December 17, 2012
Century-Old Math Provides New Understanding Of Black Holes
Brett Smith for redOrbit.com — Your Universe Online
To mark what would have been the 125th birthday of the legendary Indian mathematician Srinivasa Ramanujan, an Emory University professor has shown the math icon´s century-old writings might lead to a better understanding of the properties of black holes.
“No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,” said Ken Ono, who led a team of mathematicians in studying the mysteriously derived theorems.
Ramanujan´s work is well-known in academic circles because of its extraordinary numerical patterns that were seemingly devised without the use of proofs or modern mathematical methods. Ramanujan said his findings were a divine gift, revealed to him from the Hindu goddess Namagiri.
“I wanted to do something special, in the spirit of Ramanujan, to mark the anniversary,” Ono said. “It´s fascinating to me to explore his writings and imagine how his brain may have worked. It´s like being a mathematical anthropologist.”
Ramanujan only lived for 32 years and while on his death-bed in 1920, he sent a letter to his mentor, the English mathematician G. H. Hardy. In the letter, Ramanujan described numerous new functions that acted differently from known modular forms, yet closely copied their behavior. Ramanujan posited his mock modular forms correlated to the ordinary modular forms identified by the previous work of Carl Jacobi.
At the time, the mock forms were not understood and probably dismissed as the ramblings of a malnourished, sickly and dying individual.
“It wasn´t until 2002, through the work of Sander Zwegers, that we had a description of the functions that Ramanujan was writing about in 1920,” Ono said.
Building on that more recent work, Ono and his colleagues used modern mathematical tools to show a mock modular form based on Ramanujan´s writings was possible. The mock modular form shoots off into enormous numbers; however, it corresponded to Jacobi´s modular form that expands at close to the same rate.
Ono used the analogy of “magic coins” to describe how Ramanujan´s form acts alongside the ordinary modular form. He equated both Ramanujan´s and Jacobi´s forms to a coin each man spends in a shop. Each coin then travels on its own journey–from the shop, to the bank, to another shop, and so on.
“For months, the paths of the two coins look chaotic, like they aren´t doing anything in unison,” Ono says. “But eventually Ramanujan´s coin starts mocking, or trailing, Jacobi´s coin. After a year, the two coins end up very near one another: In the same town, in the same shop, in the same cash register, about four inches apart.”
The results of the team´s finding could have ramifications for computing the entropy of black holes that do not behave according to modular form models.
“We developed a theorem that shows that the bizarre methodology he used to construct his examples is correct,” Ono says. “For the first time, we can prove that the exotic functions that Ramanujan conjured in his death-bed letter behave exactly as he said they would, in every case.”