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Riemann Zeta Function

Imagine a sum that keeps getting bigger: 1 + 2 + 3 + 4 + ... but somehow, its "value" is said to be  \[1+2+3+4+5… = -\frac{1}{12}\] This equation isn’t a joke. It isn’t a typo. And it definitely isn’t like one of those internet tricks that try to prove that 2 = 3. But how? Doesn’t calculus dictate that the sum should diverge? How can an infinite sum of positive integers result in a negative fraction? This equation is one of the paradoxical results of the famed Riemann Zeta function: \[\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\] First Encounters My first encounter of the Riemann Zeta function was in a problem introduced to me in Math circle India:\(\newline\)               What is the probability that two random numbers x & y are coprime i.e. gcd(x,y) = 1 for \(  x,y \in \mathbb N\) At first glance, it seems like an impossible question to compute. We’re picking 2 numbers at random — how do you even define a probability over an...

Black Hole Information Paradox

 

What is information? A word that we use every day but has a deeper meaning than we comprehend. Take an example of a diamond, a stone with exorbitant prices commonly used to make jewellery. Now, take a look at graphite, ordinary, everyday, mundane. These two are made of the same substance, carbon, but are so wildly different. Information sets them apart, something abstract, but a thing that holds the universe together in the most elegant of manners. Much of our modern physics and ultimately all sciences are based on one law: Information can never be destroyed. When we go near black holes, the most extreme objects in the universe, everything starts to get hazy...

Black holes are essentially an area of gravitation so infinitely dense that it is like a trap door for the stage that is space-time. They are the opposite of information. While it sets things apart, black holes take different things and make them the same. This is deleting the information of the matter that the particles were once part of. This creates the information paradox. This theory can be visualized by a simple thought experiment below:

Imagine two fluids, water and oil which floats on top of it. Now imagine yourself and all matter in the universe to be waves of different shapes and sizes in the oil. The oil represents the Higgs field, a field where various fluctuations (waves) result in the formation of fundamental particles. The water represents the space- time continuum in which we live. Space-time is affected by the density and strength of the waves in the oil that represents matter. A black hole is represented by a whirlpool going through both the liquids. This whirlpool cancels out the matter (waves in the oil). Thus the information about the matter is lost.

If this were true, we would have to go back and redo much of our sciences, postponing any new inventions by centuries. It forces us to rebuild our civilisation and rethink our understanding of our universe. This world would be quite post-apocalypse with many people taking the opportunity to spread ignorance like in the pre-enlightenment period in Western Europe. While this is all very dark, it is not the only possibility.

Black holes are full of unicorns and treasure pots at the end of

rainbows, prove me wrong! You can’t. Since anything that enters a black hole can never leave, who knows what is inside a black hole? Maybe there is an entire universe inside a black hole that we just can’t access unless of course, we are willing to never come back. And who knows, maybe we can bend science just enough that we can actually get out of a singularity. This can open many possibilities for us such as using black holes as colossal hard drives, entire universes to expand to, and even from a realtor’s side, more land!!!

The universe is a hologram:
How the black holes store data is a topic that leads to new and interesting conclusions. It is theorized that a black hole expands by a tiny bit each time it devours any matter. This could mean that the black hole is actually saving the matter's information at its surface. If you fall into a black hole would meet with all that had previously fallen into the black hole and experience the 3 dimensions. Everything you see would be information spread on the surface of a black hole. This is sort of like a hologram or a 3-D image. This could also open the possibility that we live on the surface of a black hole
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