a-cosmology-group/acg#358

Back to your introduction, black holes are not considered one of the traditional “pillars” of the Big Bang model.

The Big Bang universe started as a singularity, they say. Now it seems to me that such a (literally) incredible concentration of matter at a single point IS a black hole.

My understanding is that there is no pressure in a neutron star, it’s the Pauli exclusion principle that prevents new neutrons from going to the ‘already-filled’ low-energy states.

Very reasonable viewpoint I would say. But it seems to me this is not the picture from which TOV limit is calculated. Ok, I don’t have ready links for explanation how TOV limit is derived. I will try to look for some.

Back to your introduction, black holes are not considered one of the traditional “pillars” of the Big Bang model. So eliminating them won’t help arguing against the Big Bang. The simple argument that shows that black holes don’t exist is that as matter falls into a dense mass, time slows down as seen by an observer (us) outside the dense mass. This produces a ‘Zeno paradox’ where the infalling particle can never reach the speed of light and never reaches a hypothetical horizon.

Good point, frozen stars would work just as well for Big Bang model. So I should instead specify that I see problem with runaway gravitational collapse irrespective of end result of that collapse (black hole or frozen star).

The Big Bang universe started as a singularity, they say. Now it seems to me that such a (literally) incredible concentration of matter at a single point IS a black hole.

Look, it does not work. There is definitely outside for black hole, but there is no outside for Big Bang universe. So it’s clearly different concepts.

Runaway gravitational collapse is self-limiting under General Relativity because the speed of light diminishes inversely with the gravitational field strength. Since E/m=c^2 a collapsing body will, of necessity, shed mass by emitting excess radiation, resulting in a luminous compact body, which is what we observe - quasars.

The strong gravitational field of the quasar will also produce a significant gravitational redshift which, if interpreted as part of the cosmological redshift, will make the quasar appear to be more distant than it is. This is consistent with Halton Arp’s observations of discordant quasar redshifts and explains why there do not appear to be any cosmologically nearby quasars. Quasars are physics, Black Holes are just math.

Runaway gravitational collapse is self-limiting under General Relativity because the speed of light diminishes inversely with the gravitational field strength.

Gravitational collapse is runaway if there are no means to reverse this collapse. Diminishing speed of light does not seem to provide any means to reverse collapse.

Since E/m=c^2 a collapsing body will, of necessity, shed mass by emitting excess radiation, resulting in a luminous compact body, which is what we observe - quasars.

Well of course, collapsing body is shedding mass by radiating out some binding energy but I’m not sure about “necessity” in that process. I believe binding energy is limited by depth of gravitational well and some other factors too (say Pauli exclusion principle). But to the last point about quasars being physics and black holes being just math I would certainly agree.

The two sentences you quote are interrelated. A unit of matter, a proton for instance, has a mass-energy that is dependent on c. If c is diminished in the vicinity of the matter-unit it has to reduce its mass-energy content in accord with the reduced value of c. The only known way for it to do this is to emit electromagnetic radiation. As long as E/m=c^2 holds radiative emission is an inescapable consequence of an increasing gravitational field strength.

That emission process would in turn reduce the particles mass-energy content. The strength of the associated gravitational field is dependent on the mass-energy of the gravitating body’s composite particles; if the composite particles are shedding mass the gravitational field of a collapsing body will stop increasing, stabilize, and eventually weaken - preventing a runaway collapse. The interim state of a stable, highly compact body with a strong gravitational field describes a quasar. Mathematics runs off to infinities, physical systems do not.

My understanding is that there is no pressure in a neutron star, it’s the Pauli exclusion principle that prevents new neutrons from going to the ‘already-filled’ low-energy states.

Very reasonable viewpoint I would say. But it seems to me this is not the picture from which TOV limit is calculated. Ok, I don’t have ready links for explanation how TOV limit is derived. I will try to look for some.

Oppenheimer and Volkoff in “On Massive Neutron Cores” say: “If, however, under extreme compression, phenomena occurred which have the effect of repulsive forces, i.e., of raising the pressure for a given density above the value given by the Fermi equation of state, this could tend to prevent the collapse.” They clearly consider pressure as the only mechanism that can oppose gravitational collapse. And in the text they consider relationship between pressure and density of Fermi gas. More detailed derivations about properties of Fermi gas seems to be in references. But it seems quite clear that they have taken some textbook exercise about particles in infinite potential well and used it as a baseline for calculations of pressure in volume element of degenerate neutron gas in neutron stars.

In wikipedia page on Fermi gas I found this sentence: “For this reason, the pressure of a Fermi gas is non-zero even at zero temperature, in contrast to that of a classical ideal gas.” I think it nicely illustrates understanding of degeneracy pressure in astrophysics.

@RedshiftDrift So it seems that unfortunately you (and me) are minority with your understanding. Btw just noticed. You said: So I would reword your last sentence as: “higher energy neutrons in cold (degenerate) neutron gas can fall into the core of the neutron star but they can not stay there because the low-energy states are already taken by the cold bosons of the star.” but it’s incorrect. Bosons can occupy the same quantum state. Its fermions that can’t occupy the same quantum state. And besides quantum states are separate for each type of particle. Replacing “bosons” with “neutrons” would make it correct.

The two sentences you quote are interrelated. A unit of matter, a proton for instance, has a mass-energy that is dependent on c. If c is diminished in the vicinity of the matter-unit it has to reduce its mass-energy content in accord with the reduced value of c. The only known way for it to do this is to emit electromagnetic radiation. As long as E/m=c^2 holds radiative emission is an inescapable consequence of an increasing gravitational field strength.

That emission process would in turn reduce the particles mass-energy content. The strength of the associated gravitational field is dependent on the mass-energy of the gravitating body’s composite particles

We are on the same page so far. But I would like to point out that General Relativity accounts for that reduction of mass but in mathematicaly very contrieved way.

if the composite particles are shedding mass the gravitational field of a collapsing body will stop increasing, stabilize, and eventually weaken - preventing a runaway collapse.

In classical system it’s pressure that prevents runaway collapse. Virial theorem gives you a number how much potential energy of collapsing mass can be radiated out of the system as binding energy - it’s exactly half. Taking into account rest mass reduction might slightly alter the numbers. As far as I can tell Oppenheimer and Volkoff are correctly taking into account GR corrections (rest mass reduction). As I understand they get runaway collapse when they take into account Special Relativity effect on pressure - that relativistically moving particles can produce less pressure proportionally to the energy they have (but that is not in the paper, I suppose it comes from derivations in references). Then they counteract that by adding pressure from particles being degenerate (Fermi gas), but that in turn increases energy (and mass by energy/mass equivalence). As they scale solution to more and more particles, energy trapped in energy levels of Fermi gas gives more gravity than the pressure of Fermi gas (diminished by SR effects) can counteract. And then they get runaway gravitational collapse.

So I would say you are skipping a lot of details.

Mathematics runs off to infinities, physical systems do not.

No disagreement here.

@js2zonde Indeed, I meant ‘fermions’, thanks for noticing. (Text is corrected.)

Look, it does not work. There is definitely outside for black hole, but there is no outside for Big Bang universe. So it’s clearly different concepts.

A Google AI overview:

The “grapefruit universe” concept refers to a cosmological analogy stating that shortly after the Big Bang, the observable universe was compressed into a space smaller than a grapefruit. During the rapid expansion of inflation, this ultra-dense, small-scale state existed for a fraction of a second.

A little problem is the operational definition of a “size” of that space smaller than a grapefruit. Because what people usually do is employ space outside it for measuring, with a ruler or so. In short, I do not understand “size” of a volume without an “outside”.

But I would like to point out that General Relativity accounts for that reduction of mass but in mathematicaly very contrieved way.

Whether GR accounts for the mass reduction due to the speed of light reduction is an open question that is dependent on which solution of the field equations you are referring to. The Schwarzschild solution for instance treats c as a constant and so therefore it does not incorporate the mass reduction - which is why it generates black holes.

In the TOV article @RedshiftDrift cited there is this mention: >In terms of Planck units, with the reduced Planck constant ħ, the speed of light c, and the gravitational constant G all set equal to one, there will be a corresponding pressure given roughly by…

So the speed of light is again being treated as a constant which means no mass-energy decrease associated with the reduced SOL is going to be generated in the model. TOV also generates black holes if the Wiki entry is accurate. The GR stipulated decrease in SOL in a gravitational field is the critical detail skipped in both Schwarzschild and TOV. Do you know of any solution that incorporates that detail? I don’t.

A Google AI overview:

The “grapefruit universe” concept refers to a cosmological analogy stating that shortly after the Big Bang, the observable universe was compressed into a space smaller than a grapefruit. During the rapid expansion of inflation, this ultra-dense, small-scale state existed for a fraction of a second.

A little problem is the operational definition of a “size” of that space smaller than a grapefruit. Because what people usually do is employ space outside it for measuring, with a ruler or so. In short, I do not understand “size” of a volume without an “outside”.

The key there is “observable universe”. It is sort of a ruler that shrinks as you go back in time. There can be outside for observable universe because its limited in size and not infinite.

But I would like to point out that General Relativity accounts for that reduction of mass but in mathematicaly very contrieved way.

Whether GR accounts for the mass reduction due to the speed of light reduction is an open question that is dependent on which solution of the field equations you are referring to. The Schwarzschild solution for instance treats c as a constant and so therefore it does not incorporate the mass reduction - which is why it generates black holes.

Schwarzschild solution is exterior solution so it says nothing about mass. It basically uses mass as a point mass at the center. For interior solution you have sort of redshift of gravity so that while you have the same rest mass at different gravitational potentials, gravity it produces for an observer at infinity depends how deep it is in gravitational potential well.

I found this page about neutron stars by university professor. Its quite nice IMO. It says: First, the typical mass of a neutron star is about 1.4 solar masses, and the radius is probably about 10 km. By the way, the “mass” here is the gravitational mass (i.e., what you’d put into Kepler’s laws for a satellite orbiting far away). This is distinct from the baryonic mass, which is what you’d get if you took every particle from a neutron star and weighed it on a distant scale. Because the gravitational redshift of a neutron star is so great, the gravitational mass is about 20% lower than the baryonic mass.

Unfortunately there is no justification given for this statement, but at least it shows mainstream person claiming the same thing.

In the TOV article @RedshiftDrift cited there is this mention:

In terms of Planck units, with the reduced Planck constant ħ, the speed of light c, and the gravitational constant G all set equal to one, there will be a corresponding pressure given roughly by…

So the speed of light is again being treated as a constant which means no mass-energy decrease associated with the reduced SOL is going to be generated in the model. TOV also generates black holes if the Wiki entry is accurate. The GR stipulated decrease in SOL in a gravitational field is the critical detail skipped in both Schwarzschild and TOV. Do you know of any solution that incorporates that detail? I don’t.

Reduced SOL and gravitational redshift is just two different perspectives of the same physical phenomena.

Btw here is interior solution: Interior Schwarzschild metric And notice what it says: However, M is the mass measured from the outside, for example by observing a test particle orbiting the gravitating body (the “Kepler mass”), which in general relativity is not necessarily equal to the proper mass.

Because the gravitational redshift of a neutron star is so great, the gravitational mass is about 20% lower than the baryonic mass.

Well that about sums up the inanity of the mathematicist approach to doing physics - redshift causes mass loss. Sure, why not? In some loopy model maybe that make sense, but in physics it is nonsense. Maybe someone might object that the professor has strayed a little bit off the reservation of the approved “physics” nonsense, but nonsense is nonsense. So…

There is indeed a Schwarzschild exterior solution and a separate interior solution - if you restrict yourself to using the Schwarzschild metric. In that case the ext solution has a coordinate singularity where the radius of a spherical mass = the Schwarzschild radius r(s) and so the ext solution says nothing about the interior. For that you need the int solution which (according to the Wiki entry cited),

is an exact solution for the gravitational field in the interior of a non-rotating spherical body which consists of an incompressible fluid (implying that density is constant throughout the body) and has zero pressure at the surface.

So by definition a black hole is precluded because the fluid mass is incompressible, and then we are told,

For a complete metric of the sphere’s gravitational field, the interior Schwarzschild metric has to be matched with the exterior one…

So fine, no black holes according to the exterior + interior solutions, right? Wrong of course. Because according to the Wiki entry on the exterior solution:

The Schwarzschild solution, taken to be valid for all r > 0, is called a Schwarzschild black hole. It is a perfectly valid solution of the Einstein field equations…

At that point it is reasonable to state that this mathematicist account of physical reality is illogical, incoherent, and absurd. It bears no resemblance to the physical reality we actually observe and measure. This is the end result of an academically enforced inability to differentiate between physical reality and some mathematical formalisms that are believed, by a consensus of the mathematically confused, to be the determinants of physical reality. It is the reason someone can write inane math babble like, “…the baryonic mass, which is what you’d get if you took every particle from a neutron star and weighed it on a distant scale.”

Reduced SOL and gravitational redshift is just two different perspectives of the same physical phenomena.

I would put it somewhat differently:

Reduced SOL, slowing of clocks, reduced mass (matter-energy), and gravitational redshift are all physical aspects associated with an increasing gravitational field strength. An increasing gravitational field strength is associated with an increasing matter-energy density. Increasing matter-energy densities are offset by increasing radiation-energy emissions which are necessitated by the reduced SOL. If the SOL is reduced by half, the matter-energy (mass) of the gravitating body will be reduced by 3/4 - because E/m=c^2. The process is self-limiting.

However, M is the mass measured from the outside, for example by observing a test particle orbiting the gravitating body (the “Kepler mass”), which in general relativity is not necessarily equal to the proper mass.

Notice the sloppy thinking going on here that is characteristic of mathematicism. A “proper mass” is invoked even though it is not included in the interior or exterior solution, both of which have only a constant mass value M for the gravitating body. A constant M and a constant SOL c are in the definition of the Schwarzschild radius [r(s)=2GM/c^2].

To say that M is measured from the outside is a vapid truism used to imply that there is some proper mass measurable internally which differs from the external measurement. But masses are always measured externally. And measurement isn’t exactly the right word - the mass is calculated in this case from the observation of the orbital speed of an external body using a mathematical formula which may or may not be reliable in all cases. GR, for instance, has a great deal of difficulty calculating a galaxy’s mass from its orbital dynamics.

There are also the constantly shifting modifications to the simple term mass. There is gravitational mass, and inertial mass and Keplerian mass and baryonic mass and non-baryonic mass and proper mass. This gives rise to a lot of math babble. What does not exist in the wonderful world of mathematicism is a straightforward definition for the fundamental concept of mass.

There is gravitational mass, and inertial mass and Keplerian mass and baryonic mass and non-baryonic mass and proper mass.

You forgot to add nuclear mass. I suppose ‘baryonic’ could fit this though. We already know from nuclear fusion that M+M<2M which means all of the above is not entirely just ‘babble’.

I’m not sure if a 20% loss of mass is plausible for a neutron star, but yes, if you were able to break it up the sums of all the masses would be more than the gravitational mass of the whole neutron star. The same would go for a black hole (not singularity), to some extreme point where you throw in extra mass, the total measured mass would not increase at all anymore. M+m=M. Not just because of rounding down, but saturation of inertia.

As for the SOL and redshift arguments, these account for observations from ‘outside’. An observer within the gravity well sees everything blueshifted. Does SOL increase for him? No, his clock is just slow so everything else looks ‘fast’. Now he might not be able to come out of the black hole to exchange observations, but I see a paradox here. If observed time is going to stand still for the outside observer, time must appear infinitely fast for the inside observer. Problem.

Reduced SOL, slowing of clocks, reduced mass (matter-energy), and gravitational redshift are all physical aspects associated with an increasing gravitational field strength.

Could you provide example where reduced gravitational redshift should be combined with reduced SOL/slowing of clocks/reduced mass to get the correct results?

@js2zonde I did not say anything about a reduced gravitational redshift. Gravitational redshift, the SOL, mass, and clock speed are all correlated with the strength of the gravitational field. Gravitational redshift increases with an increasing field strength, while the other three are inversely correlated.

@Francois-Zinserling Black holes are not physics, they are the imaginary outcome of some incomplete math. The math is incomplete because it does not take into account the variation of light speed in a gravitational field. I’m not interested in imaginary arguments that assume black holes have some scientific viability; they do not.

There are plenty of people out there who believe that black holes are real physics but I’m not one of them. In empirical reality stuff does not disappear irretrievably behind imaginary black curtains and the singularities of the Schwarzschild solution are incoherent mathematical nonsense. You can’t just wave those singularities away and pretend the math is just fine otherwise. It is not - because it represents an incomplete account of gravitational physics.

I did not say anything about a reduced gravitational redshift.

Ah yes, my mistake.

If the SOL is reduced by half, the matter-energy (mass) of the gravitating body will be reduced by 3/4 - because E/m=c^2.

This does not seem right. Say we have electron and positron near the surface of gravitating body with not much kinetic energy worth considering. Now they undergo annihilation and emit two gamma-ray photons. A distant observer catches these two photons and looks at their energy. And we conclude that gravitating mass-energy of that gravitating body is reduced by that energy/c^2 (speed of light is that of the distant observer). But energy of these photons is reduced by gravitational redshift (compared to energy distant observer would observe from electron positron annihilation local to him). And the only mass that was taken from gravitating body is mass of that electron positron pair. So it seems to me that mass of that electron positron pair was reduced by the same factor of gravitational redshift as these gamma-ray photons. @budrap00 Do you see any flaw in my reasoning?

@js2zonde Simple mathematics that says if the local value of c in the vicinity of a gravitationally collapsing body is reduced by half, then because E/m=c^2 the mass of the body must be reduced to 1/4 of its initial value. I don’t see how your argument addresses that specific claim. At issue is the physical behavior of a gravitationally collapsing body. I don’t see the relevance to that specific situation of an analysis of a theoretical detection by a remote observer of an electron-positron annihilation which is taking place somewhere in the vicinity of the collapsing body.

Simple mathematics.

$$ \frac{E}{m} = c^2 \quad \Longleftrightarrow \quad m = \frac{E}{c^2} $$

Therefore if $E$ assumed constant

$$ m = \frac{E}{c^2}\rightarrow \frac{E}{(c/2)^2} = 4\times \frac{E}{c^2} \quad \Longrightarrow \quad m \rightarrow 4\cdot m $$

Hence the mass of the body must be increased to four times its initial value. Not?

That makes sense in mathland but not in physics where E and m are known to be variables.

Of course, but how?

@HanDeBruijn I don’t understand what you are asking. How what?

@budrap00 . Yes, you don’t understand what I am asking.

Ah I see, your wisdom surpasseth all understanding. I’ll pass on the question in that case.

@HanDeBruijn is correct that by that reasoning the mass should increase if you want to keep the energy constant.

This is all bogus though. Even if you wish to keep the mass constant, then the energy must reduce to the same ratio as $c^2$. Where does this energy go? Black holes should be glowing brighter than the brightest star if ‘all’ mass is converted to energy.

On the other hand, back to Han’s post, if E is constant and $c^2$ is reduced, then m - as a measure of inertia - must increase. It will take more force to accelerate the mass, to the hypothetic point where $c=0$ it will be impossible to accelerate a mass. This may explain why some SMBH have such ridiculous high masses? I don’t really think so but it’s a weird math result which I’m sure will send @budrap00 over the edge again. :)

The problem stems from equating what an observer sees with reality. It is like seeing a lamppost in the distance and insisting that far-away things are small.

At least for black holes and a black hole universe, there could be some other factors that could be in play. I consider the basic structure of black holes and BHU to be the same. For black holes, the Schwarzschild radius Rs is proportional to the mass, by Rs = 2GM/c^2. So if the radius were to shrink the mass would have to decrease also. Similarly, if the radius were to increase the mass would have to increase too. How could this happen? We have the general relationship through Tryon and others that the mass of a body is equal to its gravitational potential energy U with respect to the universe as a whole. The total quantity U for the whole shell universe would then be GM^2/Rs. When you factor in the mass-radius relationship again, we see U is proportional to M also. Now consider a BHU or black hole that is decreasing in size. A loss of mass is implied. This comes through the conversion of gravitational potential energy to radiation (added to the CMB). Mass energy is being converted to radiation. Conversely, if the BHU were increasing in radius, this happens because CMB energy is increasing U again and thereby M. The CMB would be cooled in the process. Ordinary black holes would have an equivalent interior radiation and process.

Note to Han @HanDeBruijn , so you see you can have an increase in mass over time, if the system you are in is expanding!

The problem stems from equating what an observer sees with reality.

No, the problem stems from equating mathematical models with reality. It’s like thinking there are physical black holes just because you have a black hole model or thinking there is an Expanding Universe just because you have an Expanding Universe model. Math is like Silly Putty, you can mold it into all kinds of fanciful shapes but most of them have nothing to do with empirical reality. And if you pay no attention to physical reality it’s impossible to sort the wheat from the chaff.

Therefore if 𝐸 assumed constant

@HanDeBruijn The simple answer is that E is not constant and is not conserved. Conserved quantity is m + E/c^2. Energy can transform into rest mass and vice versa so it’s not conserved. And as c decreases energy increases but this sum I wrote remains the same. This should be correct if we believe that any internal process of gravitating object should not change external gravity.

The first one involves alternative solutions for the interior of dense objects first discovered by Jun Ni and then developed by others. In these solutions, there is a repulsive gravitational field at the centre of the object, which causes them to form a stable shell with an interior void.

I sort of don’t see the point of inventing new physics when people can’t correctly apply existing well tested physics. I would rather stripe existing physics of strange philosophical embellishments it currently has and instead would put it on some solid philosophical grounds. Then would look if it helps.

That is exactly what the authors Ni, Neslusan and DeLyra are trying to do with those equations. They say that the integrations of equations of state for compact objects were not properly done in the past. When done properly there is a new term which appears at the origin, which is directed outwards.

On the Lambda luminosity, if you prefer the ‘old physics’, then maybe you can explain why neutron stars have luminosities in the range of 10^32–10^34 erg s^−1? Why do these supposedly dead stars radiate with the same power as ordinary stars?

That is exactly what the authors Ni, Neslusan and DeLyra are trying to do with those equations. They say that the integrations of equations of state for compact objects were not properly done in the past. When done properly there is a new term which appears at the origin, which is directed outwards.

Can you tell more about this? Or give some links? This seems counterintuitive. As you yourself where saying in that article in Phys.org: This runs counter to the standard picture in general relativity, which features a flat, Minkowski spacetime inside a spherical shell of matter.

The new approach was started by Jun Ni in 2011 and since then has been studied a lot by Lubos Neslusan, Jorge DeLyra and others. Here are a few links that could help. The first one is by deLyra, the second by Neslusan and the last one is my paper describing a shell black hole universe, where I give an overview:

https://arxiv.org/abs/2101.02012 https://www.astro.sk/~ne/neslusanCAOSP.pdf https://www.mdpi.com/2674-0346/3/3/14

Here is Neslusan’s home page, where you can find other articles: https://www.astro.sk/~ne/

I have also attached a paper by Neslusan: Neslušan-2019-the-second-rise-of-general-relativity-in-astrophysics.pdf

I have also attached a poster by Neslusan, which could give a good overview. NESLUSANposter.pdf

While I think the Ni solutions could form part of the story, I also think other factors could be at work, such as a finite range of gravity. If we think of a shell universe and us being close to the centre, then light must have a range of only the Schwarzschild radius. If gravity has an electromagnetic basis, which I subscribe to, then the shell theorem of GR and Newtonian gravity would then not apply: the two sides of the shell don’t ‘see’ each other.

I looked at deLyra paper. There is no way I could follow all the math in the paper but some basic points are simple enough, that they do not require high math skills. In equations (10) they specify assumed boundary conditions at inner and outer surfaces of shell and in particular for inner surface: P(r1)=0. In simple terms this seems to say that they assume there is outward gravity or in other words antigravity (from direction of the center of cavity) at the inner surface. And the only interesting thing is that this assumed configuration can still be described using Einstein field equations. That is if one does not consider singularity at the center as a flaw of solution.

I’m certainly not the best one to shed light on these papers. The paper by deLyra is the hardest one to read of the ones I sent. Neslusan gives a better handle on it. About Eq. 10, the two pressure terms being 0 on both the inner and outer surface arise from the the interface between a vacuum and a liquid state, according to deLyra. DeLyra called the force from the origin a repulsive force but Neslusan cautions to call it an outwardly attractive force of matter towards the shell. There is also an accompanying gravitational redshift, which I link to the Hubble redshift.

The point is that there seems to be a class of GR solutions which favour compact objects having shell-like configurations. We can also think of the observable universe as a shell black hole. If we say that the observable galaxies and dust comprise 5% of the total required for a black hole (or the critical density), then we can do away with dark matter and dark energy.

Solutions of mathematical equations are not necessarily physical, e.g. $x^2 = -1$, black holes, expanding universe, etc.

I should have included the original paper by Jun Ni, which perhaps offers the clearest expression. I have attached it now.

Ni shell paper 2011.pdf

Jun Ni says the same, but I found that assumptions in deLyra paper where stated more clearly. To reiterate what RedshiftDrift already said. Bluntly speaking your claim sounds a bit like that: Lets assume pigs can fly. Now can we describe this situation mathematically? Yes we can. So that means that pigs can fly.

No need to discuss further then.