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Posted by Aarit Jhindal on

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...
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Posted by Aarit Jhindal on

From the big bang to the end of the universe

 


When Einstein developed the general theory of relativity, he was not satisfied with it as he believed that the universe should be stable. After a lot of thinking he added a new cosmological constant which is the opposite of gravity, if gravity attracts things it repels things. It worked out well until scientist discovered that the universe was expanding. Edwin Hubble found out that two different galaxies were moving apart from each other using the doppler effect theory which states that wavelengths get shorter the closer the object moves towards you. Einstein was puzzled by this. Although he strongly believed in a more stable universe, his work on the expansion of the universe ended up making him the rightful owner of the theory, even if he didn’t want to be. This theory led scientists to believe in a ‘big bang’ and that everything started out from a single singularity and spread out to the universe we know and believe in today. They even conceptualized a new energy particle corresponding to this- Dark energy (Anti-Gravity).  A few years later, to the surprise of many, it was calculated that the rate of expansion is speeding up, which led theoretical physicists  to postulate three ways the universe might end:

·    Big Crunch

The scenario in which gravity overcomes dark energy and results in everything coming together into a dense, massive singularity (much like the opposite of big bang).

·    Heat death

When dark energy stretches gravity to the point it loses its pull and the universe continuously expands at a constant velocity for ever losing all sources of heat.

·    Big Rip

When gravity is overcome by dark energy by so much that it also overcome the nuclear and electromagnetic forces that hold together atoms ripping all matter in the universe.

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