The information paradox is perhaps the most puzzling problem in fundamental theoretical physics today. It was discovered by Stephen Hawking 43 years ago, and until recently has puzzled many.

Starting in 2015, Stephen, Andrew Strominger and I started to wonder if we could understand a way out of this difficulty by questioning the basic assumptions that underlie the difficulties. We published our first paper on the subject in 2016 and have been working hard on this problem ever since.

The most recent work, and perhaps the last paper that Stephen was involved in, has just come out. While we have not solved the information paradox, we hope that we have paved the way, and we are continuing our intensive work in this area.

Physics is really about being able to predict the future given how things are now. For example, if you throw a ball, once you know its initial position and velocity, then you can figure out where it will be in the future. That kind of reasoning is fine for what we call classical physics but for small things, like atoms and electrons, the rules need some modifications, as described by quantum mechanics. In quantum mechanics, instead of describing precise outcomes, one finds that one can only calculate the probabilities for various things to happen. In the case of a ball being thrown, one would not know its precise trajectory, but only the probability that it would be in some particular place given its initial conditions.

What Hawking discovered was that in black hole physics, there seemed to be even greater uncertainty than in quantum mechanics. However, this kind of uncertainty seemed to be completely unacceptable in that it resulted in many of the laws of physics appearing to break down. It would deprive us of the ability to predict anything about the future of a black hole.

That might not have mattered – except that black holes are real physical objects. There are huge black holes at the centres of many galaxies. We know this because observations of the centre of our galaxy show that there is a compact object with a mass of a few million times that of our sun there; such a huge concentration of mass could only be a black hole. Quasars, extremely luminous objects at the centres of very distant galaxies, are powered by matter falling onto black holes. The observatory Ligo has recently discovered ripples in spacetime, gravitational waves, produced by the collision of black holes.

The root of the problem is that it was once thought that black holes were completely described by their mass and their spin. If you threw something into a black hole, once it was inside you would be unable to tell what it was that was thrown in.

These ideas were encapsulated in the phrase “a black hole has no hair”. We can often tell people apart by looking their hair, but black holes seemed to be completely bald. Back in 1974, Stephen discovered that black holes, rather than being perfect absorbers, behave more like what we call “black bodies”. A black body is characterised by a temperature, and all bodies with a temperature produce thermal radiation.

If you go to a doctor, it is quite likely your temperature will be measured by having a device pointed at you. This is an infrared sensor and it measures your temperature by detecting the thermal radiation you produce. A piece of metal heated up in a fire will glow because it produces thermal radiation.

Black holes are no different. They have a temperature and produce thermal radiation. The formula for this temperature, universally known as the Hawking temperature, is inscribed on the memorial to Stephen’s life in Westminster Abbey. Any object that has a temperature also has an entropy. The entropy is a measure of how many different ways an object could be made from its microscopic ingredients and still look the same. So, for a particular piece of red hot metal, it would be the number of ways the atoms that make it up could be arranged so as to look like the lump of metal you were observing. Stephen’s formula for the temperature of a black hole allowed him to find the entropy of a black hole.

The problem then was: how did this entropy arise? Since all black holes appear to be the same, the origin of the entropy was at the centre of the information paradox.

What we have done recently is to discover a gap in the mathematics that led to the idea that black holes are totally bald. In 2016, Stephen, Andy and I found that black holes have an infinite collection of what we call “soft hair”. This discovery allows us to question the idea that black holes lead to a breakdown in the laws of physics.

Stephen kept working with us up to the end of his life, and we have now published a paper that describes our current thoughts on the matter. In this paper, we describe a way of calculating the entropy of black holes. The entropy is basically a quantitative measure of what one knows about a black hole apart from its mass or spin.

While this is not a resolution of the information paradox, we believe it provides some considerable insight into it. Further work is needed but we feel greatly encouraged to continue our research in this area. The information paradox is intimately tied up with our quest to find a theory of gravity that is compatible with quantum mechanics.

Einstein’s general theory of relativity is extremely successful at describing spacetime and gravitation on large scales, but to see how the world works on small scales requires quantum theory. There are spectacularly successful theories of the non-gravitational forces of nature as explained by the “standard model” of particle physics. Such theories have been exhaustively tested and the recent discovery of the Higgs particle at Cern by the Large Hadron Collider is a marvellous confirmation of these ideas.

Yet the incorporation of gravitation into this picture is still something that eludes us. As well as his work on black holes, Stephen was pursuing ideas that he hoped would lead to a unification of gravitation with the other forces of nature in a way that would unite Einstein’s ideas with those of quantum theory. Our work on black holes does indeed shed light on this other puzzle. Sadly, Stephen is no longer with us to share our excitement about the possibility of resolving these issues, which have now been around for half a century.