Practical understanding of let’s say Newton’s Physics is of the type which is offered by the text books. Students understand the statements of laws along with mathematical depiction thereof and become able to solve numerical or exercise questions. After completing the study in this way, they may become good engineers or professors of physics.

But those who become scientists, they seek understanding of a different kind that may be regarded as intellectual understanding. They not only pursue to understand book contents within the context of daily real life observations, more importantly they also try to investigate the history and logic behind theories of science that they read in text books. Rather than trying to solve text books exercise questions, their efforts are usually focused at finding how the theory in question was actually derived or discovered by the original scientist. The practical understanding makes one able to apply theory in practice whereas the intellectual understanding would make one able to not only improve or refine the existing theories but to propose or formulate new theories as well.

Father of Physics is Aristotle. He made physics a separate discipline. Modern Physics developed only out of critical inquiry of the Aristotelian ideas.

It is pointless to say that Aristotle was wrong and Galileo was right. Galileo received all his raw materials for the critical inquiry from Aristotle. Aristotle’s ideas were wrong. But in case there were no ideas on Physics at all then there would have been no Galileo or Newton as well.

It is supposed that dark matter is the source of extra gravity which cannot be traced to detectable sources. Fact is however that there is no extra gravity at all. Scientists applied irrelevant Newton’s Theorem XXXI to the galactic rotations and observed that actual effect of gravity was greater than what could be calculated using Theorem XXXI.

But
Theorem XXXI was applicable to systems like solar system or planet moon
systems. For the galactic settings,
Newton had another Theorem XXXIII and scientists terribly missed to apply the
actually relevant Theorem. In case Theorem XXXIII is applied, then calculated gravity
would reasonably tally with observed gravity and there will be no need of dark
matter. And it is to be noted that Theorems XXXI and XXXIII are different
variations of Shell Theorem. [i]

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.

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.

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:

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.