kurdewiusz

For null geodesics, the integral of the Schwarzschild metric can be reduced to the same form as the integral of the FLRW metric, which gives the time integral of the reciprocal of the scale factor. The only difference will be its equation as a function of time. We have a black hole or a star and a gravitational change in the wavelength of a photon falling on the event horizon of the hole or emitted by the star. We can calculate its path by integrating over time the inverse of the scale factor calculated from the Schwarzschild metric as a function of time. Along with passing time, the radial distance between the photon and the hole or star decreases or increases, and the value of the scale factor and the photon's wavelength change with the distance. The problem is that we use the same integration to calculate the photon's path through the expanding space. Expanding, intergalactic, time-scaling space differs fundamentally from gravitationally curved space, scaled with radial distance, because in the former, there is no stage of the background photon's travel, when there are simultaneous sections of its path with different spatial and temporal scales, as there are in the latter. The entire expanding space always scales uniformly. At all times, the scale factor has the same value in the entire expanding space, while in gravity-curved space, the scale factor has different values in different places at all times. For this reason, when using the same integration to calculate the path of a photon moving through the expanding space, we calculate its path in non-existent space, which is why we get a wrong result.

Retaining the scale factor value distribution from the past along the path traversed by CMB photon not only implies the different expansion rates at different distances at the present moment. This integration requires the different time flow (that is also defined as a value of the scale factor a(t) or its reciprocal) in the place of the emission of CMB that reach us today, and in our place. This integration requires the different time flow in these two distant places Now, but the current flow of time in these two distant places is currently the same.

Yet another, similar but different objection: By definition, the scale factor is always equal to 1 at the present moment. Somehow, nobody takes into account the fact that it is always Now (except maybe Eckhart Tolle). Everyone treats the past as if it was carved in stone, in which the function of the scale factor has been carved. However, a plain fact, that the point (Now, 1) moves in time with us, implies that this function is changing its shape. If we calculate the current size of 13.8 billion years universe now and in 1 billion years (by calculating its size 1 billion years backwards) using the same integration, we will get two different results, because the function's shape will change, and we're calculating the area under the curve of its reciprocal.

If you try to argue, that the scale factor (a) is defined to be equal to 1 just for the present age of the universe, I will ask you, what about (z+1)=1/a? What happens with the redshift (z), if the scale factor (a) exceeds 1?

We were done a few times already

21 hours ago, andrew s said:

I suspect I did not understand that either. In comoving coordinates there is no time dilation. 

Exactly! But this integration requires it.

We wouldn't go in circles, if you didn't run away from this error.

Moreover, this numerical integration, which you linked, was my start point. My plot is their plot, and I've also added this link in sources.

1 hour ago, andrew s said:

I have read and reread your concerns about the integration and I don't understand it. It seems metaphysical.

Long ago, at the beginning of this conversation, you wrote: "The integral is well defined and I believe your concept of "retaining the scale factor from the past" is mistaken or at least I don't know what it means."
I answered: "It means, that the current time flow in the place of emission of background photons is much greater than ours. If the scale factor is retained, time dilation is retained as well."
You didn't reply to my answer.
I want to clarify, that this is the condition for this integration to be correct. This condition is not correct, thus the integration is not correct.
This integration requires the different time flow (that is also defined as a value of the scale factor a(t) or its reciprocal) in the place of the emission of CMB that reach us today in our place.
This integration requires the different time flow in these two distant places CURRENTLY, but the current flow of time in these two distant places is currently the same.
This integration is applicable for the path of the gravitationally redshifted photon. Gravitationally curved spacetime retains its distributions of time-scale factor (time dilation) and space-scale factor (length contraction) at all times at all distances, so we can integrate it over time or the redshift. That is not the case with expanding, intergalactic space.

As to your calculations: I dare to claim, they're are wrong for the same reasons I'm giving you and repeating.

And don't think I will forget about a0/a = a(tr)/a(te) = 1 for a = 1 in case of the particle horizon calculation. I will keep reminding you this error, until we're done.

btw, that was my update before they deleted my post, so you know, that my reasoning was your reasoning with the metric:

Thank U, this is the best thing you could write right now. Nevertheless, physics and maths will never be a popularity contest for me, unless a voter gives the explanation of his vote.

It doesn't matter how simple is the equation that we're integrating over time. Numerical integration only shows the problem with it.

Just now, andrew s said:

This is as I have repeated is nonsense unless the tr= te.

Andrew, it's your nonsense. It follows from your formula.

I also described two cases, where this integration is indeed applicable.

1. Indeed, it was for your benefit, but also for my detriment. You wrote it all, to cover the problem with a0/a = a(tr)/a(te) = z+1 for a =1, that implies a(tr)/a(te) = 1 for a=1, that can't be true for te equal to the time of emission of CMB in case of the calculation of the particle horizon, that is the observable universe size.  If there is one case, when the formula is wrong, it's no longer a valid formula, period.

2. Almost everything you wrote is true and your reasoning was also my reasoning. Everything, except one thing, and I wrote about it:

On 12/08/2023 at 16:54, kurdewiusz said:

In my opinion, the source of inconsistency between general relativity (which is supposedly a basis of calculations giving 46.5 GLy and 3.2 c values) and the Doppler is in calling the integration-based calculated recession velocity (as well as the proper size of the universe) a part of GR. Friedmann equations, as a solution of Einstein's equations, are a part of GR. Friedmann–Lemaitre–Robertson–Walker metric is a part of GR. Explicit form of the scale factor, derived from the Friedmann equations, is a part of GR. In my opinion, what is not a part of GR, is the integration of the general metric that gives the current, proper distance. This metric equation uses the explicit form of the scale factor derived from GR and may be valid for every single spacetime frame, but if you integrate it, you get this:  *[Back to my first message with my images and the description.]

You wrote again, that this integration is not an issue, (implicitly, because there are no issues), so I'll repeat myself too. No matter how many times you'll repeat, that this integration has no issues, you will not get rid of the issue with the integration described by me, unless you refute my argumentation regarding the integration itself and also the problem with a0/a = a(tr)/a(te) = z+1 for a =1, from which you ran away.

@andrew s please, don't, there's nothing more to argue about, unless you want us both end up in a lunatic asylum. Nevertheless, I am ready for it if necessary.

@andrew s
a0/a = a(tr)/a(te) = z+1 (we both agree on that)
a0/a = 1 for a =1
THEREFORE
a(tr)/a(te) = 1 for a = 1 (can't be true in this case, so this formula a0/a = a(tr)/a(te) is wrong for a=1)

1 hour ago, andrew s said:

No it means there is zero time and distance between the time of emission and reception. That is the source and defector are next to each other. Totally irrelevant to the detection of the CMB. 

So we agree, that the case of the calculation of the observable universe size is not the case, when the emitter and the receiver are next to each other. In case of the calculation of the observable universe size, te is the time of CMB emission, so it can't be equal to the reception time tr = Now, but that's what follows from the formula a0/a = a(tr)/a(te) = z+1 for a =1, that implies a(tr)/a(te) = 1 for a=1. 

9 minutes ago, andrew s said:

That is not my formula. I have never posted that. It's yours.

a0/a = a(tr)/a(te) = z+1 (we both agree on that)
a0/a = 1 for a =1
a(tr)/a(te) = 1 for a = 1 (can't be true in this case, so this formula a0/a = a(tr)/a(te) is wrong for a=1)

13 minutes ago, andrew s said:

This is your error tr can't equal te. The CMB we see now was emitted some 13.6 Gyr ago and we observe it Now.

(...)

I have been consistent on this throughout. 

Unless you accept your error there is no point continuing this discussion. 

That's not my error, that's Your error! That's what I'm trying to show You!

Constancy of te excludes the possibility, that te=tr for a=1, so the formula z+1 = a0/a = a(tr)/a(te) = 1 is no longer true, because a(tr) and a(te) can't be equal in this case, because te and tr can't be equal in this case for a=1.

So your formula a(tr)/a(te) = 1 is no longer true.

@andrew s Sorry, false suspicion. I repeat:

You wrote:
"It is a variable as it depends on what you are observing. For the CMB its zero but for Andromeda it will Now - 2.4 Myrs"

I wrote earlier: You say, that te is variable. In the observable universe size calculation, that is the integration of the reciprocal of the scale factor function a(t), te is the time of the emission of CMB and tr is the variable, that is Now.
You wrote: Correct

Is te variable or a constant IN CASE OF THE CALCULATION OF THE OBSERVABLE UNIVERSE SIZE? I remind you, that you need tr=te for a(tr)=a(te) for a=1.

@andrew s Did you edit your comment with your claim, that te is a variable?

@andrew s You say, that te is variable. In the observable universe size calculation, that is the integration of the reciprocal of the scale factor function a(t), te is the time of the emission of CMB and tr is the variable, that is Now.

Indeed, you need tr=te for a(tr)=a(te) for a=1.

Bye

5 minutes ago, andrew s said:

I don't think I changed anything tr is always the time received and te the time emitted. 

If you don't change the definition of te or tr in a=1, you will never get the equality a(te)=a(tr), but you need it in a=1.

te=0 for both a=0 and a=0.99999, it's not a variable, but constant. What you wanted to teach me is that t=0 for a=0, no need.

5 hours ago, andrew s said:

No.

a(te) is the scale factor for when the light was emitted I.e. in our past. a(tr) is the scale factor at the time it is received. 

That was your answer to my question, how is it possible, that a(tr)=a(te). Back then you had the old definitions of te and tr in mind, but you changed them.

Please...

For a = 0.99999, te = 0 and tr is the age of the universe. For a = 1.00001, te becomes equal to tr, that is the age of the universe, or both te and tr become almost zero, that is the time of the emission. You change the difinitions of te and tr in a = 1 to explain the equality of a(te) and a(tr), but it's still the same function. If it's the same function, you can't change the definitions of time parameters.

z+1 = a0/a = a(tr)/a(te) = (z/x)/(y/x) = z/y = 1 for a=1

a(tr) = (z/x) = a(te) = (y/x) for a = 1

z/y = 1 for a = 1

Changes nothings, adds nothing, explains nothing, a(tr) is still equal to a(te) for a = 1.

@andrew s a is normalized for a<=1, a(tr) may be normalized, a(te) may be normalized, their quotient a(tr)/a(te) is > 1, therefore not normalized. And the scale factor function (on the plot given in this post) ranges from 0 to infinity for time as well as for value, therefore it's not normalized.

Normalized and unnormalized values need to be equal for a=a0=1. You didn't explain the fact, that for a=1.00000001 the values of a(tr) and a(te) are practically equal, since we're already in unnormalized range for a > 1.

z+1 = a0/a = a(tr)/a(te) = 1 for a=1

It just ocurred to me, that i explained it, but as a mistake, in the description of my drawing: "then the photons reaching us would have the original wavelength form the time of their emission".

@George Jones i get it. Thank you for your neutral attitude and your choice of words - my respect.