Hawking has still to provide a fully worked mathematical proof

His popular science book A Brief History of Time was a publishing sensation, staying at the top of the best-seller lists longer than any other book in recent history.

But behind the public face lies an argument that has been raging for almost 30 years.

Hawking shot to fame in the world of physics when he provided a mathematical proof for the "Big Bang" theory.

This theory showed that the entire Universe exploded from a singularity - an infinitely small point with infinite density and infinite gravity. Hawking was able to come to his proof using mathematical techniques that had been developed by Roger Penrose.

However, Penrose's techniques were developed to deal not with the beginning of the Universe but with black holes.

Science had long predicted that if a sufficiently large star collapsed at the end of its life, all the matter left in the star would be crushed into an infinitely small point with infinite gravity and infinite density - a singularity.

Hawking realised that the Universe was, in effect, a black hole in reverse; instead of matter being crushed into a singularity, the Universe began when a singularity expanded to form everything we see around us today - from stars to planets to people.

Hawking realised that to come to a complete understanding of the Universe, he would have unravel the mysteries of the black hole.

Leaking holes

Hawking and his fellow physicist embarked on an extraordinary intellectual expedition - to tame the black hole. The period from the early 70s to the early 80s became known as the "Golden Age" of black hole research.

Slowly physicists were coming to understand its nature.

But Hawking realised that there was something missing from the picture that was emerging. All work on black holes to that point used the physics of the large-scale Universe.

The physics of gravity - first developed by Newton and then refined by Einstein's general theory of relativity. Hawking realised that to come to a full understanding of black holes, physicists would also have to use the physics of the small-scale Universe; the physics that had been developed to explain the movements of atoms and sub-atomic particles known as quantum mechanics.

The only problem was that no one had ever combined these two areas of physics before. This didn't deter Hawking. He set about developing a new way to force the physics of quantum mechanics to co-exist with Einstein's relativity within the intense gravity of a black hole.

After months of work, Hawking came up with a remarkable result. His equations were showing him that something was coming out of the black hole.

This was supposed to be impossible - the one thing that everyone thought they knew about black holes was that things went in but nothing, not even light itself, could escape.

The more Hawking checked, the more he was convinced he was right. He could see radiation coming out of the black hole. And it led him to the realisation that this radiation (later called Hawking radiation) would cause the black hole to evaporate and eventually disappear.

Shocking revelation

Although Hawking's theories about black hole evaporation were revolutionary, they soon came to be widely accepted. But Hawking felt that this work had far more fundamental consequences.

In 1976 he published a paper in Physical Review D called, "The breakdown of predictability in gravitational collapse". In this paper, Hawking argued that it wasn't just the black hole that disappeared.

He said that all the information about everything that had ever been inside the black hole disappeared, too.

The understanding of black has accelerated in recent years

The reason physicists cling on to the idea that information can't be lost is that it's their link with either the past or the future. If information is lost then science can never know the past or predict the future. There are limits to what science can know.

For many years, no one took much notice of Hawking's ideas until a fateful meeting in San Francisco.

Hawking presented his ideas to some of the world's leading physicists, and in the audience were two particle physicists, Gerard t'Hooft and Leonard Susskind.

They were shocked. They both grasped that Hawking's "breakdown of predictability" applied not only to black holes but to all processes in physics.

The long search

According to Susskind, if Hawking's ideas were correct then it would infect all physics; there would no longer be any direct link between cause and effect. Physics would become impotent.

Since that meeting the "information paradox" has come to be seen as one of the most fundamental and most difficult problems in physics.

Arguments effectively boiled down into two camps. On the one side, Susskind and those who believed that Hawking was wrong and that information could not be lost - and on the other, Hawking and those who believed that physics would have to be re-written to take into account the uncertainty about information that Hawking had uncovered.

For 20 years, arguments raged. No side was willing to admit defeat... until a paper emerged written by a brilliant young Argentinean mathematician known as Juan Maldacena.

This paper claimed to be a rigorous mathematical explanation of what happened to information in black holes - and it showed that information was not lost. Hawking, it seemed, was on the losing side. But Hawking was not convinced.

Hawking set to work with a young research student, Christophe Galfard, to try to pick apart the Maldacena paper.

They thought they could use the same mathematical techniques employed by Maldacena to prove that information was in fact lost. But after two years' work, they still could not prove their thesis.

Then disaster struck, Stephen Hawking was taken ill with pneumonia and rushed to hospital; doctors feared for his life.

Hawking was kept in hospital for over three months. But whilst others fussed over his health, Hawking was thinking. Finally, on what many feared might be his death bed, he thought he'd come across what had eluded him for the past 30 years - a solution to the information paradox.

Bold claims

Once again, Hawking defied doctors' dire predictions and was soon at work, working on a new proof for the information paradox.

Then in July last year, at one of the most prestigious conferences in physics, Hawking made a dramatic announcement.

Hawking presented his new position in a Dublin lecture

Hawking presented the outline of a proof that he hoped would at last solve the problem that he had posed almost 30 years earlier. However, despite the bold claims, some physicists remain unconvinced.

Over a year has passed since the conference and Hawking has still not presented a fully worked mathematical proof to back up his ideas.

But Hawking is a stubborn man. If he is going to change his mind on a belief he held for almost 30 years then it will be with his own proof, in his own time.

In spite of failing health and increasing problems communicating with his colleagues, Hawking is still working on the proof.

If he succeeds in completing a proof that convinces his colleagues, he will not only have solved one of the most difficult problems in physics but he will have produced ground-breaking work at the very end of his career.

That would be a feat that even his hero Einstein could not accomplish.

Horizon: The Hawking Paradox was broadcast on BBC Two at 2100 BST on Thursday, 15 September.