An Indian mathematician's genius formula from a century ago might explain the dark secrets of black holes

In 1914, Indian mathematician Srinivasa Ramanujan arrived at Cambridge with a notebook containing 17 extraordinary infinite series for 1/π. They were not only efficient but also yielded accurate digits of the world's most famous irrational number. Yet, despite these formulas long being considered the pinnacle of number theory, no one could actually explain for a century why they worked so perfectly.

But researchers now at the Indian Institute of Science have found an unlikely bridge between Ramanujan's celebrated formulas for pi and the cutting-edge physics behind black holes and turbulent fluids. The research implies that Ramanujan was unwittingly working on the very mathematics that describes how matter behaves on the verge of extreme change.

Ramanujan's spectacular formulas for π

Before taking off for Cambridge from Madras in the year of 1914, Indian mathematician Srinivasa Ramanujan published one paper listing 17 formulas that were used in the calculation of pi. The formulas proved conspicuous because they were far more efficient compared to the methods at the time. Surprisingly, using only a few terms in mathematics, it could generate a colossal amount of correct digits of pi.

More than a century later, their influence remains strong. Ramanujan's ideas constitute the bedrock of modern techniques that are used to compute pi on powerful computers today. "Scientists have computed pi up to 200 trillion digits using an algorithm called the Chudnovsky algorithm," says Aninda Sinha, professor at the Centre for High Energy Physics (CHEP) and senior author of the study. "These algorithms are actually based upon Ramanujan's work."

A deeper question behind the mathematics

For Sinha and Faizan Bhat, the study's first author and a former PhD student at the Indian Institute of Science (IISc), the interest was not just in how fast these formulae work. They wanted to understand why such powerful formulas exist at all. Instead of looking at them as purely abstract mathematics, the researchers sought a connection with the physical world.

“We wanted to see whether the starting point of his formulas fit naturally into some physics,” says Sinha. “In other words, is there a physical world where Ramanujan’s mathematics appears on its own?”

Where pi meets scale invariance and extreme physics

Their study brought them to a class of theories called conformal field theories and, particularly to logarithmic conformal field theories. These are theories describing systems with scale invariance, meaning that the system is the same no matter at what scale one observes it, whether zooming in or out.

A well-known example is water at its critical point, a specific temperature and pressure where liquid water and water vapour become indistinguishable. At this point, water shows scale-invariant behaviour which can be described using conformal field theory. Similar behaviour appears in processes such as percolation, the early stages of turbulence in fluids, and in certain theoretical descriptions of black holes. These are all areas where logarithmic conformal field theories are used.

Using Ramanujan’s structure to solve physics problems The researchers have now found that the same mathematical structure underlying Ramanujan's pi formulas also crops up in the equations of these logarithmic conformal field theories: by exploiting this shared structure, they could compute key quantities in these theories more easily. It may help scientists study better some real complex phenomena, like turbulence and percolation. The approach here mirrors closely enough Ramanujan's own style, wherein compact mathematical expressions lead quckly to precise results. "[In] any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics, says Bhat. "Ramanujan's motivation might have been very mathematical, but without his knowledge, he was also studying black holes, turbulence, percolation, all sorts of things." A century-old insight of current significance "The study shows that Ramanujan's work, completed over a century ago, still provides new tools for making modern high-energy physics calculations faster and easier. Beyond the technical benefits, the researchers say the findings underline the extraordinary depth of his ideas. "We were simply mesmerised with the fact that a genius who worked in early 20th century India, completely cut off from all contact with modern physics, could actually have anticipated structures which are now at the core of our knowledge concerning the universe", says Sinha.