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 theSolar System.[d] —- [d]>“This is a consequence of theshell theoremand 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.**