# TakenItSeriously

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## Posts posted by TakenItSeriously

### Are relativistic effects directional?

1 minute ago, iNow said:

What’s the point? You’re wrong and many people here have corrected you. Why waste their or our time any further?

I don’t completely agree with his results.

That’s why I’d like to use a trusted third party results in the Wikipedia example.

### Are relativistic effects directional?

35 minutes ago, Janus said:

You have fallen into a common trap: assuming that there is only one correct way to explain the time difference between the two twins when they meet up again.

According to Bob, Alice's clock always runs slow by the time dilation factor both on the outbound and return trip.  He see Her clock red-shifted fro the outbound trip and blue-shifted  during the return.  However, he will see the red-shift for a longer than he sees the red-shift.  He has to wait until the light from Alice's turnaround to reach until the he starts to see the Blue-shift.

Thus at 0.8c, it would take Alice 10 yrs to travel to a turn around point 8 ly distant. This means that Bob sees a red-shift coming from Alice for 18yrs at a factor of 1/3 ( during that 18 years, he would visually see Alice age 6 yrs.  From the time of turn around, it will take another 10 yrs for Alice to return to Bob, Ergo, Alice arrives back at Bob just 2 yrs after he first see's Alice's light shift from red to Blue shift. Thus he visually watches Alice age at a factor of 3 for 2 years or another 6 years.   After twenty years by his own clock, he sees Alice age 12 yrs.

No length contraction involved.

Alice, also sees Bob red-shifted on the outbound trip and Blue-shift during the return.  For Alice, she travels out to a distance of 6 ly from Bob before returning.  Thus she sees Bob aging at a rate of 1/3 for 6 yrs or aging 2 yrs before she turns around.  Right after turn around She will see Bob age at a factor of 3, this holds for the entire 6 years for the return trip, and sees him age another 18 yrs.   After returning to Bob he will have seen him age a total of 20 years while she aged 12 yrs. The difference being that Alice see the shift from red to blue shift immediately upon turnaround, and doesn't have to wait.  This is because Alice is the one undergoing an acceleration in order to change velocity and sees the effect of this change immediately.

As far as what happens to Bob's clock during the outbound and return trips:  According to Alice it runs slow during both legs due to time dilation. ( it is important here to distinguish between time dilation, which is the comparison of clock rates between relatively moving frames, and Accumulated time difference, which can be the result of time dilation, length contraction and the relativity of simultaneity).

Bob, indeed measures less elapsed time due to length contraction, as the 6 ly distance he measures is shorter than the 10 measured by Bob.  But this applies to both legs of the trip, and has no bearing in terms of red or blue shift.

The reason that Alice ends up agreeing that less total time accumulated for her is due to the fact that She had to undergo an acceleration in switching between outbound and inbound legs,  and doing so invokes a new set of rules as to what she concludes happens to Bob's clock.  She will conclude that during this acceleration Bob's age advanced by a great deal (12.8 years), so that even though he only aged 3.6 years during the outbound leg, and 3.6 years during the return leg, He aged a total of 20 yrs for the whole trip.

The empirical data only tells us the difference in their ages when they meet up again. It say nothing about how that difference came about.

Bob will say that it was because Alice aged slower during the whole trip, While Alice will say that Bob aged slower for large part of the trip, but aged a great deal for a brief part of the trip.

Both of these views are equally correct.

The upshot is simply that Alice and Bob measure time differently during the different parts of the trip, but come up with the same end conclusion.

Time measurement is frame dependent, and there is no absolute measure of who's clock was "really" ticking slower than the others at any one point.  There is no single "reason" why Alice returns younger than Bob, every frame will have a different reason and they all are equally valid.

This is simply the way things are.

22 minutes ago, studiot said:

A little bit of editing required ?

I agree. (respectfully)

To keep things relatively simple can we use the Wikipedia example so that the problem has a known reliable baseline that we can check results to:

This example problem uses a proper distance of 4 light years each way and traveling at 80%c.

If you like, I could outline the solution to save you some time or if you’d prefer to adapt your solution that’s fine as well.

### Are relativistic effects directional?

I’d like to keep this thread focused on the Janus discussion if you don’t mind which I left off here:

### Are relativistic effects directional?

27 minutes ago, studiot said:

I think you will be (mis) remembering the part where Einstein showed that time dilation/ length contraction is zero at right angles to the direction of travel and increases to its full value as the direction vector rotates to become parallel to the direction of motion.

I think Janus has linked to some striking videos showing the results of this effect.

I don’t see any links posted by Janus.

He had stated that relativistic effects were true in the horizontal and vertical access and I was going to argue that assumption was in error.

I honestly can’t say anything about what I can distinctly remember about SR at least in regards to directionality. I first learned about it in the early 70’s when I was a child and my memories about it are spread across the entire range of time until now.

Scratch that. I can remember learning about SR in school years later.

The teacher had stated “length contraction in the direction of travel” and I always interpreted that as literally only in the direction of a moving ship.

I also recall asking him if he meant to say that the dimension of space in the direction of travel not just the length of physical objects. We disagreed on that point and I quietly conceded that he was wrong because it does no good to argue with your teacher in grade school.

### Are relativistic effects directional?

1 hour ago, Janus said:

with Relativistic blue-shift, you visually a higher frequency arriving from the approaching source, but that does not mean you conclude that time is proceeding faster for the source.

I agree.

1 hour ago, Janus said:

With classical Doppler shift, you also would see a higher frequency arriving from the approaching source but you would not conclude that time runs faster for the source, ...

Again I agree. I wasn’t trying to assert that time must be running faster. I was only trying to assert that you can no longer link time dilation as the cause for the relativistic blue shift.

1 hour ago, Janus said:

once you factor out the decreasing distance effect, you would conclude that time runs at the same rate for the source as it does for the receiver.    With Relativistic Doppler effect, once you factor out the decreasing distance effect you are left with time dilation and events occurring slower at the source.

You mean classical distance effect? In that the distance is decreasing classically due to motion?

I wasn’t trying imply anything about that. I was trying to say that relativistic blue shift must be due to length contraction of the distance. i.e. from Alices point of view at the turn around, her distance to the Earth is 0.6*4 ly or 2.4 ly due to length contraction.

1 hour ago, Janus said:

With Relativistic Doppler effect, once you factor out the decreasing distance effect you are left with time dilation and events occurring slower at the source.

Ok, here is where it gets super tricky.and deviates from what most people assume is happening so please try to keep an open mind.

From Alices point of view, is time still dilated on the return leg when Alice is now approaching Bob?

Empirically, we can’t conclude this because both clocks now seem to be running faster so I’m saying that time is not progressing faster or slower.

I’m saying that length contraction is nothing more than time dilation except as seen from the other side.

Again, relativistic redshift is due to time dilation of the frequency.

Relativistic blueshift is due to length contraction of the wavelength.

If this has got you thinking, there are more consequences to it that can resolve a whopping mystery, but you kind of need to accept this step first.

### Are relativistic effects directional?

I’ll need some time to reply to this but will reply to Janus first.

In the mean time, how do you split up quotes in a reply. I can’t recall off hand as it’s been a long time since I last posted here.

12 minutes ago, Strange said:

For example:

and:

As you can't remember the quotation or the source, I think it is more likely you have misremembered it than that the mathematics of SR is wrong.

Never mind, I figured it out.

I need to grab a bite and run some errands but I’ll try to reply today.

### Are relativistic effects directional?

3 minutes ago, Strange said:

You just have to look at the derivation.

That is not what you said originally. But that has nothing to do with the object moving towards or away from the observer. It happens in the direction the object is moving, which could be towards/away from (both are equivalent) the observer. Or any other arbitrary direction.

Let's say the +ve Z axis is away from the observer and the -ve Z axis is towards the observer. The +X his is to the right, -X to the left. +Y is up, -Y is down.

If the object is moving in either the +Z or -Z direction at 86% c, then it will be half the length in the Z axis.

If the object is moving in either the +X or -X direction at 86% c, then it will be half the length in the X axis.

If the object is moving in either the +Y or -Y direction at 86% c, then it will be half the length in the Y axis.

If it is moving in some other direction, then you just need to decompose it into the velocity vectors in each axis.

Who’s derivation? can you provide a link?

If I’m not mistaken, I believe the direction of travel quote was from Einstein himself. I cant recall exactly my source for this belief however length contraction is always stated in terms of the direction of travel.

I don’t think that Einstein was prone to making mistakes in hidden assumptions in language given the theory itself which is inundated with such intuitive traps.

### Are relativistic effects directional?

7 minutes ago, swansont said:

That's the wrong approach. There's nothing that says that it is. You have to assume something extra to put directionality into it.

The directionality of the doppler effect comes into play with the sign of v. The Lorentz factor uses v^2, so the sign goes away.

(edit: and now I see Janus has given a more complete answer than my short summary)

6 minutes ago, Strange said:

You are wrong. Time dilation is independent of direction. Length contraction happens in the direction of movement. That is defined by the mathematics of SR.

I’ve never seen anything that shows that time dilation is independent of direction.

as far as length contraction goes, I agree. Length contraction does happen in the direction of travel. Not that length contraction happens in both the direction of travel and opposite to the direction of travel.

### Are relativistic effects directional?

1 hour ago, Janus said:

Relativistic Doppler shift is is a combination of Time dilation and classical Doppler shift effect.  The Classical Doppler Shift effect is simply due to the changing propagation time for the light due to the changing distance between source and receiver.

So in your example of a source approaching at 0.8c and having it light blue shifted by a factor of 3:

Start with a source 1 light hr from the receiver ( as measured by the receiver),  The source is sending out  signal at 1 hz (as measured at the source).

I agree that the relativistic component of relativistic redshift is caused by time dilation.That much seems to be logically self evident.

I don’t agree that time dilation can explain the relativistic component of relativistic blue shift.

Remember, time seems to be speeding up on the return leg and I think you would agree that time dilation never speeds up no matter how you look at it.

1 hour ago, Janus said:

The tail end of the light wave will be emitted 1 sec later according to the source, and  1 2/3 sec later according to the receiver. In that 1 2/3 sec, according to the receiver, the source will have moved  1 1/3 light second closer.  Thus the tail end of the light takes 1 1/3 light sec less time to travel the distance between source and receiver.  Leaving 1 2/3 seconds later, and taking 1 1/3 sec less to make the trip means it arrives at the receiver 1/3 sec after the front end of the wave arrived. Since each successive wave follows immediately after the previous one, the receiver measures a frequency of 3 hz, compared the 1 Hz measured at the source.

Baring time dilation, this analysis no longer has a causality attached to it.

I agree with your math but I would posit that the causality for it is due to length contraction not time dilation. i.e. wavelength is decreasing due to length contraction not that frequency is increasing due to time dilation.

### Are relativistic effects directional?

1 hour ago, Janus said:

Here v is not squared, so the stipulation is that v is positive if the source is receding, and negative if it is approaching.

This is what I’m saying.

You need to assign a negative sign to the approaching direction which suggests directionality.

### Are relativistic effects directional?

13 minutes ago, swansont said:

That doesn't jibe with "positing something different"

As far as I’m aware, nothing in SR states that time dilation or length contraction is not directional. It’s only an intuitive assumption that it’s not.

Wouldn’t you at least agree that relativistic doppler effect is directional?

### Are relativistic effects directional?

I’m not contending that special relativity is wrong, only that our understanding of it is not fully realized.

Nothing in my post is inconsistent with the theory that Einstein discovered.

### Are relativistic effects directional?

1 minute ago, Strange said:

Then you need to show how this new theory can be derived from first principles and how it is supported by evidence.

My reasoning is not circular. It is simply a statement of fact.

This discussion is theoretically based not experimentally based. I don’t have the bank roll to fund an experiment of space ships that travel at relativistic speeds, not that we have the technology to do that anyway.

### Are relativistic effects directional?

1 minute ago, Strange said:

No, that is not what the theory says (and not what is shown by experiment). Time is dilated whatever the relative direction; in other words it is only related to speed, not direction. Length contraction occurs in the direction of motion (also unrelated to the direction relative to the observer).

I agree, that’s what we learned in school but I’m positing something different and your reasoning is therefore circular.

### Are relativistic effects directional?

We know that relativistic doppler effect is directional in terms of relative motion. That is to say that light becomes red shifted when it’s source is receding away from us while light becomes blue shifted if the light source is moving towards us.

But what about time dilation and length contraction? Could it be that time dilates only in a moving frame that is moving away from us? and that length only contracts in the sense of a frame that is moving towards us?

This was one of the important consequences I had mentioned in my explanation of the Twin Paradox which I had posted here:

In short, if you do the math, we can only say that from Bob’s perspective on Earth, time appears to be dilated due to relativistic redshift.

That is to say that if the light we are observing is a powerful radio transponder that pings at 1Hz, then from Bob’s perspective, he receives only 1 ping every 3 seconds due to relativistic redshift.

Now, this is not just due to time dilation because, as we know, relativistic redshift includes normal redshift plus the relativistic component based on the Lorentz transformation.

If we calculate time dilation alone at 80% of c the Lorentz factor (α) is 0.6 so time is only reduced to 60% of proper time.

That’s all well and good, but when we observe the ship on the return leg, Bob receives 3 pings per second! as if time were somehow speeding up in the moving frame which we know is not possible.

We already know that time dilation can never be speeding up beyond proper time and length contraction can never be expanding beyond proper length.

So what relativistic effect is responsible for this relativistic blue shift?

I would posit that length contraction is what causes the waves to compress beyond that due to normal blueshift.

Another words, we know that information must be conserved and of course waves are basically information. So we know that waves (or pings) cannot just be added to or removed from the streaming radio signal.

I know that the pings sent at 1 Hz are not waves but a composite of many waves but for all intents and purposes, we can think of them like light waves with a frequency of 1Hz.

We also know that the streaming radio signal is mostly still in the space between Earth and the ship as it approaches us as waves in space.

So my point is that if the distance between Earth in the approaching ship is length contracted, then so must be the wavelength of those waves.

The bottom line is that time dilation causes the relativistic redshift for receding light sources in terms of emitting fewer waves per second and length contraction causes the relativistic blue shift in terms of a shorter wavelength.

### Could signal encryption explain the Fermi Paradox?

This wasn’t my idea but Edward Snowdens which I heard on an episode of Startalk and it just struck me as a very compelling argument.

He posited that an intelligent species would have a relatively small window of broadcasting radio signals in the clear before they started encrypting all their radio communications to be indistinguishable from background noise as a matter of course.

Personally I can’t think of any reason why this couldn’t be true and it makes perfectly good sense.

It also makes me think that it may already be happening but that’s besides the point.

What do you guys think?

### Adding Time to 2D PFHM

On 12/6/2018 at 10:12 PM, Trurl said:

I like how your matrices did not have to rely on the previous matrices. I don’t understand the patterns you are using or what your method is. Maybe you could explain in a book format.

Each pattern designates the locations of all multiples for a given prime factor. For example the first pattern designates the positions for all multiples of 2, the next matrix reveals the positions for all multiples of 3 and so on for all given prime factors, for a given level of the matrix.

The number of prime factor patterns required depends upon the range of numbers or the level of the matrix that you wish to look at.
On 12/6/2018 at 10:12 PM, Trurl said:

What interests me now is if you can take my equation and find a pattern between Prime numbers.

N^2 = ((((p^2 * N^4 + 2 * N^2 * p^5) + p^8 / N^4) – ((1 – p^2 / (2 * N)))) * ((N^2 / p^2)))

I have other less complex equations that will prove p is Prime knowing q and N. The equation is cumbersome, but it will show if 2 numbers are Prime knowing all values.

Please define your variables and double check your use of brackets as there seems to be some redundancy included.

### Adding Time to 2D PFHM

Quote

I enjoy the idea of applying physics to Prime numbers. There should be a wave that shows a pattern in Prime numbers. I once posted the idea of having a logarithmic spiral to show a pattern in Prime numbers. I couldn’t get it to work but relating geometry to patterns does things computation can’t.

I completely agree that patterns are important in number theory but, in this particular case, rather than looking for patterns indiscriminately and then trying to explain them after the fact, I had first deduced that the harmonic patterns must exist based on the periodicity of prime factors and the resulting patterns had confirmed those deductions.

Quote

I think the entire problem of finding a pattern in Prime numbers is starting at zero. That is how we count but finding a series is near impossible. Have you ever thought of starting at a starting point other than zero? It may be impossible not to. But I do like your computation and charts. I also like your idea of relating them to physics.

Two points:
1. I discovered an interesting symmetry depending on whether a PFHM starts with 0 or starts with 1.
Try the following excercise:
Take any PFHM starting with 1 and highlight the prime numbers to reveal the harmonic patterns.

You will note that a symmetry exists in the x axis with the exception that the first column on the left is a prime column while the second from the last column is a prime column.

When we take the same matrix only start it with 0 instead of 1 the assymetry swps such that it is the second column from the left that is prime while the last column on the right is prime.
1. The second point is that when using a PFHM, one needs not begin at the beginning every time such as with a number seive. That is one of the huge advantages of using a matrix.
You can start with the first matrix, the second matrix or the millionth matrix
e.g.
for a 30x7 matrix we can calculate the primality for the ranges:
1-210, 211-420, or 209,999,791-210,000,000
without needing to calculate the primality of all preceding matrices first!
for instance here is the primality of the millionth matrix which I derrived directly without first derriving all 999,999 preceding matrices.

Note that there may be errors involved since it did require creating a composite of 1,698 prime factor patterns.

Quote
I will close with this idea. What if you stop looking at a pattern in Prime numbers and look for patterns in the way they interact with other numbers. For example, I have been trying to solve semi-Primes. If you could prove a number is a semi-Prime, its factors are Prime numbers. So if you take one known Prime number and multiply it by another number if you could prove the resulting number is a semi-Prime, the unknown number is Prime. So what I am saying is that if your charts tested for Primality based on one known Prime and a test value forming a semi-Prime, you would have a pattern. I know this is no easy task. But looking at Prime numbers for awhile now I don’t think a pattern will be formed without somehow placing Prime numbers into a known function, and then find a pattern in that function. Which I think is what you are trying to do with physics, harmonics, and time. I am just suggesting using semi-Primes to see what you can come up with.

I’ve been meaning to look into patterns of semi-primes within a PFHM, though I haven’t had the time to look into it yet. I will post any new results regarding semi-primes if I discover any in the future, however, you should note that my first priority at the moment is to find a method for unbounded data compression.

Quote

Also you’ve got to teach me how to create these matrices. And how you are getting those graphics of the patterns. That is just awesome.

Thanks, I appreciate it.

The key to understanding how to create a PFHM are the formulae that are needed to automatically derive the primality patterns within Excell for which there are too many details, in general, to discuss easily within this forum.

I may post pics showing the formulae of certain key cells at a later time.

### Adding Time to 2D PFHM

Just because it’s kind of cool to look at here is an expanded view of the animated 30x7 PFHM stack:

Figure 2: An expanded view of the prime factor patterns.

Above is actually only a partial view of a large array of prime factor patterns in Excel. It starts in the upper left with the harmonic patterns for primes 2, 3, 5, & 7.

Each row actually contains 38 PFPs and the number of rows can extend indefinitely until Excel runs out of memory and crashes. Initially, I tried to include enough prime factors to define the primality up to the millionth matrix (numbers from 1 up to 210,000,000) which requires 1700 prime factor patterns for the primality to be fully defined. Oddly it was at the millionth matrix when Excel started crashing.

The animation represents matrix levels from 10,000 to 10,033 for 33 frames. I chose 33 frames so that the fifth prime (11) would appear to be smooth when displayed in a repeating loop.

### Adding Time to 2D PFHM

I thought you guys might find this interesting.

Adding Time to a 2D Prime Factor Harmonic Matrix to demonstrate the “behavior” of standing vs moving prime factor “waves”

Previously I had introduced the Prime Factor Harmonic Matrix which showed that prime factors behaved like waves or specifically 1 dimensional waves that either behave like moving waves or standing harmonic waves within a 2 dimensional matrix of natural numbers.

A PFHM is simply any matrix of natural numbers that is dimensioned according to a primorial.

When this is done, the pattern of prime factors that make up the primorial behave like standing waves while prime factors that are not part of the primorial would behave like moving waves.

In order to better demonstrate this behavior, I added the dimension of time to a 2D 30x7 PFHM which is orthoganal to the plane.

Another words I created an animated gif which shows a progression of matrices level by level. i.e.
210 = 2x3x5x7
1. 1-210
2. 211-420
3. 421-630
4. ...

Figure 1: A series of 30x7 matrices of natural numbers such that their prime factors are distributed periodically throughout the matrix. The slot on the left represents the factors of prime number 5, the second shows the factors of 7, the third shows the factors of 11 and the fourth shows the factors of 13. Each frame of the animation shows a progression of levels in the matrix.

Note that since 5 and 7 are both factors of the primorial 210, they never move or they behave like standing waves while all larger prime factor patterns each propagate at a different rate or a different “frequency over time”.

### Finding large Primes using Standing Wave Harmonics

On 8/25/2018 at 2:55 PM, zerodrama said:

This is not a sieve.

It is a kind of map.

You're exposing patterns by changing the base.

Check this out

Yes, that’s exactly right.

A seive is one dimensional which is why they always need to be started from 0 and very innefficient in terms of storage space while the matrix can be two or three dimensional and the base can be a primorial of any size. The larger the base, the more composite number collumns are segregated out and the greater the probability that the remaining numbers are either prime or semi-prime.

There are a number of huge advantages over a number sieve.

Thanks for the link. It does indeed appear to be based on the same principles though I never intended it to be used as a method for faster prime factorization.

My aim, at least in part, was to find a method for a faster primality test for discovering large primes, especially Mersenne primes.

I can see how they might succeed using quantum probability analysis to speed up the factorization process though.

Prime factors are harmonic which was the basis of the PFHM after all.

### Why can’t we derive velocity directly from it’s doppler factor?

1 hour ago, Mordred said:

One has to apply each type of redshift for each dynamic involved.  While light crosses a gravity well, we need to apply gravitational redshift. For a galaxies actual motion relative to the observer, Doppler shift. For an expanding universe during transit times between emitter and reciever cosmological redshift.

All three can be involved on the same signal.

Yet Relativistic Redshift isn’t accounted for, for galaxies that are receding at speeds appraching the speed of light?

That suggests that time dilation isn’t accounted for at those speeds either.

Time dilation will cause luminosity to dim which could have easily been overlooked just as length contraction was overlooked for the cause of time deviation in the Twin Paradox.

What may seem obvious in hind-site is not obvious in fore-site. Please trust me on this. It has been the lifelong bane of my existance.

### Why can’t we derive velocity directly from it’s doppler factor?

8 minutes ago, Janus said:

Where did I say anywhere that light was an invalid source?  Both the examples I gave, for the Relativistic Red shift and Cosmological red shift, used light.  Nor did I ever imply that SR is wrong.

There is more than one mechanism that causes red shift.  They  each have  a different formalization, and you can't use the formulation for one to calculate the effect of the other.  You cannot use the Relativistic Doppler shift formula to calculate cosmological recession because they are different effects.

Ah, I see.

So your suggesting that if you use relativistic redshift then the numbers don’t agree with the Cosmological data is that correct?

### Why can’t we derive velocity directly from it’s doppler factor?

On 5/5/2018 at 2:23 PM, Janus said:

Let's put it this way: the Relativistic Doppler shift only depend on the relative velocity difference between the source at the time of emission and the receiver at the time of reception.

For example, if the source is moving at 0.5 c relative to you at the time of emission, but you accelerate up to 0.5 in that same direction just before the light reaches you, you will measure no Doppler shift because  relative velocity difference between you at reception and the source at transmission is 0.

However, with cosmological red-shift we are dealing with the geometric expansion of space between the time of emission and reception.  This stretches the light waves.   So lets; say there was zero expansion of the universe at the moment of emission, then during some period between emission and reception there is some expansion. the light wave will share this expansion. Now the expansion stops before reception so that it is zero again. However, this does not mean that the light wave reverts to it original length. For that to happen, there would have to been a contraction of the universe.   This means that you the observer will measure the red-shift caused by this period of expansion even though it no longer exists a the time of reception.

So to put it simply, Relativistic Doppler shift only depends on the relative velocity difference between emission and reception ( You could change your velocity to many times while the light it traveling towards you, but the only thing that counts is your velocity at the moment of reception.)

Cosmological red-shift  is dependent on what occurs during the entire period that the light is traveling and not just the conditions at the "end points".

I’m not following the basis of your argument.

Luminosity which is used to determine distance is essentially the same source and time that relativic redshift came from. So how could light be an invalid source of information in the case of redshift, but be a valid source for distance?

In fact light is essentially the only source of information used in this case.

It seems as though you are just trying to make the claim that Special Relativity is wrong.

### Why can’t we derive velocity directly from it’s doppler factor?

Since we can measure the doppler shift, relativistic or otherwise, of a receding galaxy why can’t we use the equation for calculating the Relativistic Doppler Factor (fs/fo) which I will call r for ratio, to derive the galaxies recessional velocity?

fs/fo = √[(1+β)/(1-β)]
r = fs/fo
r = √[(1+β)/(1-β)]
(1+β)/(1-β) = r²
β+1 = r²(1-β)
β = r²-r²β-1
r²β+β = r²-1
β(r²+1) = r²-1
β = (r²-1)/(r²+1)

where:
fo is the frequency that an observer sees
fs is the source frequency that we can find from its spectrographic footprint
β = v/c
r = fs/fo

If the galaxy is in redshift then r > 1
⇒ 0 < β < 1

If the galaxy is in blueshift then r < 1
⇒ 0 > β > -1

The negative velocity only means that the objects are moving closer together instead of further apart so we can see that the absolute speed should never exceed c according to Special Relativity.
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