An analogy between effective potential representing prime numbers and Schwarzschild black hole

Department of Physics, Siksha ‘O’ Anusandhan, deemed to be University - Bhubaneswar-751030, India

Received: 19 July 2023
Accepted: 2 January 2024

Abstract

The effective two body interaction is derived using the single particle Schrödinger equation for the prime numbers with probability density based on the prime counting function (which is the number of primes below a certain x). Though the speculative connection of primes to Schwarzschild spacetime or black hole has been there for a long time, here for the first time our derived potential serendipitously shows the unique similarity with the effective potential derived from general theory of relativity (GTR) for Schwarzschild spacetime. Also, we report the existence of ISCO and photon sphere in prime potential and compare with GTR. It is found that when the prime numbers are represented by a quantum system, the two-body interactions mimic the effective general relativistic gravitational interaction in black hole with the four-acceleration diverging. We also establish an analogy between Einstein's static universe and the universe of primes. The question is: “Who ordered that?”