This online source is confirming my stance that scientists had actually applied Newton’s Theorem XXXI to the Galactic Rotation Problem.

Above source is criticizing that scientists should not have modeled gravity of galaxy like that (i.e. like Theorem XXXI). However this source has not reached to the point of Theorem XXXIII.

My book Philosophy Unscrambles Dark Matter has pointed out that Scientists should have applied Newton’s Theorem XXXIII to galactic rotation problem but they had applied irrelevant Theorem XXXI which was applicable to solar system and that’s why scientists were wrongfully expecting “Keplerian drop-off” from galactic rotations.

Here is the free complete section of book that explains that actually applicable Theorem was XXXIII that naturally gives flat rotation curves for galaxies.

General Relativity was an achievement. But Universe itself is greater, deeper, stranger and different from this achievement. Our best theories … including GR do not correctly ‘predict’ the actual universe. Any such claim is false. Universe also does not laugh on such claims. We ourselves should review our claims whether they are realistic or not.

Before the observation based discovery or confirmation of the existence of galaxies in year 1924, there were three solutions to GR equations available. First by Einstein himself and second by de-Sitter (both: 1917). Both could not reach to the concept of disk shaped island universes. Then in 1922, Friedmann also presented a scheme of various types of universe models. He also failed to reach the concept of disk shaped island universes.

In his 1917 paper, after discussing limitations or problems of Newton’s theory then Einstein proceeds to describe his own theory in section 3. Title of section 3 is following:

3. The Spatially Finite Universe with a Uniform Distribution of Matter

The first paragraph of this section clearly shows that Einstein totally missed the existence of galaxies at large astronomical scales. Following is the first para:

According to the general theory of relativity the metrical character (curvature) of the four-dimensional space-time con- tinuum is defined at every point by the matter at that point and the state of that matter. Therefore, on account of the lack of uniformity in the distribution of matter, the metrical structure of this continuum must necessarily be extremely complicated. But if we are concerned with the structure only on a large scale, we may represent matter to ourselves as being uniformly distributed over enormous spaces, so that its density of distribution is a variable function which varies extremely slowly.

So density varies extremely slowly….

It means that Einstein (or GR) completely missed the concept of island universe having concentrated density and huge voids in-between.

Following is link to English Translation Einstein (1917) paper.

Volume 6: The Berlin Years: Writings, 1914-1917 (English translation supplement)

However – Newton did reach to the concept of Galaxies

In a letter to Isaac Newton, David Gregory declared in 1694: “A continual miracle is needed to prevent the Sun and the fixed stars from rushing together through gravity.” Newton pondered the issue over the years starting around 1685 and concluded:

“The fixed stars being… at such vast distances from one another, can neither attract each other perceptibly, nor be attracted by our Sun.” I. Newton, Principia (1728)

Newton reasoned that:

“if the matter of our sun and planets and all the matter in the universe were evenly scattered throughout all the heavens, and every particle had an innate gravity toward all the rest, and the whole space throughout which this matter was scattered was but finite; the matter on the outside of the space would, by its gravity, tend toward all the matter on the inside, and by consequence, fall down into the middle of the whole space and there compose one great spherical mass. But if the matter was evenly disposed throughout an infinite space, it could never convene into one mass; but some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered at great distances from one to another throughout all that infinite space. I.

Newton, letter to theologian Richard Bentley (1692)

Thus we see that Newton had accurately reached to the idea of the existence of Galaxies and that existence of galaxies actually indicate infinite vastness of space. On the other hand Einstein and his GR had terribly failed in reaching to the concept of galaxies even though observational indications were available at that time and even discussion based on observations regarding existence or non-existence of galaxies was also available.

The answer to this question is “Yes and No”. More precise answer is “No and Yes”.

No part is because Philosophers tend to transcend physical reality. They may or may not go to right side. Even if some of them reach to right destination of transcendent reality, their method is not suitable to study physical reality.

For the yes part … (i) Philosophy must not peruse transcendent destination beyond of physical reality and (ii) Philosophy must be assertive of physical reality. Yes part of Philosophy is rare these days but it is part of philosophical tradition as explained below:

Preface & Complete 1st Chapter – Book: “Philosophy Unscrambles Dark Matter”

Preface & Complete 1st Chapter – Book: “Philosophy Unscrambles Dark Matter”

Preface thoroughly outlines the development and status of dark matter theory at the time of publishing this book. First chapter is like a combat between mathematical counterintuitive physics and human commonsense and explains that human commonsense equipped with proper philosophical approach is capable to deal with the problem of dark matter. Thus the first chapter makes a case for human commonsense and philosophical method.

Distance Anomalies – Book: “Philosophy Unscrambles Dark Matter”

This section of the book points out that rather than ‘dark matter’, the actual anomalies are in the official distances of galaxies. All the luminosity based distances of galaxies are systematically wrong. All the galaxies having redshift greater than 3 are located at (light travel) distance of more than 14 billion light years. The farthest visible galaxies may be located at (light travel) distance of hundreds of billions light years.

Sections about Galactic Rotation – Book: “Philosophy Unscrambles Dark Matter”

Scientists did apply Newton’s Shell Theorem to the Galactic Rotation Problem. These sections of book explain that scientists applied irrelevant part of Shell Theorem (Theorem XXXI) that was applicable to Solar System and terribly missed to apply the relevant part of Shell Theorem (Theorem XXXIII) that was applicable to the galaxies. These sections then properly explain the flat rotation curves of galaxies in the light of Theorem XXXIII without leaving room for dark matter or ‘modification of theory’. As flat curves are naturally explained, Spiral Structure of Galaxies also gets explained. These sections also explain dwarf galaxies with respect to attribution of greater quantity of dark matter towards them and so-called ‘no dark matter galaxies’ also have been discussed. It is concluded that theory was complete enough to explain flat rotation curves of galaxies. Dark Matter was not the failure of theory rather it was the failure of counterintuitive regime of Physics and Cosmology.

Sections I.IV – Book: “A Philosophical Rejection of The Big Bang Theory”

This section of the book analyses and rejects the claim that Georges Lemaître (1927) had mathematically predicted Hubble’s Law. It is explained that he did reach to this law in year 1927 but he had learnt Hubble’ Law from Hubble himself and had not derived it from GR equations. GR had not ‘predicted’ this law. Lemaître published manipulated translation of French article (1927) in year 1931. The manipulated translation removed the reference to Hubble and it appeared as if Lemaître had mathematically predicted Hubble’s Law in year 1927.

Sections I.V to I.VII – Book: “A Philosophical Rejection of The Big Bang Theory”

These sections of the book analyze and reject the claim that Friedmann (1922) had already mathematically reached to the Hubble’s Law or Hubble type redshift-distance relationship. The source of the idea of (Big Bang) initial singularity is also traced within the works of Friedmann and it is pointed out that Friedmann was talking about monotonic world of which the initial singularity is a distorted and misunderstood form. 

In response to my recent blog post that briefly explains how scientists applied irrelevant Newton’s Theorem XXXI to the galactic rotation problem and missed to apply correct Theorem XXXIII, a PhD physics person pointed out that even if Theorem XXXIII is applied, the discrepancy remains. To this, my reply was that blog post was brief. The remaining discrepancy has been acknowledged by the book. Two things directly come from Theorem XXXIII i.e. (i) Inverse square distance law is replaced by inverse (linear) distance law within the disc of galaxy and; (ii) For any depth ‘r’ within the disc of galaxy, the outer layers have no gravitational effect. Now only these two things really do not cover the full discrepancy. Third thing, which has been explained in sections II.II.IV and II.II.VI of the book “Philosophy Unscrambles Dark Matter” is that luminosity is not actually reduced towards the edges. Yes along the line of diameter, or radially, the luminosity is decreased … but total luminosity of the complete circumference of outer layers is not decreased. These sections did satisfy him and he only raised one further objection to which I satisfactorily replied (will share it in next blog post).

II.II.IV. Flat Rotation Curves of Galaxies – Proper Interpretation

After having seen that galactic rotations should have been described in the light of Theorem XXXIII of Newton’s Principia Mathematica, let us therefore try to do it now.

Expected (A) and observed (B) star velocities as a function of distance from the galactic center. (Credit: Wikimedia: Commons)

The prerequisite here is that we must completely forget the so-called ‘Expected’ line (A) because within the right context of Theorem XXXIII, we simply do not expect line (A). Line (B) is the actually observed line and the same is anticipated by applying Theorem XXXIII. Regardless of what official theory tells us about the existence of super massive black hole at the center of galaxy, this graph is actually telling that closer to the center, orbital speed is lowest. Within a disk of uniform density of mass, we should expect zero orbital velocity at the center of the disk. The lowest orbital velocity at point close to the center is consistent with this theory which means that law of inside of sphere (or disk) is being demonstrated. Non-zero but lowest orbital velocity near the center of disk may or may not indicate the presence of super massive black hole at the center. Afterwards, over a very short distance, there is substantial increase in the orbital velocity as the velocity curve moves up quite sharply. Our interpretation is that this area is the central bulge of the galactic disk and over this short distance, actual mass is substantially increasing layer upon layer such that density of each layer almost remains the same. Following actual graph confirms the idea that area of sudden increase of orbital velocity approximately relates to central bulge of the galaxy M33.

M-33 Galaxy Rotation Curve, Credit: Wikimedia Common – Source link[i]

M33 is not very large galaxy as the diameter of galactic disk is only about 60000 light years. We see (or assume) in this picture that radius of the central bulge of the disk spans about 5000 light years and within this distance of 5000 light years, there is sharp jump in the velocity curve. This actual graph is showing gradual upward movement of velocity curve even beyond this point but for the sake of simplicity, we shall assume that after this point, velocity curve becomes flat.

Basically there are two distinct portions of the Rotation Curve of Galaxy. Up to the distance from center towards the edge of the central bulge, there is sharp increase in orbital velocity of stars within disk. The lowest orbital speed is found in the area closer to the center of the disk. It means that area close to the center is subject to lowest gravity and this thing is in harmony with the Shell Theorem as applicable within the sphere (or disk). In the example of galaxy M33, we see that radius of central bulge is almost 5000 light years. For the sake of our analysis regarding why orbital velocity is increasing very sharply over this distance, we suppose that there are 5 layers within the radius of central bulge and the width of each layer is 1000 light years. Our interpretation will not depend on the existence or absence of super massive black hole at the center of galactic disk. So the interpretation goes that for the five layers of central bulge, a huge quantity of mass, let’s say 1 billion solar masses, is concentrated in the innermost layer that may or may not include super massive black hole. The second layer is orbiting around inner most layer with the lowest velocity. The second layer has same width of 1000 light years but due to being outer layer of the circle, the area is far greater than the innermost layer. The second layer has almost equal density of mass which means that total mass of the second layer may be around, let’s say, 8 billion solar masses i.e. just approximate number only to explain the point.

Now the third layer is orbiting a total mass of 9 billion solar masses. Therefore, within the third layer, orbital velocity has increased quite sharply. Width of third layer is also same 1000 light years but area is still far larger than that of second layer. And again, the density of mass remains the same and thus total mass of this layer may be let’s say 16 billion solar masses.

Now this setup repeats up to the fifth layer which is subject to the highest orbital velocity of stars within the disk so far and also marks the boundary of the central bulge of the galactic disk. The central bulge area is therefore the first portion of the Rotation Curve of Galaxies. The important thing of the first portion is that mass is considerably increasing layer upon layer and reaches to, let’s say, 32 billion solar masses for the fifth and outermost layer of the first portion.

The central bulge area was characterized by layer upon layer successive and substantial increase in mass such that overall density of the bulge remained uniform. The outermost layer of the central bulge contains greatest quantity of mass so far which is 32 billion solar masses (i.e. approximate number just to explain the point). Next to the central bulge area, the second ‘flat’ portion of the Rotation Curve of Galaxies begins.

If the radius of M33 galaxy is 30000 light years wide then this second portion starts from 5000 light years from center of the disk and ends at 30000 light years from the center of the disk. For the sake of simplicity, here again, we divide this second portion into 25 layers each having width of 1000 light years.

We know that outer layer of central bulge had mass of 32 billion solar masses. Now we interpret the start of flat curve portion by saying that inner layer of this portion contains almost same mass i.e. 32 billion solar masses. In this way, the innermost layer of the second portion is having same mass as the outer layer of the central bulge had. However due to larger area, the density and luminosity (per unit area) of this layer is lower than that of central bulge. Due to the fact that previous layer i.e. the outer layer of the central bulge had the greatest mass, our present layer i.e. the inner layer of outer area has the greatest orbital velocity and the rotation curve moves still higher. Therefore, flat portion of curve has not actually started yet.

Now comes the second layer of the outer portion of galactic disk. Again mass will remain the same i.e. 32 billion solar masses and due to larger area, there will be slight reduction in the density and luminosity (per unit area) across this layer. Because previous layer had augmented a constant mass, therefore, keeping in view the applicable inverse distance law of gravity, orbital velocity curve will remain horizontally flat across the current layer.

If this pattern repeats up till 25th outermost layer, each successive layer will get equal quantity of mass however slightly lesser and lesser density and (per unit area) luminosity will be added and the overall galactic rotation, keeping in view the simplified assumptions, should show up as a flat curve on graph. It is possible that same pattern of successive layers, up to few more, may continue even after 25th layer but that outer portion of galaxy may remain invisible or normally undetectable due to low density and (per unit area) luminosity over there.

An important thing to be noticed is that let’s say when an object moves from 10th layer to 11th one, the object will be subject to gravity of the mass available in all the inner layers including central bulge and up to 10th layer (or even 11th layer). Objects placed in 11th layer will not be subject to gravitational effects of still outer layers i.e. 12th and rest of the outer layers because according to Theorem XXXIII, an object placed at certain depth within sphere (or disk) will not be affected by the gravity of outer surface area. With this setup, availability of constant mass in each successive outer layer will give the result of flat rotation curve because law of inverse square distance is also replaced with the law of linear inverse distance within the sphere (or disk).

The following is the graph of mass available in successive layers and it is similar to the rotation curve graph of galaxies.


The above scheme of the things is actually based on oversimplified assumption of two dimensional setup of mass. In reality, galactic disk has thickness that is usually more or less or almost 1000 light years. Thus within central bulge, in reality, there should be far greater increment of available mass than by the factor of just 8 which is being presented in this scheme. Moreover, onward from central bulge, the quantity of mass may get slightly increased layer upon layer i.e. only as much that density of the layer should remain lower than that of previous layer and the net effect may be slightly upward velocity curve which is the case we have seen in the diagram of M33.   

However, for the purpose of our analysis, we carry on with the simplistic two dimensional assumption and constant increase of mass for area onward from the central bulge. Following schematic diagram with inner five layers of central bulge with uniform density and outer (only) eleven layers each having mass equal to the outermost layer of central bulge shows that such a structure not only explains the observed flat rotation curves of galaxies, it also develops the spiral structure of galaxies.

Blocks placed in successive layers

The above diagram is made up of equal size squares or blocks. The central yellow mark is the innermost layer of the central bulge and the other prominent yellow square is the outermost layer of the central bulge such that this layer consists of 32 small blocks which means that outer layer of the central bulge is 32 times massive than the innermost layer. Following is close up view of above diagram up to only the fifth layer and covers the complete central bulge area.

Blocks placed in successive layers – Bulge area close up. Outermost layer is 32 times massive than central layer.

In this schematic diagram, each small square represents equal quantity of mass let’s say 1 billion solar masses. If there is mass of 1 billion solar masses in the innermost layer, then second layer contains 8 billion solar masses and overall density remains the same. The fifth layer is the outermost layer of the central bulge.

Following close up shows what would eventually look like spiral structure from a far-view:

Blocks placed in successive layers – Outside of Bulge area close up

Here we see that outer layer of central bulge had mass of 32 billion solar masses whereas the total mass of the central bulge was (1+8+16+24+32) = 81 billion solar masses.

Next to the yellow layer starts the second portion of galaxy whose just eleven layers are shown in the image that starts looking like a spiral galaxy. In this portion, each layer contains 32 billion solar masses. While density remained uniform throughout the central bulge but beyond the central bulge, now mass is constant per layer and density per layer is getting reduced layer upon layer. A random placement of 32 blocks in each successive layer would give the overall shape of a spiral structure.

Blocks placed in successive layers – total 11 layers after Central Bulge

Note that this schematic diagram is based on square blocks and yet the basic shape of spiral has been achieved. Here, equal number of blocks have been randomly placed in each successive layer of the second portion of galaxy which is outside of the central bulge and the result is a crude or basic shape of galaxy. In a real galaxy, matter is not randomly arranged as the actual shape is determined by the overall scheme of the larger structure as well as quantity and placement of nearby mass or the availability of local structures. After eleventh layer, if we add next layers up to 25th layer by placing the blocks in accordance with the already emerging shape, the following final shape is achieved.

Blocks placed in successive layers – total 25 layers after Central Bulge

The real galaxies are often arranged in spiral shapes such as following.

Pinwheel Galaxy. Image Credit: NASA/ESA

In real galaxy, there is no empty space between spiral arms. But it does not mean that Spirals are merely illusions. In the schematic diagram, one billion solar masses was represented by just one square box. But in a real galaxy, mass of one billion solar masses is spread out in the form of fog of stars. Secondly, one box actually represents the compacted mass of central bulge area. For the outward area, mass should remain the same layer upon layer but one billion solar masses, being non-compact area, actually takes space of more than one box and this would be the reason why in-between spiral arms areas are not empty for the real galaxies. The in-between spaces of spiral arms are not empty or devoid of matter but however spiral arms are the places where greater mass is concentrated and thus spiral arms are real (i.e. not illusion) and assume their shape due to slightly greater mass but overall reduced density of the successive outer layers of galaxy. Within an actual galaxy, each successive layer may get more than slightly greater mass which seems to be the case with M33 galaxy where flat rotation curve is actually a slightly upward curve. It is also possible that in any galaxy, each successive layer may get slightly reduced mass than the previous layer and dark matter regime ‘scientists’ may identify such a galaxy as ‘dark matter free’ galaxy. Scientists do have identified two such galaxies so far but firstly they have not measured the rotation speed of stars within galaxies rather they have taken the velocity dispersions of globular clusters around them therefore inside of sphere or disk rule does not apply. Secondly, they also assert that these are not the confirmed cases of dark matter free galaxies as with ‘latest’ observations, they have considerably reduced the distance of those galaxies[ii] and have started saying that these are not dark matter free galaxies. Therefore it seems appropriate to not discuss this issue here at length.

As for as mainstream Astrophysics goes, standard interpretation accepts that there seems to be increase of available mass as one moves from inner parts of galaxy towards the outer ones. But within the standard interpretation, the total mass of galaxy is theorized to be concentrated at the center and test particles (stars) are orbiting around the center. Test particles are facing full gravity subject to inverse square distance law while the source of gravity is the central point of galaxy and there is no distinction between inner or outer layers and also it is not deliberated that mass belonging to outer layers has no actual gravitational bearing on this setup and thus, due to non-consideration of important factors, Keplerian drop-off is expected for this system. But since actually observed rotation curve is flat therefore they theorize (or hypothesize) that extra mass, over and above the total mass of galaxy is increasing with increase of distance from the center and to this supposed extra mass they assign the name ‘Dark Matter’.

Following is section II.II.VI  …. The relevant part where I openly disagree that luminosity decreases towards edges of disk is written in bold font. My point is that TOTAL LUMINOSITY of every complete outer layer or BAND remains the same … then flat curves are obtained. And also … then Spiral Structure is obtained.

II.II.VI. Is Dark Matter the failure of Theory?

We conclude that Newton’s Theory, subject to correct application, would have rightly described the rotation pattern of galaxies. Accurate theory already existed but problem of rotation curves of galaxies was never interpreted in the light of relevant part of the available theory. By 1920, when on the basis of famous 1919 solar eclipse experiment, Arthur Eddington and co-authors wrote in their paper that Einstein’s General Theory of Relativity was found superior theory of gravity to Newton’s theory, at that point in time, Relativity Theory did not even have Shell Theorem. Relativistic Shell Theorem was presented in year 1923 or as early as 1921[i]. Yes – it should mean that relativistic shell theorem was available at the time when scientists were dealing with the problem of dark matter. But it seems like the Birkhoff’s Theorem i.e. the Relativistic Shell Theorem does not consider the specific case of gravity field experienced by a test particle which is placed inside a sphere having uniform density which mean that till date relativistic counterpart of Newton’s Theorem XXXIII does not exist. But overall implication of this Birkhoff’s Theorem is that general relativity reduces to Newtonian gravitation in the Newtonian limit[ii].

The problem of rotation curves was within the Newtonian limit and the theory to be applied was Newton’s Theory thus we can accept that, in principle, theory was complete; rotation patterns could have been rightfully interpreted without invoking the need of dark matter. But – it did not happen; rotation curves were not rightfully interpreted. Theoretical Physicists did apply Newton’s theory but missed an important aspect i.e. Theorem XXXIII of the theory. Instead, they applied irrelevant Theorem XXXI. The wrong application of theory was dubbed as incredible discovery of ‘dark matter’ which was basically a ghost object; an unprovable hypothesis that was also found out to be seemingly supportive of few other unprovable conjectures relating to the Big Bang Cosmology and credit of those farfetched findings was assigned to the ‘more accurate’ theory of General Relativity. In this way, Theoretical Physicists extended the wrong application of (Newton’s) simple theory to their so-called ‘precise’ theory (GR) without realizing that they merely interpolated the results of incorrect application of simple theory to their ‘precise’ theory and this thing casts serious doubt on their claim that they do understand their counterintuitive theories.

Dark Matter was thus not the failure of theory. Precisely, it was the failure of correct application of the theory whereas the theory itself was capable for the task. What happened was like that while first time noting the rotational pattern of galaxies, scientists were naturally anticipating Keplerian drop-off in the rotation curves because by that time, it was the only observed pattern. But deviation of actual finding from the expectations did not spark the willingness to review the dynamical considerations even though Babcock (1939) had pointed out the need for the same. Scientists focused their attention towards getting better accuracy of observed data regarding rotation of galaxies but no one questioned in official papers concerning why Keplarian drop-off should be expected at all when galaxy is a whole different structure than solar system. Experimental Scientists were doing their job well as their task was really to gather correct observational data. But Theoretical Physicists were not using their commonsense because commonsense is a despised thing which they officially do not use. At least they should have seriously reviewed the relevancy of Keplerian drop-off for the dynamics of the galaxy.

Experimental scientists were doing their job well and they were presenting their findings along with judgments regarding what they had observed. In 1939, Horace Babcock reported in his PhD thesis that measurements of the rotation curve for Andromeda suggested that the mass-to-luminosity ratio increased radially[iii]. Yes – it was accurate judgment because at least gravitational mass does increase radially in terms of Theorem XXXIII. Babcock was accurate also because he pointed out that new dynamical considerations were required; a right proposal that was not taken seriously. Off course, whole new theory was not required; only requisite thing was to get rid of the Keplerial drop-off anticipations and to reach to the relevant Theorem XXXIII of the already available theory. Likewise, following quote out of Wikipedia article titled “Galaxy Rotation Curve”[iv] also informs that MS Vera Rubin (1970) not only reported her observations but also came up with accurate judgment that observations had the implication that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars.

In the late 1960s and early 1970s, Vera Rubin, an astronomer at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington, worked with a new sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a greater degree of accuracy than had ever before been achieved.[13] Together with fellow staff-member Kent Ford, Rubin announced at a 1975 meeting of the American Astronomical Society the discovery that most stars in spiral galaxies orbit at roughly the same speed,[14]and that this implied that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars (the galactic bulge). Rubin presented her results in an influential paper in 1980.[15] These results suggested that either Newtonian gravity does not apply universally or that, conservatively, upwards of 50% of the mass of galaxies was contained in the relatively dark galactic halo. Although initially met with skepticism, Rubin’s results have been confirmed over the subsequent decades.[16]

Here we note that MS Vera Rubin said in year 1970 that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars.

We know that according to Theorem XXXIII, a test particle placed at a particular depth within a sphere of uniform density will not be gravitationally affected by the outer layers of the sphere (or disk). It means that ‘gravitational mass’ of outer layers can be regarded as non-existent. Now suppose that test particle was placed at the edge of the galactic bulge and then starts moving towards outer area of the disk. This movement towards outer surface will cause regular ‘growth’ in the gravitational mass which according to MS Rubin, will be approximately linear with increase in radius. And yes, MS Rubin was talking about regular linear growth in mass over and above the total luminous mass that, for the purpose of determining the influence of gravity, was already theorized to be located at center. Theorem XXXIII, on the other hand, have the implication of regular linear growth in gravitational mass such that at every depth, the available (gravitational) mass is exerting full gravity from the center. This gravitational mass is not over and above the luminous (observable) mass. One thing Experimental Scientists missed was that they only radially determined the luminosity of disk. Yes radially the luminosity decreases over large distances but great distance with low (per unit distance) luminosity when projected in complete circumference of the outer belt, band or layer then ‘total’ luminosity also should remain the same layer upon layer just like total mass also remains the same layer upon layer. For example, Roberts and Whitehurst (1975)[v] also concluded the same that mass increases linearly towards the outer edge of the M31 galaxy. They had studied southern end of M31 and observed rotation and luminosity, off course, relating to only that southern end and observed, for that part of the galaxy, that luminosity decreases with no decrease of rotational velocity. The Astrophysical Journal (Aug:2011) has published a paper titled “The Luminosity Profile and Structural Parameters of The Andromeda Galaxy”[vi]This paper presents bell shaped graphs of luminosity of Andromeda as recorded along major and minor axis of the disk. Thus luminosity is decreasing only along the line of diameter and so far there is no realization that total luminosity of the outer bands or layers should be almost equal to total luminosity of inner bands or layers. Therefore, there may actually be no increase of mass to luminosity ratio taking place for the outer parts of galactic disks.  

Now we can recall our schematic diagram where mass increased linearly with radius well beyond the central bulge.


Outer edge of central bulge have 32 square boxes (representing mass). Each succeeding outer layer also has 32 boxes which means that mass is increasing linearly with radius i.e. exact wording of MS Vera Rubin.

This schematic diagram is based on idea that in accordance with Theorem XXXIII, after central bulge, mass should linearly increase so as to give flat rotation curve like graph. The outer layer of the central bulge consists of 32 equal size boxes. Now onward mass should increase linearly therefore each succeeding layer also consists of exact 32 boxes. By random placement of boxes in succeeding outer layers up to 11th layer (after bulge), the basic shape of spiral started to emerge. Rest of the layers, up to 25th, were arranged by placing the boxes in accordance with already emerging shape of spiral.

Here basic spiral shape was achieved but actual spirals of real galaxies are denser and in-between spiral areas are also not empty. Therefore, in real galaxies, mass increases more than linearly and ‘flat rotation curves’ may actually be slightly upward curves throughout most of the disk as we see in the case of M33 which seems to be usual case and these curves are accomplished due to offsetting caused by the inverse distance (from center) law of gravity as applicable within the sphere (or disk). The galaxy rotation is actually an excellent confirmation of the astonishing accuracy of Newton’s Theory. Here we are dealing with the inside of sphere or disk scenario and if we wrongfully consider inverse square distance law, we shall get Keplerian drop-off even though gravitational mass grows linearly. The flat or slightly higher rotation curves and usual spiral structures of galaxies are in great harmony with Theorem XXXIII of Newton’s Principia.

‘Dark Matter’ is thus not the failure of the Theory but can be regarded as failure of counterintuitive regime. It is failure of overrated understanding level of the theory and it is the failure of the idea that counterintuitive ideas are correct and are actually understood when they, intrinsically being ‘counterintuitive’ were not actually comprehensible. Failure was in the unscientific method that assigns reality status to ghost objects. For example, following paragraph from Wikipedia article titled ‘Dark Matter’ shows that they do not treat this ghost object just as a placeholder only to denote a shortage of proper explanation but they take it for a real object that cannot be traced in the real world.

Dark matter is a form of matter thought to account for approximately 85% of the matter in the universe and about a quarter of its total energy density. The majority of dark matter is thought to be non-baryonic in nature, possibly being composed of some as-yet undiscovered subatomic particles.[a] Its presence is implied in a variety of astrophysical observations, including gravitational effects which cannot be explained by accepted theories of gravity unless more matter is present than can be seen. For this reason, most experts think dark matter to be abundant in the universe and to have had a strong influence on its structure and evolution. Dark matter is called dark because it does not appear to interact with observable electromagnetic radiation, such as light, and is thus invisible to the entire electromagnetic spectrum, making it undetectable using existing astronomical instruments.[1]   

[i] – This Wikipedia article about Relativistic Shell Theorem says following:

“The theorem was proven in 1923 by G. D. Birkhoff (author of another famous Birkhoff theorem, the pointwise ergodic theorem which lies at the foundation of ergodic theory). However, Stanley Deser recently pointed out that it was published two years earlier by a little-known Norwegian physicist, Jørg Tofte Jebsen.”


“Thus, this part of the theorem is just what we would expect from the fact that general relativity reduces to Newtonian gravitation in the Newtonian limit.”

[iii] Babcock, H. W. (1939). “The rotation of the Andromeda Nebula”.





[ii] — “A more recent study on NGC 1052-DF2 suggests the previously reported distance of the galaxy was greatly exaggerated. Consequently, the galaxy now looks “normal” in every way. Using five independent methods to estimate distances of heavenly bodies, a team of researchers from the Instituto de Astrofísica de Canarias (IAC) found the correct distance of NGC 1052-DF2 to be 42 million light years (13 MPc), not some 64 million light years (19 MPc) from the Earth.[10] The total mass of the galaxy is around one-half of the mass estimated previously, but the mass of its stars is only about one-quarter of the previously estimated mass. This implies a significant part of NGC 1052-DF2 could be made up of dark matter, like any other galaxies.”

Scientists expected that star velocities of galaxy must follow Kepler’s 3rd law which required that galaxy should rotate slowly from edges than from center.

However, Kepler’s 3rd law was a specific law that was applicable to solar system or planet-moon systems where greater mass is concentrated at the center. But galaxy is a different system where mass is spread out across whole of the galaxy. Basically Kepler’s 3rd law, being specific law, was not applicable to galactic rotations.

Scientists also had general theories like Newton’s Theory and General Relativity (GR). The question before me when I started writing book “Philosophy Unscrambles Dark Matter” was that why did Physicists get same result from a particular law i.e. Kepler’s 3rd law and a ‘General’ Theory (GR) about faster than expected rotation pattern of galaxies?

The obvious answer to this question was that general theories should have given different result. But somehow scientists actually received results from applying general theories which were consistent with Kepler’s 3rd law. The galactic rotation computed by using general theories was consistent with the results of Kepler’s 3rd law.

So what was the fault with the general theories? Which General Theory was wrong? GR or Newton’s?

It turned out that according to the relativistic Birkhoff’s theorem (relativity) – Wikipedia, GR reduces to Newtonian Theory within Newtonian limits. Galactic Rotation was within Newtonian limits and thus Newton’s Theory was applicable to the galactic rotation problem.

If any general theory was wrong, that was not GR. That had to be Newton’s Theory.

So was Newton’s Theory really wrong? Why were Newton’s (general) Theory’s results about galactic rotation found out to be consistent with the results of a solar system specific law i.e. Kepler’s 3rd law?

Eventually I also reached to the correct answer to this question. I found the answer to the question as to why scientists so comfortably accepted the results of galactic rotation obtained from applying general theory of Newton such that a specific law i.e. Kepler’s 3rd law was also giving the same result.

For example please consider following point from “Galaxy Rotation Curves” section from Wikipedia’s article about Dark matter – Wikipedia.

If luminous mass were all the matter, then we can model the galaxy as a point mass in the centre and test masses orbiting around it, similar to the Solar System.[d]

In the above given quote, there is footnote [d] at the end which reads as “This is a consequence of the shell theorem and the observation that spiral galaxies are spherically symmetric to a large extent (in 2D).”

Actually it was the catching point. I already had reached to the (wrong) conclusion that scientists had actually missed to apply shell theorem that’s why they expected Keplerian drop-off in the galactic rotation problem. Not only I, actually few other people were also thinking like that. For example, Nikolay Sones asked following question on 14–04–2019:

If we have Newton’s shell theorem then why do we need dark matter to explain why galaxies stay together?

At that time my (wrong) answer was that scientists really missed to apply shell theorem to the galactic rotation problem perhaps because galaxy is disc and not sphere etc.

The actual thing that surfaced later on was that scientists did apply shell theorem but in a wrong way.

What we (I and Nikolay Sones) were thinking in April-2019 was that scientists missed to realize that stars rotate within galactic disk and shell theorem as applicable within sphere (like within earth) was applicable which was missed by scientists. We were right in this thinking!

However it turned out that scientists did not miss to apply shell theorem. However they wrongfully applied shell theorem as it was applicable to solar system and they terribly missed to apply shell theorem as it was applicable within disc of galaxy.

Actually in Newton’s Principia, there are more than dozen Theorems that all deal with gravitational effects of spherical bodies under different situations. These are different Theorems but some of them are collectively known as ‘Shell Theorem’.

Having said that, now I again refer to above quoted wikipedia portion – I quote it again:

If luminous mass were all the matter, then we can model the galaxy as a point mass in the centre and test masses orbiting around it, similar to the Solar System.[d] —- [d]> “This is a consequence of the shell theorem and the observation that spiral galaxies are spherically symmetric to a large extent (in 2D).”

Now the situation gets cleared. Scientists applied shell theorem as it was applicable to the solar system. In terms of Newton’s Theorem XXXI (i.e. Shell Theorem as applicable to Solar System), they modeled gravity of galaxy as point mass located at central point and test masses orbiting around the center as per following diagram:

And – following is the screenshot of Newton’s Theorem XXXI i.e. Shell Theorem as applicable to Solar System:

Since scientists had applied shell theorem as applicable to solar system for galactic rotation – so they were expecting Keplerian drop-off in galactic rotations.

And that’s why general theory and specific law were giving the same result for galactic rotations. Actual observations were showing flat rotation curves and mass (over and above luminous mass) was seeming to linearly increase with increase in distance from galactic center (Vera Rubin:1970).

Actually scientists had missed to apply exact relevant Newton’s Theorem XXXIII (Shell Theorem as applicable to galaxy) which was applicable to test masses located within sphere (or disc) of uniform density. Following is the screenshot of Theorem XXXIII:

Following diagram shows how this Theorem XXXIII functions:

Orbits in the setup that fall under Theorem XXXIII are NOT subject to Keplerian Drop-off. Therefore Theorem XXXIII (i.e. a special case of Shell Theorem) naturally gives flat rotation curves for galaxies. With Theorem XXXIII, there is no need of dark matter. First thing is that within this setup, gravity is not subject to inverse square distance law. Here gravity is subject to inverse distance (linear) law. Secondly, at any depth within the disc, the outer layers have no gravitational effect. Its meaning is that from center to edges, mass will seem to linearly increase though actually the total mass remains the same.

Scientists are fully aware of the implications of Theorem XXXIII and they know that gravity drops linearly inside earth and reaches to zero at the center.

By noting that Theorem XXXIII was applicable to galaxy and that by applying this Theorem, we naturally get flat rotation curves for galaxies – dark matter is actually resolved.

MOND is not the proper alternative interpretation. It would be viable if scientists had not actually committed the mistake of applying solar system specific theorem to the problem of galaxy.

But what if scientists did actually commit this mistake?

Then MOND is not viable even if it works.

Note: after reading the blog post about Theorem XXXIII, a PhD Physics person had pointed out that even after applying Theorem XXXIII, the discrepancy remains. To this I replied that blog post is brief. The remaining discrepancy has been acknowledged and also solved by the book. For this reason, I also need to share free sections II.II.IV and II.II.VI from book.

To be Philosophical means looking things in totality. To have a totalitarian approach. Rather than focusing a particular problem, trying to understand overall general structure, scheme or theme. To have a systematic approach and unwillingness to accept any stance within that general scheme without having solid logical justification.

This philosophical approach is good because non Philosophical people are often trapped in particular problems that may not even have sound logical basis. Having totalitarian approach is not a hindrance for solving particular problems either. Philosophy is more interested in big picture. A big picture philosopher can be dealing with particular problems but that would be his practical activity rather than philosophical.

Kepler’s 3rd law was not relevant to Galactic Rotation:

Idea of Dark Matter is like an offshoot of wrongfully expecting the relevancy of Kepler’s 3rd law for rotation of Galaxies.

In our solar system, Mercury is closest to Sun and has greatest orbital velocity. (Let’s say) Pluto is farthest from Sun and has lowest orbital velocity.

In simple terms, Kepler’s 3rd law is that a planet farther from (central) Sun revolves slowly in orbit than the one closer in. In other words, a planet closer to the center has greater orbital velocity and there is ‘drop-off’ in the orbital velocity with increase of distance from the center.

Off course, this behavior of planets was due to underlying general principles that finally Newton extracted but the Kepler’s 3rd law itself was a specific law that was applicable to systems where greater mass is concentrated in the center whereas test masses orbit around that central mass.

Galaxy is a different system where greater mass is not concentrated at the central point. In principle, there was no genuine reason for scientists to expect Keplerian-drop-off for the galactic rotations.

Why actually did scientists expect Keplerial drop-off for galactic rotation had been a mystery for me because galaxy was a different system and Keplerian drop-off was specific to only Solar Systems or Plant-Moon Systems etc. But somehow scientists wrongfully linked Keplerian drop-off with rotation of galaxies.

For example please consider from Dark Matter article of Wikipedia:

Galaxy rotation curves[edit]

The arms of spiral galaxies rotate around the galactic center. The luminous mass density of a spiral galaxy decreases as one goes from the center to the outskirts. If luminous mass were all the matter, then we can model the galaxy as a point mass in the centre and test masses orbiting around it, similar to the Solar System.[d]

At the end of this quote, there is reference to footnote [d] which reads as follows:

This is a consequence of the shell theorem and the observation that spiral galaxies are spherically symmetric to a large extent (in 2D).

Galaxy was modeled in the light of irrelevant part of Shell Theorem:

Now it is clear as to what scientists actually did with the galactic rotation problem and how did they reconcile galactic system with 3rd law of Kepler. They modeled galaxy as a point mass located at center of galaxy where stars were orbiting around that central mass and they did it on the basis of ground or justification that this is a consequence of shell theorem.

In this way, for the purpose of studying the gravitational effects, they modeled galaxy exactly like solar system and justified the expectation of Keplerian drop-off.

But in reality, galaxy is a different system and could not be equalized with Solar System. Yes – Newton’s Shell Theorem was applicable to the disc of galaxy but there are various independent components of the so-called Shell Theorem. Basically various Theorems of Newton’s Principia relating to gravitational effects of spherical bodies are collectively known as ‘Shell Theorem’ such that name ‘Shell’ is not assigned to them in the Principia.

The investigation into the matter revealed that (which I have explained in details in book “Philosophy Unscrambles Dark Matter”) scientists, by modeling galaxy as point mass located at center and test masses orbiting around the central mass, actually applied Newton’s Theorem XXXI that was irrelevant part of the Shell Theorem as it was applicable to Solar System or any Planet-Moon System and NOT to Galaxy.

The official Gravity Model for Galaxy was wrong. Galactic rotation means rotation of stars within the disc of galaxy. There was another relevant part of Shell Theorem i.e. Theorem XXXIII that deals with the cases where test masses are located within sphere (or disc) of uniform density.

What scientists actually did was that they applied irrelevant Theorem XXXI in following way:

Diagram: Theorem XXXI that is applicable to Solar System but NOT applicable to Galaxy

Scientists modeled gravity of Galaxy within the above depicted meaning and scope of Theorem XXXI. With this setup, they were expecting that Keplerian drop-off will be observed but that was not observed. Thus they were trapped into a fictitious problem to which they eventually settled with ghost solution of ‘dark matter’. Following is the snapshot of Newton’s Theorem XXXI that was wrongfully applied by the scientists:

Real scientists can make mistake of applying irrelevant theorems. But real scientists should never settle with ghost solutions.

Scientists Terribly Missed to apply relevant Theorem XXXIII:

Following diagram shows the scenario of Theorem XXXIII which was rightfully applicable to the galactic rotation but that was terribly missed by the scientists.

Diagram: Theorem XXXIII that is rightfully applicable to Galaxy

Following is the snapshot of Newton’s Theorem XXXIII that was applicable to galactic rotations but was terribly missed by scientists:

Theorem XXXIII is not a usual case of gravity. There are two important points:

  1. The test mass is NOT subject to inverse square distance law. Now the test mass is subject to inverse (linear) distance law and;
  2. If test particle is located at a particular depth within sphere (or disk), the outer area of sphere (or disk) will have NO GRAVITATIONAL EFFECT on test mass. Its meaning is that if test particle moves from center to outer edge, the (gravitational) mass will seem to increase.

Given the above two points along with few others which I have explained in the book (Free PDF link), Theorem XXXIII naturally gives flat rotation curves for galaxies; Kepler’s 3rd law becomes totally irrelevant and Dark Matter problem is solved from root.

Following are free excerpts from section II.I.II of my recent book “Philosophy Unscrambles Dark Matter”.

II.I.II. The actual Anomalies

Not only that redshifts of far off galaxies are not due to Doppler’s Effect and thus do not represent radial velocities, there are also genuine problems in the established distances of those galaxies. Official distances of nearer objects have been determined by using common techniques of geometry and these distances are correct. The case of far off galaxies was challenging where finally a method to use luminosity of certain kind of stars (i.e. Cepheid Variable) as indicator of distance was adopted. Theory of redshift-distance relationship was already published where distances had been determined on the basis of luminosity alone and it was afterwards that important works of Fritz Zwicky (1933 and 1937) emerged where he analyzed a huge cluster of galaxies with respect to the total size as well as luminosity and redshift profiles of individual member galaxies of that cluster. The type of Zwicky’s analysis offered a great signal concerning the applicability of straightforward geometrical technique for the determination of galactic distances but unfortunately the hint was taken up by the Expansionists (i.e. relativists) and never brought to the general view in the original simpler form because after sensing the anomaly in this aspect, Expansionists distorted the facts and implemented the hint within twisted expansionist terminology and framework to keep the anomalous results hidden from view. Before discussing the hint, it is better to apply simple technique of geometry to determine the distances of (i) Moon and; (ii) the Sun.

Here we have taken standard values for ‘angle of view’ and ‘diameters’ of Moon and the Sun from online sources[i]. Our calculated distances of both the objects are only slightly different from official distances therefore we regard this method as accurate for the purpose of evaluating the distance. The hint that we get from the works of Zwicky is that he has estimated or determined the total diameter of the Coma Cluster. In his 1937 paper, he has taken radius of Coma Cluster to be only 2 million light years which is actually wrong as the up-to-date official estimate is 10 million light years and the reason of underestimation was that by that time, distance of Coma Cluster was also underestimated to be only 45 million light years which, by modern and ‘finalized’ standard is almost 321 million light years. The hint that now we can get is that the diameter is 20 million light years[ii] that should form an ample measurable angle of view on the sky. Now, instead of relying on measurements of the distances on the basis of luminosity, let us here calculate the distances of few prominent astronomical objects on the basis of simple geometry.

We have seen that the above method of distance estimation of astronomical objects requires (i) angle of view on sky and; (ii) actual or approximate diameter of the object. Historically, the estimates regarding distances of beyond Milky Way galaxies started in 1920s on the basis of luminosity of certain kind of stars because perhaps no data, calculation or approximation about diameters of those astronomical objects was available at that time. In 1930’s, works of Fritz Zwicky and others featured estimates regarding diameters of astronomical objects located far beyond Milky Way. Initially those estimates were wrong but they were improved and corrected over time. Angles of view on sky of those astronomical objects were also not difficult to figure out that were determined eventually but matters were in the hands of Expansionists who contaminated simple techniques of geometry with formulas of redshifts,[iii] possibly after having sensed the type of anomalies that must have surfaced in case the straightforward methods were implemented. As a matter of fact, so far simple distance determination method has not been applied[iv] even for the case of Andromeda which is the nearest large galaxy; whose official diameter in light years is known[v] and angle of view on sky is also known[vi] to be slightly larger than six times the angle of moon. Likewise the estimate regarding diameter of Coma Cluster is available and angle of view on sky is also known to be almost four times the angle of moon[vii]. Here, for our analysis, we select another astronomical ‘object’ i.e. the famous Hubble Deep Field image that belongs to tiny section of sky whose angle of view on sky is almost 10 times smaller[viii] than that of moon but contains ten times more galaxies than Coma Cluster due to which we can get a rough but safe (lower side) estimate of diameter of almost 60 million light years because with almost 1000+ galaxies, from edge to edge, there should be average 33 galaxies in Coma Cluster and with 10000 galaxies in deep field image, the number of edge to edge galaxies should be 100 which is three times greater therefore we take diameter of deep field image to be three times greater than that of Coma Cluster. We however regard it as lower side safe estimate because deep field is not a cluster of relatively compacted galaxies (having compressed in-between distances) which is the case with Coma Cluster.

Hubble Deep Field Image – Credit: NASA/ESA

By applying the straightforward geometrical method we get following ‘anomalous’ estimates of distances of these astronomical objects:

For Andromeda Galaxy, we notice a large discrepancy in the distance of 3.9 million light years calculated through straightforward geometry in comparison with the official distance which is only 2.5 million light years. Likewise, the official distance of Coma Cluster is only 321 million light years but geometry is telling it should be located at almost 509 million light years. And the case of Deep Field Image is particularly ‘anomalous’ because the calculated distance is located far beyond the permitted zone of the so-called standard model. Readers are requested to recalculate these figures by themselves to see the genuineness of the results. For instance, how come a huge cluster of galaxies that contains almost 1000 separate galaxies having large distances in-between as well, forms a smaller than a single galaxy Andromeda’s angular view on sky and yet located at distance of only 321 million light years when Andromeda, a single galaxy, itself is officially located at 2.5 million light years? A very large object that is larger by ratio of many thousands and not by ratio of only many hundreds is appearing smaller – it means that distance of the object must not be as low as only 321 million light years. It is making perfect sense that Coma Cluster’s actual distance is more than 500 million light years.

It is stated earlier that so far straightforward geometric method has not been applied for distance estimation even for the case of Andromeda which is the nearest large galaxy and whose diameter as well as angle of view on sky is known. Following section of Wikipedia article explains which methods have been applied so far and anyone should wonder why simple geometry has not been applied so far[ix]:

Distance estimate

At least four distinct techniques have been used to estimate distances from Earth to the Andromeda Galaxy. In 2003, using the infrared surface brightness fluctuations (I-SBF) and adjusting for the new period-luminosity value and a metallicity correction of −0.2 mag dex−1 in (O/H), an estimate of 2.57 ± 0.06 million light-years (1.625×1011 ± 3.8×109 astronomical units) was derived. A 2004 Cepheid variable method estimated the distance to be 2.51 ± 0.13 million light-years (770 ± 40 kpc).[2][3] In 2005, an eclipsing binary star was discovered in the Andromeda Galaxy. The binary[c] is two hot blue stars of types O and B. By studying the eclipses of the stars, astronomers were able to measure their sizes. Knowing the sizes and temperatures of the stars, they were able to measure their absolute magnitude. When the visual and absolute magnitudes are known, the distance to the star can be calculated. The stars lie at a distance of 2.52×106 ± 0.14×106 ly (1.594×1011 ± 8.9×109 AU) and the whole Andromeda Galaxy at about 2.5×106 ly (1.6×1011 AU).[4] This new value is in excellent agreement with the previous, independent Cepheid-based distance value. The TRGB method was also used in 2005 giving a distance of 2.56×106 ± 0.08×106 ly (1.619×1011 ± 5.1×109 AU).[5] Averaged together, these distance estimates give a value of 2.54×106 ± 0.11×106 ly (1.606×1011 ± 7.0×109 AU).[a] And, from this, the diameter of Andromeda at the widest point is estimated to be 220 ± 3 kly (67,450 ± 920 pc).[original research?] Applying trigonometry (angular diameter), this is equivalent to an apparent 4.96° angle in the sky.

From the above quoted text from the Wikipedia article, the very last sentence is particularly important. This sentence is not supported through citation and it seems that this sentence has been added by some curious individual who actually applied trigonometry on accepted diameter and distance of Andromeda galaxy and found that angle of view on sky should be 4.96°instead of official and observed value of 3.167°. Therefore we now put the value of 4.96° in our straightforward geometry for the case of Andromeda and note that with 4.96° angle of view on sky, the calculated distance of Andromeda galaxy tallies with the official distance of 2.54 million light years.

Here we see that someone’s independent calculations using official Trigonometry have confirmed the results of our commonsense based straightforward geometric formula. It is clear that if Andromeda is actually located at distance of just 2.54 million light years then it should form a larger angle of view of 4.96°which is larger than actually observed angle of view of only 3.167°. Therefore, the official distance of Andromeda galaxy is not supported by the known diameter and angle of view on sky as the official distance defies official Trigonometry.

The case of Deep Field Image is anomalous and complex as well because this image covers galaxies located at wider range of distances along line of sight. But if ‘nearer’ galaxies or objects are included in this image whose total angular diameter on sky is just one tenth that of moon then those ‘nearer’ galaxies also must be located very far away. For example up to the distance of Coma Cluster, only 20-30 galaxies should fill this image in complete. The angular diameter of deep field image is almost 45 times smaller than that of Coma Cluster yet contains 10 times more galaxies within a very small angle on sky. Roughly there should be only few hundred ‘foreground objects’ such that the ‘foreground’ also should be located very far away. The difficulty of ‘foreground objects’ however has been greatly solved through ‘Extreme Deep Field Image’ which is actually a close-up assessment of the core or nucleus of the same Deep Field Image where 5500 galaxies are assessed but angle of view is also reduced to become almost 14th the angular size of moon. The overall implication regarding distance estimation should remain almost same. Having total 10000 galaxies with margin of few hundred ‘foreground objects’, the diameter from edge to edge has been taken only at 60 million light years which is a safe lower side estimate because official estimate of diameter of Coma Cluster with only 1000 and squeezed galaxies is 20 million light years. Furthermore, if we remove the foreground objects from the deep field image then even greater number of background objects will be exposed and total number of galaxies in the image will be increased. It is safe lower side estimate also because if we consider another perspective that distance between small dot at one edge and another small dot on opposite edge should be at least the distance between Milky Way and one of the galaxies of Coma Cluster which is actually located at 500 million light years but still lies within our cosmic neighborhood then the estimate of distance of the farthest galaxy in the deep field image reaches to almost 583 billion light years from earth. The moderate estimate can be something like 20-40% of the higher side estimate thus those farther galaxies of deep field image, with moderate estimate, may be located at distance of 100 to 200 billion light years. We can attempt to get more precise result on our moderate estimate of distances. The Deep Field Image officially contains 10000 galaxies that include many foreground larger galaxies also. If we remove foreground galaxies then even more small looking galaxies are expected to be revealed from behind the foreground objects. But for the sake of our moderate estimate, we say there are only 10000 background small looking galaxies. From edge to edge, only 100 galaxies (each 80000 light years across) exist and galaxies are separated by the moderate distance of two million light years each. With these settings, we get edge to edge diameter of 208 million light years of the background visible extent of Hubble Deep Field Image. With this precise moderate estimate of diameter, the distance of those farthest visible (small looking) galaxies in Deep Field Image comes at 238.254 billion light years. Even at safer lower side estimate of 68 billion light years, this is serious discrepancy of the standard model where the viewable galaxies must not cross the distance of 13.2 billion light years only to remain conforming to the standard age of the universe of 13.8 billion years.

With huge distance scale of many hundred billion light years, the farthest galaxies ‘look’ small due to obvious reasons. But NASA loves to tell us that those galaxies are actually smaller in size and their ‘standard’ reason is also obvious because at distance of only 13+ billion light years, large galaxies should not have appeared so small. NASA very conveniently informs us that earlier galaxies are actually smaller in size in following words[x]

When we look at very distant galaxies, we see a completely different picture. Many of these galaxies tend to be small and clumpy, often with a lot of star formation occurring in the massive knots. 

In my opinion, the farthest visible galaxies, being located at distance scale of many hundred billion light years are typically very large galaxies as smaller ones simply could not be seen from such huge distances. NASA insists that they are smaller in size only to project them on a little and unrealistic but ‘standard’ distance scale of just 13+ billion light years. When a galaxy actually located at distance of many hundred billion light years is declared to be located at only 13+ billion light years then ‘yes’ it is smaller in size and may also be ‘half manufactured’ sort of. In case background small looking galaxies of Deep Field Image are located at 13.5 billion light years then edge to edge diameter of background visible extent of this image should be only 11.8 million light years which is almost equal to the diameter of our small local group[xi] that contains only three large galaxies along with just 50 other dwarf galaxies. Wikipedia article about current record holder of farthest galaxy[xii] informs us that diameter of this farthest galaxy ‘GN-z11’ is only 25000 light years. If diameter of visible background extent of Deep Field Image is only 11.8 million light years then from edge to edge there are 100 small galaxies each having diameter of only 25000 light years and each separated by distance of only 93000 light years and result would be what NASA wants to tell that those background smaller galaxies are located at distance of only 13+ billion light years. These results do not match with the actual Deep Field Image whose careful glimpse reveals very sparse density of galaxies such that edge to edge smaller looking galaxies are seemingly separated by more than our previous moderate estimate of 2 million light years each. In fact the claim of standard model cosmology that early universe was ‘denser’ is not actually confirmed as farthest galaxies in Deep Field Image are not denser than our local density of galaxies. For this reason official people often say that early ‘dense’ universe can be seen in CMB only because ‘early’ galaxies do not show the desired high density. The actual background small looking galaxies are in fact very large galaxies and from edge to edge they are separated by very large distances – far more than our previous moderate figure of 2 million light years. By no means can they fit within diameter of only 11.8 million light years and thus by no means they can reasonably demonstrated to be located at distance of only 13+ billion light years.  

The primary objective of this book is not to highlight the discrepancies of the Big Bang Cosmology. Basically two reasons persuaded me to include these anomalies in this section of the book – firstly the readers should forget the so-called ‘anomaly’ of dark matter and should think about the actual anomalies. Secondly, it seemed appropriate to repeat the pattern of Fritz Zwicky in presenting apparently out of context anomaly within the discussion of a separate topic. As far as clusters of galaxies are concerned, there is no genuine anomaly of ‘dark matter’ because redshifts do not mean ‘radial velocities’ and actually there is no ‘velocity dispersion’ within the cluster; Zwicky was also trying to assert the same thing. The actual anomaly whose hint comes from study of cluster as a whole is the anomaly in the official distances of astronomical objects because they extensively differ from the distances that can be calculated quite easily by employing simple technique of geometry. The discrepancy starts right from Andromeda Galaxy thus any excuse of the ‘curved spacetime’ over very long distance will not work. Actual and strict finding of Edwin Hubble was only that there is linear relationship between ‘redshift’ and ‘distance’ and to be precise, for Hubble, the reason of redshifts is not known[xiii]. But unfortunately, official science has adopted ‘velocity’ as a valid reason of redshifts. With redshifts being interpreted in terms of ‘velocity’, the formula of those redshifts contains ‘c’ i.e. the value of speed of light. With ‘c’ included in the formula, ‘v’ (velocity) will never reach closer to ‘c’ or the results will be twisted may be in some other way.

Now within next few years[xiv], NASA is going to launch James Webb Space Telescope which is said to be 100x more powerful[xv] than highly successful predecessor Hubble Space Telescope (HST). The strange aspect is that despite 100x power of upcoming new space telescope, NASA is dead sure that no galaxy beyond 13.6 billion light years will be seen[xvi]. NASA explains that Big Bang occurred 13.8 billion years before – although the upcoming telescope will not be able to see Big Bang itself but the very first galaxies belonging to the distance of 13.6 billion light years will be resolved whereas nothing will be seen beyond that distance because actually there was no light at all before that era.

The fact is only that due to twisted formulas, actually the ‘distance’ will never be shown greater than certain value. The reason behind the absolute surety of NASA that any galaxy older than 13.6 billion years will not be seen by the 100x more powerful telescope is the fact that NASA is fully aware[xvii] that limit on distance is imposed by the formula itself. Please see the following table of different values of redshifts (Z) and corresponding distances of galaxies in light years:

Redshift-distance relationship that should be expected to be tabulated here in simple linear format where certain increment in redshift should result in regular (linear) increment of distance, actually has been implemented in a twisted form such that with increase of value of ‘z’ (redshift) after the value 2, there is decreasing trend of distance which means that distance is not being increased properly in official tables. For example with increment of 1 in the value of z from 1 to 2, the corresponding increment in the distance is almost 3 billion light years. But afterwards with the increments in z from 4 to 5 to 6 to 7; not a single billion light year is incremented on the distance scale. Clearly this is the consequence of including value of ‘c’ in the formula of redshift. At redshift 10, galaxies are ‘moving away’ at speed close to ‘c’. When receding speed (if really receding) of galaxy will further approach towards ‘c’, the galaxy will no more be visible. While the formula intends to restrict visibility within the range of below luminal receding speeds but another factor is on the play. The sort of cosmic horizon beyond which HST cannot see is not actually determined by receding speeds of galaxies because galaxies are not in fact receding away like that. Actually there is region beyond (official) 13.2 billion light years where galaxies are considerably redshifted to near infrared zone that HST cannot see. James Webb Space Telescope is able to see infrared portion objects but that also has limit. With these hard compulsions that come mainly from calculation methodology, NASA conveniently asserts that beyond 13.6 billion light years, there will be complete darkness and the darkness will be due to absolute absence of galaxies. Galaxies did not exist prior to 13.6 billion light years and the Big Bang Theory is directly confirmed through a powerful telescope, even before the launch of the telescope. With too expensive project of prestigious space telescope that is not even going to have long functional age, the maximum they are going to find or deliberately want to show is that galaxies do not exist beyond the distance of 13.6 billion light years; age of universe i.e. 13.8 billion years is correct; Big Bang Theory is therefore ‘confirmed’ at ‘observational level’. Furthermore, we have already seen in first chapter that these formulas serve as colored spectacles and result is that if real or even hypothetical galaxy is located beyond 14 billion light years, the formula will tell that it is not located at distance more than 13.6 billion light years. So here need is to look at the reality with clear objective eyes and vision which is not contaminated by the colored spectacles.  

MS. Tamara M. Davis is an official voice who tells these things slightly differently. In a paper titled “Superluminal Recession Velocities”[xviii] she and co-author write that official redshift formulas are taken within the context of Special Theory of Relativity (SR) that requires that visibility of galaxies should stop when ‘v’ becomes equal to ‘c’ i.e. when receding velocity equals velocity of light, then the galaxy permanently goes out of sight.

Thus, galaxies with distances greater than D = c/H are receding from us with velocities greater than the speed of light and superluminal recession is a fundamental part of the general relativistic description of the expanding universe. This apparent contradiction of special relativity (SR) is often mistakenly remedied by converting redshift to velocity using SR.

Being Physicists who prefer General Relativity (GR) over SR and who are straightforward in their assertions, the authors of this paper reveal the secret that galaxies having value of redshift more than 3 are actually receding away at superluminal speeds.

Here we show that galaxies with recession velocities faster than the speed of light are observable and that in all viable cosmological models, galaxies above a redshift of three are receding superluminally.

Afterwards this paper proceeds to explain the mechanism by which galaxies ‘recede away at superluminal velocities’ but still remain visible in terms of ‘curved spacetime’ model of GR.   

Now we come back to our book where the point is not SR or GR. Cosmic Redshifts do not in fact mean ‘velocity’; galaxies are not moving away at all. The actual fact is that galaxies having redshift more than 3 are located beyond the official time of Big Bang. Confidence of NASA that even 100x powerful telescope will not be able to see anything beyond the distance of 13.6 billion light years indicates that NASA is fully aware that limit is imposed by the formula itself. That is, even if they find lot of galaxies located at very far off actual distances, they will conveniently say the distance is not more than 13.6 billion light years by showing the ‘calculated’ distance as proof.

Mr. Marco Pereira[xix], MSc (Nuclear Physics), PhD (Physical Chemistry) and a Professor of Molecular Bio-Physics has also noted the anomaly of non-linear ‘observations’ of redshift-distance in Sloan Digital Sky Survey (SDSS) data[xx]. He claims to have found fourth spatial dimension in the Universe through his self-created theory of ‘Hypergeometrical Universe’[xxi]. He claims that his theory has rightly predicted non-linear redshift-distance pattern of Super Novas 1a which is actually observed by SDSS.

Image added with permission of Mr. Marco Pereira

Above type of non-linear observed redshift-distance relationship is claimed to be rightly predicted by his theory of Hypergeometric Universe and the same is said to be the proof for the existence of extra spatial dimension of the Universe. We have noted already that actual non-linear redshift-distance relationship is something which mainstream physicists avoid to mention and only few people like MS. Tamara M. Davis would dare to expose this kind of secret. Our finding is that Mr. Marco Pereira has not found some reality which was already not calculated by Special Relativity but he does reach to a position which is normally not told openly by the mainstream physicists. After sensing this anomaly, MS. Tamara M. Davis rejected SR based calculations and favored GR based explanation. After finding that same anomalous looking SDSS observations are consistent with his Theory of Hypergeometric Universe which accommodates SR formulas in its development, Mr. Marco Pereira declares that SDSS observations are the proof of the existence of extra fourth dimension of the Universe. Underlying fact is that SDSS has also calculated distances of 1a Super Novas using the formulas of Special Relativity. The Lorentz Transformation factor is the reason behind non-linear plotting of this data. According to simple Hubble Law, the plotting should have been linear. If it is actually linear which becomes possible if we remove Lorentz Factor from the formulas of redshifts then distances of farthest visible galaxies come at the scale of many hundreds of billions light years which are consistent with direct simple geometric calculations of distances. The ground for removing Lorentz Factor from formulas of redshifts is the fact that redshifts do not actually represent velocities. But if redshifts do represent velocities (i.e. the position which is official but not likely) then non-linear plotting of real redshift-distances data is actually an anomaly that can rightfully be accounted for by proposing a fourth dimension of the Universe. In case Universe is actually expanding and farther galaxies are more redshifted at those distances which are lesser than the expected linear distance then extra redshift might have been accumulated during the course of passage of those galaxies through the “fourth dimension” proposed by Mr. Marco Pereira. The result is that either farthest visible galaxies are located at the distance scale of many hundred billion light years or Mr. Marco Pereira may be right in his proposal of extra fourth dimension of the Universe.

But reality is not that complex as suggested by Mr. Marco Pereira who only apparently favors simplicity by attributing his complex theories as conforming of Occam’s razor[xxii]. The proof of the assuredly far greater distances of astronomical objects as presented in this section is as straightforward as it can get and thus conforms to the Occam’s razor in true sense. Readers are requested to recalculate these distances by themselves and also recheck the results with official Trigonometric Formulas whose results, with lower side estimates of diameters involved, are only slightly different as given in table below.

It is stated already that Expansionist regime is not entirely blank about these anomalies. They know these things and they hide the actual things by presenting them within twisted terminology and formulas of their favored framework. The Wikipedia article titled “Angular diameter distance”[xxiii]contains following important, though twisted, confession about this topic:

However, in the ΛCDM model (the currently favored cosmology), the relation is more complicated. In this model, objects at redshifts greater than about 1.5 appear larger on the sky with increasing redshift.

This is related to the angular diameter distance, which is the distance an object is calculated to be at from ɵ and x, assuming the Universe is Euclidean.

We have seen already that in official tables, with the increase of redshift, the increment in the distance scale becomes shorter and shorter. Formula tells that astronomical object is located at nearer than the actual distance and thus the object ‘appears’ (within standard model) larger on ‘sky’. Appearance on sky of anything does not depend on any model. If something is looking larger on sky within Lambda CDM model, then it only means that calculated size of object is larger than what can be actually observed on sky. We also have seen earlier in this section that just how Andromeda ‘appears’ larger on sky. This confession, though made in twisted words, automatically validates, in principle, the calculations about tremendously larger distances of visible galaxies presented in this section. Therefore the only issue remained unsettled so-far is to check whether Universe is really Euclidean or not. The dilemma of the official cosmology is that now they have reached to the finding that at least observed universe is flat and thus the actual geometry of the observable universe is Euclidean. In a flat universe which is representable using Euclidean geometry, the two parallel lines will always remain parallel no matter how great distance is covered. To a question “Is the universe really flat, or is it just very slightly curved?” – Mr. Erik Anson, Physics/Cosmology PhD student (University of Washington) provided following insightful reply[xxiv]:

Yes, it’s entirely possible that the Universe is only almost flat on large scales, as is acknowledged by the (scientific)[xxv] community. There is a cosmological parameter, Ωk, that relates to the amount of large-scale curvature, and observations can constrain it to be within a small range including zero, but can never show it to be exactly zero.

However, if there is any curvature, it’s so small that it’s effectively irrelevant, so we may as well model it as flat (which is simpler) unless and until we know otherwise.

Symmetry magazine[xxvi], which is a joint publication of ‘Fermi National Accelerator Laboratory’ and ‘SLAC National Accelerator Laboratory’ published an article titled “Our Flat Universe – Not a curve in sight, as far as eye can see”[xxvii] on date 07-04-2015. The following introductory lines say it all that observable universe is actually found out to be flat and thus representable in Euclidean geometry:

Mathematicians, scientists, philosophers and curious minds alike have guessed at the shape of our universe. There are three main options to choose from, in case you’d like to do some digging of your own:

The universe could be positively curved, like a sphere.

The universe could be negatively curved, like a saddle.

The universe could be flat, like a sheet of paper.

As far as scientists can tell, this third option is correct. But what do people really mean when they talk about “flatness”? Your high school math teacher would be overjoyed to tell you that it’s all about geometry.

In a flat universe, Euclidean geometry applies at the very largest scales. This means parallel lines will never meet, and the internal angles of a triangle always add up to exactly 180 degrees—just like you’re used to.

In Lambda CDM model, as we have seen, the distances of far off galaxies are not the actual physical distances as they are superimposed and artificially constrained by the twisted formulas. In simple geometric calculations of distances, there is no artificial or twisted superimposition at work and thus actual distances of visible galaxies really are on a much larger distance scale than could be permitted by the standard model which means that the actual physical reality is not truly ‘modelled’ by this ‘standard model’. With extremely greater distances of astronomical objects, the problem is not that there should be more than observable matter; the implication is that density of matter within the universe is far lower than the available assessments of the so-called standard model who has false claim of having explained all the observed reality because the model does not even know the right density of present day Universe and still claims to know all the details of minute fractional parts of so-assumed very first moment after the ‘Big Bang’. Secondly, it is due to ‘velocity’ interpretation of redshifts that whole need of using ‘c’ in the redshift formulas arise. It is value of velocity of light ‘c’ which compels science authorities to stay blind with wrong lower side estimates of distances of remote galaxies. Velocity interpretation of redshifts inescapably leads towards flawed calculations of the distances of those galaxies which is sort of mathematical proof that redshifts do not represent receding velocities of galaxies because with velocity interpretation, ‘c’ will be added in the formulas of redshifts; consequently the estimates of astronomical distances would be bound to be outright deceitful as no galaxy will be shown located beyond a certain distance. And although better estimates of astronomical distances have been presented in this section but this book will keep on referring to the distances of remote galaxies with ‘standard’ values or estimates unless otherwise specified.

[i] , and diameters of Sun and Moon taken from simple google search.

[ii] Britannica article on Coma Cluster says that diameter is 25 million light years. [The main body of the Coma cluster has a diameter of about 25 million light-years]. Wikipedia article seems silent on diameter but google search show up from reference of Wikipedia that diameter of Coma Cluster is 20 million light years.  

[iii] “Angular Diameter Distance” section on this Wikipedia page:

[iv] “Distance Estimates” section of Andromeda article on Wikipedia —




[viii] Based on comparison of section of sky of deep field image with size of moon on this and many other pages:






[xiv] Right now, the space telescope is expected to be launched in year 1921.



[xvii] I had little conversation (over Internet) with a NASA Information Officer. I told him that official galactic distances are underestimated. I did not explain or prove my point; just told that proof will come with book (i.e. this book). He denied the existence of any such anomaly and also denied that NASA is trying to hide this anomaly. To this I replied that even then I will write in my book that NASA fully knows and only hides this fact because otherwise it will be more embarrassing for NASA. Readers should judge by themselves whether or not NASA knows about it given the simple fact that their formulas do impose upper limit on distance of galaxy.


[xix] profile of Mr. Marco Pereira –






[xxv] Bracket (scientific) added by me.