A little while ago I read Leonard Susskind’s book The Black Hole War (subtitle: My Battle with Stephen Hawking to make the World Safe For Quantum Mechanics). It’s an interesting and mostly quite readable popular science book about the black hole information paradox. Susskind thinks that information isn’t destroyed when stuff falls into a black hole, and his book is about why.
The first part of the book has some useful thought experiments about black holes, some of which I’ll take you through below. After that it starts to talk about string theory, whereupon it becomes as utterly incomprehensible as any other book on the subject.
However, I think Susskind makes an important logical error just before he turns to string theory. I think that if you correct this error then it leads to a much more elegant resolution of the information paradox — one that doesn’t require the use of string theory. I won’t get as far as talking about that in this post, but I will point out the error I think Susskind makes, and show how resolving it leads to a simpler explanation of what happens when something passes an event horizon.
Just a quick note on my background before we get started. I’m more of a theoretical biologist than a physicist. I do have a good understanding of basic quantum mechanics and special relativity (the links point to Susskind’s excellent online lectures on the subjects) but I don’t have a mathematical understanding of quantum field theory or general relativity — that’s why this is a blog post and not a paper.
Susskind’s first thought experiment is an analogy, which gets across the point that an event horizon is not really a special part of space-time. We’re often given the impression that the event horizon is like a black wall that can be crossed in one direction but not the other, but it isn’t, and this analogy illustrates this rather well. Susskind asks us to imagine a very large, very shallow fish tank full of water. The tank is populated by blind fish who can only communicate using sound, and can’t swim faster than the speed of sound. In one place there is a hole in the bottom of the tank, and the water is constantly flowing out of the hole. In fact it’s flowing so fast that near to the hole it’s flowing faster than the speed of sound. However, as you get further away the water moves slower and slower until when you’re far enough away it’s not really moving at all.
Next we imagine a fish, Bob, swimming towards this hole. His friend, Alice, stays safely far away from the hole, where the water isn’t really moving. There is an invisible line — the point of no return — beyond which it would be unwise for Bob to swim. If he does so, he can not only never escape but can also send no signal to Alice. If he crosses this line, Alice will see his sound signals gradually slow down due to the Doppler effect, until eventually they slow by an infinite amount and Bob appears frozen at the moment he passed the invisible line. This is very similar to the effects that we would experience when watching someone pass the event horizon of a black hole.
But from Bob’s point of view there’s nothing special about the horizon he’s just passed — it’s just a piece of water like any other. He won’t even feel the motion particularly, as he’s moving with the water. He’ll notice that Alice is moving rapidly away, but he’ll still be able to hear her.
Event horizons are the same. From the point of view of someone far from a black hole, an event horizon is a very special part of space-time, but from the point of view of someone falling in to the black hole, there’s nothing special about it at all. In this respect the word ‘horizon’ is an apt name.
A little later, Susskind gets on to actual black holes, and asks us to consider lowering a thermometer towards one on an impossibly strong rope. As the thermometer approaches the black hole it will begin to register a higher temperature. As it gets close to the event horizon it will converge to the temperature of the Hawking radiation, which is a function of the black hole’s mass.
But here’s the thing, says Susskind: from the point of view of someone falling in to a black hole, the event horizon is just a piece of space-time like any other. Like Bob approaching the hole in the fish tank, someone falling into a black hole should notice nothing special as they approach the event horizon, and that includes any increase in temperature. Susskind sees an inconsistency here: our thermometer, dangling on its rope, registers one temperature whereas the one that falls into the black hole registers another.
It’s around this point that Susskind makes what I consider to be a mistake. He assumes that if we were to simply drop a thermometer into a black hole and watch it fall in, we would see it register the same temperature as the one dangling on the rope. But, he says, someone falling with the thermometer (let’s call him Fred) would see it register a different temperature. This is a big contradiction: two people observing the same object see different things.
Susskind thinks the resolution of this paradox must lie in the fact that once Fred has passed the horizon there’s no way we can compare notes about it. Thus, he says, both of these contradictory chains of events can happen (in a multiple universes kind of a way) but it doesn’t cause a paradox because the two timelines never interact. This is an extraordinary claim, but Susskind believes there is extraordinary evidence for it.
But, logically, this resolution to the paradox cannot work. Fred’s thermometer registers (or doesn’t register) the higher temperature before he crosses the event horizon. It’s theoretically possible for him to escape in the split-second before he crosses it. He’d need a carefully timed rocket boost, because he can’t see the horizon coming. Alternatively, we could send him in attached to a slack rope and then pull him back out once we’ve observed the reading on his thermometer. If we were to perform such an experiment, it has to be the case that the reading we observe is the same as the one Fred does.
So if Susskind’s resolution of the paradox doesn’t work, how should it be resolved? I think the answer has to do with something called the Unruh effect. This is a result in quantum field theory that, for reasons I don’t understand, predicts that an accelerating observer should experience an increase in temperature. I can’t remember if Susskind talks about the Unruh effect in his book, but he doesn’t make the points that follow.
Wikipedia tells me that the Unruh effect temperature has the same form as the Hawking radiation temperature. The Unruh temperature is given by (where =acceleration, =Planck’s constant, =speed of light, =Boltzmann’s constant), whereas the Hawking temperature is , where is the magnitude of the gravitational field at the event horizon of a black hole.
But now let’s re-consider our thermometer, dangling on its rope just above the event horizon. There is tension in that rope, a lot of tension, as it has to balance the huge gravitational force acting on the thermometer. Let’s imagine we’re there, dangling with the thermometer on the end of the rope. Dangling on the end of a rope in a gravitational field is very much like being accelerated by a rope in empty space — in fact, by Einstein’s equivalence principle it’s exactly the same (as long as we can neglect tidal forces). So, dangling on the end of the rope with our thermometer, we experience an acceleration of away from the event horizon, due to the pull of the rope. Because of this we experience a temperature due to the Unruh effect of Kelvins.
This is the exact same temperature that the thermometer is predicted to measure due to Hawking radiation. This is neat, but not necessarily particularly surprising. However, it does have an important implication: the temperature measured by the thermometer on the end of the rope can be entirely explained by the pull of the rope. If (having once again retreated to a safe distance) we cut the rope and let our thermometer freefall towards the event horizon, it will no longer be experiencing this pull and hence (I claim) it will no longer register the increased temperature.
But now we’ve solved the paradox. An thermometer falling towards a black hole doesn’t register an increase in temperature, regardless of the speed or acceleration of the person observing it. A thermometer dangling on a rope near the horizon does experience the Hawking temperature, but only because of the acceleration caused by the rope.
In a follow-up post I will make some (slightly) less well-founded claims about how the black hole information paradox can be solved based on these ideas. (Spoiler: information never gets destroyed because nothing ever actually passes the event horizon.)
 edit: it turns out that this trick with the rope won’t work, because the influence of our tug on the rope can only travel at the speed of light, and by the time it reaches Fred (in his frame of reference) he will already be past the event horizon, making it impossible for the rope to pull him out. Therefore general relativity does not permit the existence of unbreakable ropes. But a carefully timed rocket boost could still get him out just before he passes the event horizon.