# Minguizzi's Simultaneity Method

Discussion in 'Alternative Theories' started by Mike_Fontenot, Feb 6, 2020.

1. ### HalcRegistered Senior Member

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Agree. The question comes up why a 'method' is needed at all, especially all these complicated contrivances that seem to serve zero purpose.

There seem to be two camps of physicists: Those that support a preferred foliation of spacetime (absolutists) and those that do not (supporting relativity of metaphysical simultaneity).
In the former camp, what Bob does has zero bearing on what Alice's age is at a given moment in his travels. Hence they would never support any of said methods, none of which are objective.
In the latter camp, the wording you use suggests that Alice has an actual age simultaneous with Bob's turnaround event, which any relativist would say is a frame dependent relation, and Bob has not explicitly specified any frame. The frame used by CMIF seems an unintuitive choice for a person undergoing acceleration.
Neither group of physicists would say that Alice actually ages backwards because of what Bob is doing, so none of them should find anything repugnant.

Yes, in any accelerating reference frame, sufficiently distant events from the accelerating worldline are beyond its event horizon and thus don't have meaningful current times. This is not repugnant, it just demonstrates that accelerating frames do not foliate all of spacetime. Stars 20 BLY away have a negative age in Earth's inertial frame, so they don't really exist in that frame despite the fact that we can see them. Hence the cosmologists needed a better coordinate system than the one SR gives them, but in doing so, light no longer has constant speed. It is only c locally, and SR becomes a local theory. So CMIF is a local method in a way, and if Alice is aging backwards, it's because she's not in the local region where the method doesn't do stuff like that.
Minguzzi similarly uses a method that is frame independent, but does not foliate events that have space-like separation from the reference event, and hence cannot state the current state of any worldline outside that region simultaneous with an event inside the region.

SR posits that light speed will be locally measured to be a constant, which is not quite the same as your statement here. Einstein is very careful to point out that it is an assumed convention that light speed relative to an inertial frame is constant. That convention is not part of the premise of the theory. It essentially assumes for convention purposes, that, relative to some frame, light travels the same speed one way as it does the opposite way. This of course cannot be empirically demonstrated, despite the fact that the earliest measurement of light speed used one-way methods, not any sort of round-trip signal scenario.

Last edited: Mar 3, 2020

3. ### Mike_FontenotRegistered Senior Member

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You probably remember how Einstein explained how an inertial frame can be constructed in principle. He basically said that mutually stationary clocks could be distributed along an infinitely long line in both directions, with some fixed spacing between them. The clocks could then be synchronized by sending light pulses to each given clock, from two clocks on opposite sides of the given clock, each equidistant from the given clock. The light pulses would show the time on the sending clock at the instant the pulse was sent. When the clocks were adjusted so that the two signals arrived at the same instant, and showed the same time on the two distant clocks, Einstein said they were properly synchronized. That process has to be carried out throughout all space. How long do you think that synchronization process would take? It would take an infinite time. When Einstein talked about "an inertial observer", or "an inertial frame", his implication was that it was a "perpetually-inertial observer" or a "perpetually-inertial frame".

I don't know what you're referring to there. I DON'T assume that the CMIF method is valid, anywhere in my paper. I originally believed I had proven that CMIF ISN'T valid, but I fairly quickly realized that I hadn't actually succeeded in doing that. I now suspect that it may be impossible to prove which simultaneity method is the correct one. I DO believe that there IS one and only one correct simultaneity method in special relativity, but we probably just can't know what that correct method is. All we can do is choose which of the known methods we each want to use, based on which one we believe has the most desirable characteristics. I think my method's characteristics are the most desirable among the four simultaneity methods that I'm aware of (CMIF, Dolby&Gull, Minguizzi, and mine): mine is (1) causal, (2) it produces an ACD that is always continuous and piecewise-linear, and (3) it never produces negative ageing of the home twin ... i.e., she never gets YOUNGER, according to him. Item 3 also means that my method produces a one-to-one mapping between the her current age, versus his age, according to him ... CMIF's mapping is NOT one-to-one: he can have multiple ages corresponding to the same current age for her.

5. ### phytiRegistered Senior Member

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471
Truth is never determined by opinion polls, nor by authority figures.
The radar method is not new, since it is measurement by em signals, contained in SR by Einstein, Poincare, and others.
The aos/los are imaginary.
This is where I get off the bus.

7. ### Neddy BateValued Senior Member

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2,114
I suggest you read Einstein's 1920 book, Relativity the Special and General Theory. It is an easy read, and free online:
https://www.bartleby.com/173/

In chapter 4 he very carefully describes how he determines an inertial frame, which he calls a Galileian system of coordinates. "A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a Galileian system of co-ordinates. The laws of the mechanics of Galilei-Newton can be regarded as valid only for a Galileian system of co-ordinates."

Then in chapter 5 he refers back to a train and a railway embankment that he had talked about earlier. Surely the train could not have been perpetually inertial. The amount of time it would take to synchronise clocks along the train is the time it takes for light to travel the length of the train twice. It does not follow from that that the train must have been perpetually inertial.

A quote from chapter 5:
"If K is a Galileian co-ordinate system, then every other co-ordinate system K' is a Galileian one, when, in relation to K, it is in a condition of uniform motion of translation. Relative to K' the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K."

Here is a quote from chapter 9:
"UP to now our considerations have been referred to a particular body of reference, which we have styled a 'railway embankment.' We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig. 1. People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. "

And a quote from chapter 12, where he uses an equation to calculate the rate of a moving clock:
"As a consequence of its motion the clock goes more slowly than when at rest."

That also applies to the stay-home twin's clock, when measured from the inbound inertial frame.

Sorry I should not have put that in this thread, as it is off topic. I will approach it in the other thread.

Last edited: Mar 3, 2020
8. ### Jonathan DoolinRegistered Member

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As far as CMIF, I'm not sure what that is.

I'm used to another abbreviation "MCIRF" for momentarily comoving inertial reference frames. The idea is that at any given moment, any given object has a well-defined velocity, which is zero from it's own perspective, and from that event and velocity, you can cover all of spacetime (x,y,z,t) with an inertial reference frame. Now, does SR require the use of MCIRF's? I vote yes. In general, SR requires the use of two inertial reference frames, and maps the space and time labels from one frame to the other. In general, you will have at least one clock stationary in each frame. These frames are momentarily, (if not eternally) comoving with those clocks.

You can imagine/visualize/render the universe from any perspective you want. However, when it comes to actually making observations of the universe with your own eyes, you can only see it from your own perspective--that is, facing forward, with the top of your head up, your right arm to the right, and with zero velocity relative to yourself. Alternately, you could use a camera, but then you're using that camera's definition of forward and zero velocity.

Now, for a rock in space, which direction is forward, backward, upward, downward, left, and right, is arguable. But at any given time, it will have one, and only one velocity. That velocity, at that moment, fully determines the body's momentary inertial reference frame, which, in turn, defines a locus of simultaneous events. Does SR require the rock to be concerned about that locus of events? No. But they are still perfectly well defined for anyone who is concerned about which events they are. And if you were keeping score of such things, you could note that as your camera or eyes accelerate, events swing forward and back in time.

What your camera or eyes are concerned about, though, are the events actually being seen, which are the locus of events in the past-light-cone of the observation event. No matter how one accelerates, events will only cross this space-time surface once. (They also only pass the future light-cone once)

9. ### Jonathan DoolinRegistered Member

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With a bit of difficulty, there should be no trouble in creating an accelerating frame that foliates all of spacetime of an accelerated frame using SR. The problem is that they're trying to use "simultaneity" as their foliation-surface, when they should either use the past or future light-cone.

Doing so would put every event in spacetime into R^4 without any missing or repeated events.

Last edited: Mar 14, 2020
10. ### HalcRegistered Senior Member

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227
This makes no sense to me. Kindly elaborate on how using light cones somehow eliminates missed or repeated events. Examples perhaps.

Another thing is the bit about 'all spacetime' ... 'using SR', which runs into trouble when SR does not model real spacetime. For instance, GN-z11 (a very distant thing) seems not to exist in the inertial frame of our solar system. That's what I mean by saying that even an inertial frame doesn't foliate all of (actual) spacetime.

11. ### Jonathan DoolinRegistered Member

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Alright; an example, then. Consider a very powerful electromagnetic event at x=1 light-year from here, t=1 year ago. Now, if we "look" in the positive x direction, we should be seeing that event right now, because the light from that event should currently be reaching you. This event is on your "past-light-cone", which is the locus of every event in the universe that you can see "right now".

With a powerful starship we'll pretend we have, we decide to do a sudden acceleration toward that event. Say 0.9c, over a course of a few days. Such a boost leads to the following Lorentz Transformation:

$\begin{pmatrix}2.294&-2.064\\-2.064&2.294 \end{pmatrix} \begin{pmatrix}-1\\1 \end{pmatrix}= \begin{pmatrix}-4.354\\4.354 \end{pmatrix}$

What happened to the event that was on your past light-cone? Well, over the course of a few days, the event slipped from being 1 light year away, and 1 year ago, to being 4.354 light years away, and 4.354 years ago. The event moves into the past and further away, but it does not leave the past lightcone.

Now, you might notice, here I just said there would be no repeated events, and now with the first example I gave, I tell you the same event (pre-boost, and post-boost) is at two different coordinates. However, what I mean to say is that the boost takes place over the course of some time. For instance, even in the very limit, so far as I know, a boost must be limited by Heisenberg's uncertainty principle $\Delta t > \frac{\hbar}{\Delta E}$, so there are no truly instantaneous changes of velocity, even for the smallest of particles. Which means, in turn, from the perspective of any accelerated particle, every event in the universe will cross it's past light-cone in one and only one location.

12. ### Jonathan DoolinRegistered Member

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Now, regarding GN-Z11, according to Wikipedia it has a redshift around 11, and a distance somewhere around 32 billion light years. Now, the common question about this is "how can anything be 32 billion light years away if the universe is only 13.7 billion years old, and things are limited to moving at the speed of light?"

Most answers to such questions discuss "inflation" in rather opaque terms. But I would say that in combination with basic thermodynamics, inflation is fully explained via Special Relativity.

Then apply the equation $\frac 3 2 k_B T = U=(\gamma - 1)m c^2$

Using a proton's mass, the equation simplifies to $\gamma=1.38\times 10^{-13}T+1$

So with an early temperature, say of $T=10^{32}$ Kelvin, the time-dilation factor is $1.38\times 10^{19}$. In other words, for each time-dilated second the protons spent banging about at this temperature, the universe around these protons would age by $1.38\times 10^{19}$ seconds, which is 436 billion years. As it turns out, they didn't spend nearly a whole second at that high a temperature, and in total, we appear to have a universe that aged about 25 billion years during inflation ($\frac{3 k_B T}{2 m_p c^2}>>0$), and 13.7 billion years post-inflation ($\frac{3 k_B T}{2 m_p c^2}\approx 0$).

Last edited: Mar 15, 2020
13. ### HalcRegistered Senior Member

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227
I know what a light cone is, and it is defined for an event, which seems to be 'here and now' above.

You already saw the light from the distant event, so choosing to accelerate now isn't going to make you see something that you've already seen. The light has already passed you by.
Yes, in the different inertial frame you describe (but still here and now), that distant EM event is further away. Not sure what relevance that has to the question of foliation based on an accelerating frame.

It's still on that event's past light cone. But your worldline is not since it is no longer 'here and now'.

So still not getting you. We were talking about foliation by light cones, and you're describing Lorentz transformations with are relevant to translating one inertial foliation to another, but not to foliating spacetime in the first place.

Pretty easy. They're not using an inertial coordinate system when computing those figures. They need to use a different one, since like I said, mo inertial frame foliates all of spacetime. SR has never been a model of our universe. It is only a local model, and GN-z11 is not local.

Last edited: Mar 15, 2020
14. ### Jonathan DoolinRegistered Member

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It seems you have all the parts of what I'm saying. You take the past light cone of your current "here and now" and then stack the light cone from the next "here and now" right on top of it. Do this again and again, and you have created a stack of past light-cones that fully foliates the space. Is your concern that this does not foliate all the space, but only foliates the space out to $t-t_0 \le -\sqrt{x^2+y^2+z^2}$.

Here, I'm not getting you. To me, simply saying "They're not using an inertial coordinate system" is a very ambiguous statement, and could mean just about anything. I will say, any treatment of the data that says distance is a simple function of z-shift, I don't trust. z-shift + correcct identification of absorption/emission lines can tell you the velocity. Magnitude + knowledge of the object's true brightness tells you the distance.

What do they have in regards to GN-Z11? They have,
(1) roughly, it's shape; whatever they can make out in the telescope; a 2 dimensional cross-section.
(2) It's spectral data; what proportion of light it emits in each frequency, and
(3) it's magnitude; how bright it is.

Then they have to figure out (1) which spectral lines that are easily identified that they can still be distinctly made out when it's z-shifted by a factor of 11.
(2) Then you have to determine what the object is, and how much brightness it must be emitting, in absolute terms,
(3) And then they have to decide from the measured brightness for sure how much of the dimming is caused by a pure inverse square law,
(4) and how much is caused by relativistic aberration, and
(5) how much, potentially, might be caused by gas and dust between there and here. And then you want to add on top of that another assumption that
(6) the coordinate system by which to measure the distance is non-inertial... ?

Step 1, may be easy, with good enough equipment. Step 2, I could imagine, there could be multiple hypotheses on what such a distant object is composed of, and how bright it should be.. Step 4, the aberration should be present, greatly dimming a z=11 source . Step 5 would be guess-work. And step 6? Step 6 doesn't even make any sense! If you're taking a picture of a distant galaxy, you're telescope/camera has a specific velocity at that moment in time the picture is taken. If the sun and the earth were not there, but the telescope had the same position and velocity, it would take exactly the same picture.

15. ### HalcRegistered Senior Member

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OK, this is actually the most commonly used method because of its naive practicality. It is first on my list of commonly used methods here
No, it does not foliate all of spacetime for an accelerating object. Suppose a flat SR universe and I am continuously accelerating at 1G northward. At some moment T[sub]0[/sub] in my immediate inertial frame, at location 2 LY to the south, some flash event occurs. When, using this method here, does that occur in my accelerating frame? It occurs when I see it, but when is that? Because I never will. That's my problem with it.

Doesn't sound very ambiguous to me. Problem is, any inertial coordinate system assumes a flat Minkowski spacetime, which works only locally, but real spacetime isn't shaped like that, so it just plain doesn't work to use such a coordinate system.
In an inertial coordinate system, light 'currently' shining from GN-z11 will eventually get here. In reality, it never will. What we see is light emitted from quite close by (perhaps 2 BLY away) but it took 13 BY to get here. Those numbers don't make sense in an inertial frame where light takes 2 BY to get to a location 2BLY away.

Hence a different coordinate system is used (the 3rd one only my list linked above) for such vast distances, and it strangely foliates spacetime the same way that Minguzzi's method does, except it chooses a more objective reference event than the arbitrary event of the birth of a specific pair of twins.

No, they don't have to figure out a special coordinate system for it. They already had one.

16. ### Jonathan DoolinRegistered Member

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I probably shouldn't be arguing for foliation at all, given what you've said here:

I would definitely put myself in the latter category. Why? Because given position x, and velocity x=v t of any observer, the coordinates of events on his tangential worldline x'=0, and his plane of simultaneity t'=0 are well and objectively defined. What is not well and objectively defined is whether anything actually happens AT those coordinates where t'=0, or x'=0. For instance, the observer might observe that if he does not do something soon, a train will run over him at t'=5, x'=0. All he needs to do is step off the train tracks, thereby bending his worldline, and avoiding getting run over by the train. However, when it comes to events in the region $t'<-\sqrt{x^2+y^2+z^2}$ I am an absolutist. Because these are events that the observer has already seen happen. He cannot, by accelerating, cause those events not to happen. There are three regions in total. The region $t'>+\sqrt{x^2+y^2+z^2}$ where an observer can actually prevent or cause an event to happen, $t'<-\sqrt{x^2+y^2+z^2}$ where the events have become locked in, and cannot be changed, and the region in between, where you can, by accelerating, change the coordinates where you observe the events as they cross your past light-cone, but you cannot affect their occurrence.

It all depends on what your rubric is for deciding on what makes a "good" foliation. My attitude is that a foliation should be representative of what is seen by an observer, whether that observer be real or hypothetical. It is good and rational that the foliation only includes events that the observer has actually observed, and not those that he has no way of knowing whether they've happened or not. I am planning to have coffee tomorrow morning. Do I reject the existence of time because this event in the future does not yet exist? Let's say someone throws a dodgeball at me, and I see that it is coming, and accelerate so as to dodge out of the way... There was an event in my future, where the ball collided with my face. By accelerating, I prevented that event from happening. This is the same case when you're continually accelerating. You're preventing some events from happening. For instance, you know that a huge explosion is going to happen.... You begin to accelerate away from it because you don't want to get hit by the shrapnel. By continuing to accelerate, you can exceed the velocity of the shrapnel. Still you wouldn't say the explosion didn't happen. You'd say you'd avoided it's effects. You're also not technically going to be able to continue accelerating forever. You'll run out of fuel before long.

For the hypothetical accelerating rocket with a=9.8 m/s^2, the Rindler Horizon. $X_R=\frac{c^2}{a}=9.18\times 10^{15} m = 0.971$ light years. During the time you are accelerating at 9.8 m/s^2 forward, you would not see any new events occur beyond 0.917 ly behind you. The stars in that direction would be redshifted, more and more, until the electromagnetic wavelength approached infinity. Events in that region after that would not be seen by the starship, and therefore would not be mapped to coordinates, until the spaceship stopped accelerating. That's okay. Any real rocket will run out of fuel, and stop accelerating eventually. Then the events behind the starship will be seen, as the light finally catches up.

17. ### Neddy BateValued Senior Member

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2,114
Welcome Jonathan. Somehow I missed your first post here at the time it was posted, and only read it now that I noticed that you were posting.

I agree with the summary you wrote for your critique of Dolby & Gull. I mentioned in my post #38 that I thought they were purposely trying to make a problem out of something that is not a problem at all. I also agree that they seem to be actually trying to HIDE the relativity of simultaneity (ROS), rather than to elucidate in any way.

Unfortunately, I think Mike Fontenot's current method, which he calls Fontenot's theorem, is also designed for that very same specific purpose. I'm not sure if you've read it, but there is a thread on it with a link to it in post #1:

It is rather sad to see Mike forgo his CADO ideas in favor of this new method, but I can only take his word that there are some people (physicists no less) who find it repugnant that a change in simultaneity due to acceleration can shift the time currently displayed on a distant clock toward the past. I don't see any problem with it myself.

18. ### HalcRegistered Senior Member

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What you said at first made no sense until the train example. It seems that you are confusing a preferred ordering of events (absolutism) with determinism. They're different topics. A preferred ordering of events (event definition: point in spacetime) has absolutely nothing to do with the worldlines of things that may or may not trace their way through certain events. The preferred ordering simply means that given two events, it isn't ambiguous which one occurred first, even if there is no empirical test for it.

Any objective foliation of spacetime would be entirely independent of decisions of people and paths of any worldlines.

19. ### Jonathan DoolinRegistered Member

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Birkhoff's theorem says the opposite; that the universe is locally Schwarzschild, but globally Minkowski:

(1) "In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution (i.e. the spacetime outside of a spherical, nonrotating, gravitating body) must be given by the Schwarzschild metric." (2) "An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime."

What prevents the light from getting from GN-Z11 to here in an inertial coordinate system? What makes you think the light came from only 2 BLY away?

20. ### Jonathan DoolinRegistered Member

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But within the context of SR, what are the implication of a preferred temporal ordering of space-like-separated events?

If you say there IS a preferred ordering, then what you're saying is that you're NOT ALLOWED to do the Lorentz Transformation on them, because the LT changes the temporal order of space-like separated events. You'd either be saying that SR was invalid, and do a Galilean transformation, or you'd be saying that there was a preferred $v=0$ so that any changes in ordering that the LT does are somehow illusory; perhaps temporary. (Is that the view you are espousing when you claim that Minkowski spacetime is valid only "locally"?)

21. ### HalcRegistered Senior Member

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I think he is distinguishing small scale (near any massive object) with medium scale (assorted massive objects scattered about uniformly). But I'm talking larger scale where the geometry is very much distinguishable from Minkowski spacetime. In the latter, any light emitted at any event anywhere toward inertial object X will eventually intersect the worldline of object X. This is not true of our universe where light emitted from beyond the event horizon will never reach here (our inertial galaxy cluster). Minkowski spacetime does not form an event horizon for inertial objects, only accelerating ones. Hence my example of stacked light cones on an accelerating worldline failing to foliate SR space.

Nothing. In inertial space, light will eventually get from anywhere in that space to any anywhere else in that space. But our geometry is not Minkowski, so such a coordinate system does not correspond to it globally, only locally.

More like 4 BLY actually. It seems that the object has got to 32 BLY away in 13.8 BY, putting its recession speed at about 2.3c. It isn't moving that fast, but the distance between us and it is increasing at a pace of 2.3c due to expansion. The light we see is 12 BY old, and at that time, at that speed, GN-z11 would have been about 4 BLY away (proper distance, comoving coordinates). Minkowski spacetime cannot handle objects beyond the event horizon. It is beyond it now, but it wasn't back then, so we can see it.
This is due not to the expansion of space, which would not generate an event horizon, but due to the acceleration of said expansion, which does. A regular expanding space can be represented (and foliated) with any inertial frame, and can even explain GN-z11. It is an interesting exercise to map one to another. For instance, the light from something similar to GN-z11, mapped on the current frame of our galaxy, would have the light being emitted from about 6 BLY away and has thus spent only 6 BY in transit, not 12, but it would never get so far away that light cannot reach us from it anymore.

I bring all this up because the comoving coordinate system does foliate all of spacetime. It gives an objective time to every event. There are other ways to do it, but all more complicated I think.

SR doesn't posit one. It only concludes that there can be no local test for it. It says that given two such events A and B, which occurs first is frame dependent, which is little more controversial that saying that given two dots drawn on a paper plate, which is north the other depends on which way on the plate you designate as north.

Personally I don't, but there are those that do (the two camps of physicists of whom I spoke).

You are allowed, but only one of those orderings corresponds to reality. The others are just for empirical simplicity. SR is not invalidated since it does not forbid a preferred ordering. It just doesn't assume it as a premise, nor does it conclude it.

22. ### Jonathan DoolinRegistered Member

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I think I understand. You have adopted a model similar to the Lambda Cold Dark Matter model, like what Davis and Lineweaver have here: https://arxiv.org/abs/astro-ph/0310808

Let me give a sort of list, what I think are the primary differences between their model, and the kinematic models:

LCDM: the redshift of the CMBR is caused primarily by the changing scale-factor of the universe, a function of universe age.
Kinematic: The redshift of the CMBR is caused by the relativistic redshift equation, a function of velocity

LCDM: The big bang represents multiple simultaneous events that were brought together by a scale factor a(t) which was zero when the age of the universe was zero a(0)=0
Kinematic: The big bang represents a single event with shrapnel coming out at all rapidities. No scale factor is needed to explain how how the distance between such objects was zero, or why objects with velocity recede from one another.

LCDM: the past light-cone of "here and now" extends all the way back to the big bang event itself.
Kinematic: The past light-cone extends out to the shrapnel from the big bang event.

Davis and Lineweaver don't treat the Kinematic model and LCDM-model as different hypotheses, but instead describe all of the elements of the Kinematic model as "misconceptions" of LCDM.

23. ### HalcRegistered Senior Member

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Yes, the LCDM model uses the coordinate system I mention. I've not heard of the model before, and I wonder what dark matter has to do with it.
I notice a similar drawing (mapping light cones, distant objects, etc.) does not appear for this Kinematic model. I would think the Kinematic model adds velocities the relativistic way, whereas the LCDM just uses simple addition, which is why it has objects moving faster than light.

You seem to have a contradiction going, saying:
The first line speaks of multiple events (separated by no distance due to a scale factor of zero), but the second line speaks of 'the big bang event' like there is just one of them. That's just a terminology collision. I consider it to be one event.

I agree with D&L that the model isn't a different one, just a different (or the only) mapping of the events. As stated above, I don't see such a map from said Kinematic model, but maybe I'm looking in the wrong places. I've made one myself, but couldn't work dark energy into it, so I consider it a failed attempt.

Notice that none of the diagrams in the L&N pictures depict an inertial 'now'. It could I think, but the 'now' depicted is not inertial. The line would curve downward and eventually back towards the big bang as does the light cone, but further out.