What is the fundamental reason why information is on the event horizon of a black hole?

Question

I don't know the mathematics behind black holes and information theory but is there a simple explanation of why information is on the event horizon. Why can't it be elsewhere?

Answer 1

I'm going to answer the question you actually asked, but as you'll see, this is probably not quite the question you really wanted.

A possibly intuitive argument for it is information conservation. The basic idea behind this is Laplace's idea that if you knew the state of the universe perfectly at one instant of time, you could predict its entire future and deduce its entire past from that slice by applying the laws of motion. All the information about the entire 4D history of the universe is encoded in every 3D instant-of-time slice.

In general relativity, the slices can curve and bend, but the same general principle applies. If we take a surface that intersects (cuts) every worldline running from the beginning of the universe to the end exactly once, then the entire past and future of the universe can be determined from its state on that slice. This is called a Cauchy surface.

And if we have a region of spacetime surrounded by a surface such that every worldline passing through that region has to cross the surface, then what happens on the surface determines everything that happens inside the region. This is close to being the defining property of the event horizon of a black hole. All the information about what happens inside the black hole has to pass through the event horizon, and all the information that passes through the event horizon ends up in the black hole. If we know the state of the universe on the event horizon, we can use Laplace's 'Clockwork Universe' principle and the laws of physics to deduce everything that happens subsequently, inside the black hole. So all the information in the interior of the black hole is on the surface - or more precisely, passes through the surface.

That is certainly not the end of the story. The main reason physicists talk about information being 'on' the event horizon is that the Bekenstein-Hawking formula for the entropy of a black hole says it is proportional to the event horizon area. Laplace's idea tells us that all the information passes through the horizon, but gives no reason to expect it to do so uniformly. It is not clear why the information should be proportional to area. And of course we can consider concentric spheres inside the event horizon which also have the property that every worldline must pass through them, and must therefore also contain all the information, but which have a smaller surface area. What is special about the outermost of them; the event horizon? (There are special properties known, e.g. it's a null surface, but not reasons why that specialness should lead to this result.) I have not seen anyone give a clear and convincing answer, that doesn't amount to numerology (i.e. speculative deductions of deeper causes made from observing numerical coincidences). This is one of the current mysteries in physics.

There is a nice, fairly basic discussion of Cauchy surfaces and the implications for black holes and the information paradox in philosopher Tim Maudlin's (Information) Paradox Lost. Because he deliberately takes a provocative and controversial stance (which is good for science, of course), it's appropriate to point to replies, like (Information) Paradox Regained? A Brief Comment on Maudlin on Black Hole Information Loss.

In summary - all the information about the interior of a black hole is deducible (by Laplace's 'Clockwork Universe' principle) from the state of the universe on the event horizon. The information is certainly there. But when physicists say this, they're talking about something different, which is that in a lot of calculations the information inside a closed region like a black hole or an entire universe is proportional to the area of the boundary. So far as I know, the reason for this is not currently understood.