Insights on the Black Holes information paradox from the behavior of the entropy of the Euler’s totient function

At the very beginning let’s define what the entropy of Euler’s totient function is.

Let’s start with the implementation of the notion of the Euler’s totient function (Phi) density within the given set of numbers D[S(N)].

In simple words, the density of Phi function of some set of numbers is equal to the sum of all the values of Phi function of all the numbers within a given set divided by the sum of all these numbers.

It is well known that the density of the set of all the natural numbers is equal to

But the densities of the subsets of the natural numbers may take different values and what’s crucial their behavior might be very regular, depending on the “algorithm” of our choice.

More about the interesting behavior of the density function you can find in my other post here.

Choosing the set of N numbers from all the natural numbers we can make it using either some kind of pattern or make it more random.

The key feature of the density function of the chosen set of numbers is that the more random(ish) our choice is, the closer its value is to the mentioned before value for the full set of all the natural numbers (1/Dzeta(2)). And the more our choice is based on some specific pattern, the further from that value it is (values go down or up depending on the specificity of the pattern).

That’s why we can relate the entropy of the Phi function of the set of numbers directly to its density.

As the set is more regular (its Phi density is further from the average value) its entropy is lower, and as it is more random (density closer to the average value), its entropy grows.

The interesting outcome of such a relation between entropy and density is that the maximal entropy (for the random set) is achieved for the same value of the density (1/Dzeta(2)), for which the entropy achieves a value equal to 0, for the full set of natural numbers (there is 0 randomness in such the set).

Having such a straightforward relation between density and entropy of the Phi function we can go further.

Now let’s introduce the notion of the as I call it numbers field/plane being a geometric representation of the values of the Phi function of all the natural numbers.

You can read about this concept in more detail here, but in short, it can be described as a specific lattice of the unities where on the circle of radius representing every number n, there are n unities equally distributed on the given circle. The entire field/plane is built of circles representing all natural numbers from 1 up to infinity.

Within every circle every of n unities present there takes the specific state – it’s either relatively prime (marked red) or not (marked blue) to the n.

We can also say that every relatively prime unity (red) is visible from the origin of the plane, and all the others (not relatively primes, blue) are not.

This is how the field/plane for the first 6 n looks like.

And here is the view with more of n.

Having such a representation of the values of the Phi function we can go almost directly to the insights concerning black holes.

What’s important there is the fact (somehow obvious) that no matter how big n would we take (how distant circle from the origin) all the information about all the unities in the entire plane limited by this circle is present on this given circle. So, in other words – we can deduct everything about the unities below this circle knowing only the sequence of the unities on this given, boundary circle.

Of course, we’ve got here the very specific arrangement of the unities, but it might become somehow natural in very specific circumstances or conditions.

The described case can be of course shifted from the plane to 3d space – unities will be arranged then on the spheres, but still, the stable ratio (this time 1/Dzeta(3)) will be visible from the origin, while the rest will not.

Going further, we can say that the entropy of such a plane or space is located entirely on this boundary circle/sphere, and the entropy of the entire interior is equal to 0 because once the arrangement on boundary is known, the arrangement of the interior is fully defined and there is no uncertainty anymore.

One more general thing we have to assume here is that there exists a fabric of space, so the unities (however would we call them) of which the space itself is made, but this is something I think most physicists will agree on.

I would personally argue that there should be two variants of such unities (again a kind of similarity to the relative and not relative primes on the numbers plane) because it could help solve many issues like for example dark energy, but more about this idea in some another post.

 

Going further we could also assume (what’s also not a very strange assumption, and for the time being cannot be ruled out😊), that all the other existing (known and unknown yet) particles and force carriers are made of these fundamental unities (every being a kind of their specific arrangement). So, the space itself and everything else is made of these fundamental unities.

 

And now let’s finally go to the black holes.

We know that the entire entropy of the black hole is located on its surface (event horizon) which agrees with the situation known from the numbers plane.

But we’ve got this paradox of information being lost after passing the event horizon and reaching the singularity.

But let’s imagine that “thanks” to the extremal condition in the interior of the black hole everything that falls into the black hole is being decomposed to the fundamental unities (of which the space itself is made). The curvature of the space there, so the force of the gravity grows extremely, but finitely – until the arrangement of the unities reaches the finite, specific, one and only possible, and densest as possible arrangement.

Three things happen then:

-          entropy of such an interior drops to 0 (perfect, one and only possible arrangement)

-          Flow of time stops – time is space emergent (more thoughts on that you can find here), so due to the lack of possible rearrangement of such a system there is no more time flow.

-          We are avoiding the singularity – we are just reaching the finite curvature of the space and its density.

Depending on the size of the black hole, so the number of fundamental unities of which everything (including space itself) is made of in its interior, the arrangement of the unities on its boundary (event horizon) will take the specific, unique arrangement encoding its entropy.

And what’s crucial – an eventual process of black hole evaporation won’t be random – the amount and specificity (hypothetical variants) of the unities released during this process will be fully dependent on the current arrangement of the unities on the boundary since this arrangement cannot be random but is fully dependent of the structure (which is perfectly defined) of the interior. So, the "radiation" will be fully dependent of the current state/entropy of the black hole.

 

That’s how we are avoiding information loss during the process of evaporation.

 

Of course, the entire described here idea is very general, not showing how things precisely really work, but shows some insights into how it could generally work especially that it doesn’t violate any known physics laws and helps avoid some of the current major issues.