velocity) rather than as a function of time per se. Since the lead photon in the train arrives at each location before the tail photon, the lead photon experiences each velocity boost sooner than the tail photon does. Therefore, the lead photon’s average proper coordinate velocity is always relatively higher than the tail photon’s, despite (or rather, because of) the fact that the local peculiar velocity of each is always exactly c.

This effect is analogous to the increasing separation between a string of runners as they successively cross the crest of a hill at the same speed and then accelerate as they head downhill. Conversely the runners will bunch up at the start of an uphill climb. In an expanding cosmological model, the lead photon’s proper coordinate velocity is always accelerating away from the tail photon’s over the entire course of the journey.

The ‘kinematic’ gravitational acceleration of the photon I described in my last post is not additive to this observed increase in the separation of photons, except for a tiny faction of the acceleration which is attributable to cosmic gravitational tidal effects. The tidal effect result from the difference in the respective distances of the lead and tail photons from the origin, with the tail photon being slightly more accelerated than the lead photon, thereby shortening the photon train. The tidal effect should be insignificant at cosmological scales for trains of photons received over a time period of days or months, such as supernova events. Ignoring the tiny tidal contribution, the 'kinematic' gravitational acceleration is a function of time rather than location. So gravity causes the velocities of the lead and tail photons to increase almost exactly in lockstep.

. . . . . . . . . .

The analysis I described here is tied to the 'expanding space' paradigm and not the 'kinematic' paradigm because I haven't mapped it from the FRW metric to the Schwarzschild metric. Even with the 'expanding space' paradigm, the analysis I described seems more self-consistent than the traditional explanation that ‘expanding space’ stretches both the wavelength and separation of photons. Here is Alan Whiting’s take on the traditional explanation in his http://arxiv.org/abs/astro-ph/0404095v1" ‘The expansion of space: free particle motion and the cosmological redshift’, published in The Observatory:

“Misner, Thorne & Wheeler, p. 776 and Peebles (1993)12, p. 96-7, use the picture of a standing wave with expanding boundary conditions… It is not clear, for instance, why there should be a standing wave generated between comoving points in the universe, nor why it should maintain itself. More importantly, as pointed out by Cooperstock et al. (1988) (among others), electromagnetic radiation automatically tracking the universal expansion cannot be right. All our test equipment and comparisons are built of or use electromagnetic forces, and they should also expand with the universe; so any cosmological redshift would be undetectable in principle. At the very least, atoms in the Hubble flow would change their characteristic wavelengths with time (and perhaps with the state of the local gravitational field), leading to strange results indeed.”

 

[Naty1]

Since the in-falling particle continues to accelerate gravitationally after it passes the horizon and moves closer to the center, its coordinate velocity must exceed c.

well, you have just also described a photon approaching Earth from the sun..and we don't observe those at superluminal velocities...photons change energy not speed..."continues to accelerate" makes no sense as it was not accelerating at it approached the horizon...

 

[nutgeb]

well, you have just also described a photon approaching Earth from the sun..and we don't observe those at superluminal velocities...photons change energy not speed..."continues to accelerate" makes no sense as it was not accelerating at it approached the horizon...

If you consider the Earth and sun to be in a single local reference frame, then you're right that it wouldn't make any sense. And given how close they are (compared to the cosmological scale), they might as well be in the same local frame.

But at much larger distances, where differences in Hubble velocities become significant, the emitter and observer cannot be considered to reside within a single local frame. Then the photon does in fact accelerate toward the observer. It must do so in order to pass through every intervening local frame at a peculiar velocity of exactly c.

I want stress how very vast the cosmological scale is compared to an astronomical unit. The acceleration effect is significant at cosmological distances but would be utterly negligible the scale of our solar system. But in reality the effect doesn't occur within our solar system at all. There is no Hubble flow at all through which a particle would accelerate, because the Earth and sun are gravitationally bound and virialized in static Schwarzschild local space, not comoving relative to each other in expanding FRW space.

 

[nutgeb]

I may have misinterpreted Naty1's question, which seems to relate to proper coordinate velocity rather than peculiar velocity. I referred to the later when answering.

First, I stand by my original statement that inside the BH event horizon, photons will accelerate continuously to superluminal proper coordinate velocities as they approach the center, at least as measured by an observer plunging radially inward at escape velocity . The photons will locally pass by any in-falling non-relativistic particle (which also attains superluminal coordinate velocities in the time of its own reference frame) at exactly c.

It is less clear to me what the photons' coordinate velocity is when they cross the horizon. Since the velocity of a non-relativistic particle is c as it crosses the horizon, presumably that particle must calculate that a photon which passes by it at the horizon is traveling at 2c in Schwarzschild r coordinates. Which further suggests that the photon will be measured by a non-relativistic particle (in-falling at escape velocity) to have a superluminal coordinate recession velocity at all non-infinite distances outside the horizon. Apparently this is not an unusual circumstance. I think this superluminal velocity measurement is specific to 'comoving' in-falling non-relativistic particles in this situation. I'd appreciate if anyone can help me nail this down.

This issue relates to the 'kinematic' explanation for how the proper coordinate velocity of 'inbound' photons can accelerate from zero at the Hubble Sphere to exactly c when they arrive at the origin. Obviously gravity provides the acceleration energy, but these photons are 'outside the horizon' of the cosmic BH centered on the origin. In a non-BH situation, it bothers me that the actual coordinate velocity of photons would accelerate, instead of the photons just becoming more blueshifted as they gain momentum.

I am inclined to reject any suggestion that the 'kinematic' coordinate velocity of inbound photons is directly affected and controlled by the outbound motion of the Hubble flow, i.e. the need to pass by each comoving galaxy at a local peculiar velocity of exactly c. Clearly photons must be received by a comoving galaxy at exactly c. But this doesn't mean that merely passing nearby a comoving galaxy can directly affect their coordinate velocity. After all, how nearby is nearby enough? How much comoving matter density is enough to cause this effect?

Hermann Bondi commented on this question in his http://cdsads.u-strasbg.fr/abs/1947MNRAS.107..410B"on the L-T-B metric, in the inverse situation where the matter is moving toward the origin and the light ray is moving radially away: "It is a remarkable fact that while inward-moving matter can in such extreme circumstances reverse the direction of an outward-travelling ray, such a ray will always catch up with an outward-moving matter..."

As I said, I think this must be a coordinate-dependent effect which arises only when the coordinate velocity is measured locally by a comoving observer at escape velocity.

 

[Ich]

This issue relates to the 'kinematic' explanation for how the proper coordinate velocity of 'inbound' photons can accelerate from zero at the Hubble Sphere to exactly c when they arrive at the origin. Obviously gravity provides the acceleration energy

No, this effect has nothing to do with gravity. It's how you define your coordinates.

 

[nutgeb]

No, this effect has nothing to do with gravity. It's how you define your coordinates.

I am using proper distance coordinates, which are invariant under transformation. I am looking for a way to use the Schwarzschild metric to model the same effects that occur in a a series of static snapshots of the FRW metric.

The Schwarzschild metric is characterized by positive spatial curvature near the central mass, and this contributes to an expansion of radial proper distances in the metric. Perhaps that is relevant to the apparent acceleration effect. However, a cosmic BH is in some ways an inside-out version of a Schwarzschild BH - the gravitational potential decreases instead of increasing with proximity to the origin.

I think that gravitational acceleration has to be part of the story. At the very least, it blueshifts the photons, such that the gravitational acceleration does not contribute to the cosmological redshift observed at the origin.

 

[nutgeb]

A change in coordinates provides a straightforward way to describe the 'kinematic' gravitational acceleration of photons while avoiding having to describe the photons as having a coordinate velocity different from c.

The idea is to define the coordinate origin to be the individual comoving local frame that a photon is passing through (at a local velocity of c) at a given instant in time. Over the course a photon's journey, this coordinate origin continuously shifts along the radial null geodesic from the emitter to the observer. This enables a series of static Schwarzschild gravitational 'snapshots' to be taken at infinitesimal intervals, based on the spherical ball of cosmic matter defined by the radius r between the photon and the observer at each instant.

I am not referring to the 'rest frame' of the moving photon itself, if there is such a thing. Rather, I'm referring to the instantaneous comoving frame of the background cosmic matter through which the photon is traveling. I'll refer to this as the 'photon co-located origin.' Thus the photon always has a 'coordinate velocity' of c relative to the origin, despite the fact that the photon is always located exactly at the coordinate origin.

In this coordinate system, the comoving observer has a recession velocity away from the origin equal to the contemporaneous Hubble rate times her proper distance r from the origin. The observer experiences gravitational acceleration toward the origin (i.e., toward the photon's momentary position.) When she is outside the Hubble Sphere centered on the momentary coordinate origin, the observer's coordinate recession velocity is greater than c, and the gravitational acceleration she experiences toward the origin is of inside-the-horizon BH intensity. It is fine for the observer to have a non-zero coordinate velocity toward the photon co-located origin, as long as she is not in the photon's inertial frame. There is no extended inertial frame in this scenario because of the presence of the cosmic gravity.

With respect to a spatially flat, expanding, homogeneous, matter-only (Lambda=0) FRW model, the corresponding instantaneous Schwarzschild frame (as perceived from the origin) should also have flat spatial curvature. Comoving galaxies have recession velocities exactly equal to the matter parameter of the sphere of cosmic matter encompassed by their radius. The spatial distribution of the comoving cosmic matter is radially Lorentz contracted due to this recession velocity, but that curvature effect is exactly offset by the positive spatial curvature caused by the gravitation of the diffuse cosmic matter sphere centered on the origin.

It's not surprising that calculations of proper distance and proper time made in this coordinate system will be identical to those obtained by using proper distance coordinates with the origin located at the observer.

 

[stone1]

Let me first say I am just a layman interested in physics, so my question may not be quite correctly posed. One thing that majorly confuses me is that you are talking about different types of velocities. Experiments show redshift for regular velocities - if an object moves away from us (no space expanding) then there is redshift. How do physicists know that expanding space (which leads to a different type of velocity between objects) results in redshift?? Is that just an assumption? If space expansion does not affect an object's velocity in its local frame, then why is the light emitted by that object redshifted?

 

[nutgeb]

Good question stone1.

I want to be careful not to add to your confusion. My posts are trying to dig deeper into specific topics that usually are covered in only a cursory way in GR textbooks. I think it's important that you understand what the textbooks say before you get too emeshed in my attempt at an explanation.

The hypothesis that 'expanding space' causes wavelengths to physically increase is very simple: the separation between wave crests just increases as a function of time, proportional to the increase in the global scale factor a. Don't think of it as empty vacuum pushing the peaks apart; just think of it as the underlying physical geometry of space expanding, like an expanding hypersphere.

As far as I know, there is no proof that this hypothesis is correct. And some commentators have expressed skepticism about it, as I quoted in my first redshift post in this thread. But the hypothesis has not been disproved either. I think that it was traditionally used in textbooks because it is so simple and fits perfectly with the observed mathematical results.

In my opinion, 'expanding space' per se is incapable of causing particles or wave crests to physically separate. The expansion of space does not act as a 'force', so it cannot cause features which initially were at proper rest relative to each other to separate. The expansion of space will cause particles and wave crests to continue separating from each other only if they were emitted in a manner such that they were initially separating from each other (e.g., they were emitted from different comoving locations). If you are familiar with the 'tethered galaxy' problem, you will recognize this distinction.

Instead, I prefer the explanation I gave in my post, which is that the increase in wavelength reflects the loss of photon momentum caused by classical Doppler shift, which occurs progressively as the photons pass through successive local frames that have ever greater (less negative) Hubble velocities (H * D) relative to the ultimate observer.

 

[Ich]

Instead, I prefer the explanation I gave in my post, which is that the increase in wavelength reflects the loss of photon momentum caused by classical Doppler shift, which occurs progressively as the photons pass through successive local frames that have ever greater (less negative) Hubble velocities (H * D) relative to the ultimate observer.

Sorry nutgeb, I did not find out what you've been getting at in your previous posts. Does that mean that you agree with my position in the Bunn/Hoggs discussion?

As far as I know, there is no proof that this hypothesis is correct.

That's not a hypothesis, it's an interpretation. Interpretations are not correct or incorrect, they are consistent or inconsistent, useful or not. You can't proof or disproof consistent interpretations of a theory. And if you prefer a different explanation, that's ok.

 

[nutgeb]

Sorry nutgeb, I did not find out what you've been getting at in your previous posts.

See post #4 in this thread where I explain how photon momentum decays at 1/a.

Does that mean that you agree with my position in the Bunn/Hoggs discussion?

Bunn & Hogg's claim that cosmological redshift is an accumulation of SR redshifts does not accurately interpret the FRW metric, because no time dilation occurs as between comovers in the FRW metric. You debated that point with me endlessly in another thread, and we agreed to disagree. You can start your own thread if you wish to continue that argument.

 

[nutgeb]

Here is an updated version of the cosmological redshift equation I suggested. I have corrected the classical Doppler equation for a stationary observer, and I have eliminated the change in the Hubble rate due to gravitational deceleration, which cannot contribute to the redshift. I also realized that dt = dr in light travel distance, which simplifies the equation.

$$\frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}} = (1 + z) = \int_{t_{e}}^{t_{o}} (1 + H_{t}dt)$$

where the subscripts e and o respectively signify the time of emission and observation, H is in units of ly/y/ly, and t is in units of years.

 

[nutgeb]

I used my spreadsheet to check the results of the simple cosmological redshift integration equation in my prior post. Using a very rough integration (100 intervals of dt) the calculated result is within ~ 10% of the correct redshift for z=1023, for cases both with and without Lambda. I'm confident that the equation will prove to be correct with a reasonably large number of integration intervals.

This equation uses the classical Doppler shift formula for a stationary observer. I also tried calculating my spreadsheet using the SR Doppler shift formula, and the result was far less accurate: ~ 40% error. This is consistent with my analysis that there is no place for SR time dilation in the cosmological redshift.