## Falling Into a Black Hole, Part 1

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[1].  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 $\hbar a/2\pi ck$ (where $a$=acceleration, $\hbar$=Planck’s constant, $c$=speed of light, $k$=Boltzmann’s constant), whereas the Hawking temperature is $\hbar g/2\pi ck$, where $g$ 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 $g$ 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 $\hbar g/2\pi ck$ 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.)

N

[1] 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.

### 5 Comments to “Falling Into a Black Hole, Part 1”

1. It is my understanding that Hawking radiation is special as it requires an event horizon rather than just the mass. Would this view not be equally applied on all massive bodies? If this were true though then it makes it relatively easy to remove energy permanently from the vacuum and would seem to integrate over all space to be a relatively large (measurable?) amount to energy. Any thoughts?

2. Hi Jos

I’ve been thinking about this since Lucas forwarded me your email about it. I haven’t come to any firm conclusions. There seem to be two distinct possibilities, assuming my argument above is correct (which of course it might not be). I’m not enough of an expert to know which is the more plausible, so I’ll explain them both.

The first is that all massive bodies do indeed emit Hawking radiation. The argument would be that for an observer on the surface of something the size of Earth, their motion is indistinguishable from uniform acceleration (tidal forces notwithstanding), and so they should experience Unruh radiation with a temperature of about $4\times 10^{-20}$K. (That’s obviously an unimaginably tiny temperature, so let’s imagine this is in the distant, distant future, when the expansion of the Universe has driven the cosmic microwave background well below that.) This energy would heat the surface of the body, so it would then be radiated away.

This energy would have to come from somewhere, though. I don’t think it can come permanently from the vacuum, because that would break the first law of thermodynamics. It can’t come from the thermal motion of the particles that make up the body, because that could be at a lower temperature, breaking the second law – so the only place it can really come from is the mass of the body itself. This would mean that all massive bodies, not just black holes, must eventually evaporate into thermal photons.

If all massive objects do emit thermal radiation in this way, perhaps it could be connected to the notion of gravity as an entropic force which has been getting people excited recently but which I haven’t quite managed to get my head around.

However, the second possibility is that Hawking/Unruh radiation does require an event horizon, and hence non-black-hole massive bodies don’t emit it. This can be made consistent with my argument as follows: For a uniformly accelerating observer in flat space-time there is an event horizon behind them, because an object starting from far enough away cannot catch up with the accelerating observer no matter how fast it travels – so even photons can’t catch up. Now the wikipedia article on the Unruh effect sort-of implies that the accelerating observer will experience the effect from all directions, as if the vacuum itself was at a higher temperature. But it could be that this is wrong, and that the accelerating observer would actually experience these photons as coming from the event horizon behind them. (Actually, the Wikipedia page on Hawking radiation does say this is the case: “An accelerating observer sees a thermal bath of particles that pop out of the local acceleration horizon, turn around, and free-fall back in.”)

If this is the case then the temperature registered by a thermometer held on a rope near a black hole’s event horizon would still be consistent with the Unruh temperature caused by the rope’s acceleration, but in this case both theories would predict that this thermal radiation appears to come from the event horizon rather than from all around.

From an observer on the Earth there isn’t an event horizon. If you extrapolated your local motion while assuming that space-time was flat, you’d deduce that there was one far below your feet, but it would be very distant (left as an exercise: work out how distant) and hidden by the bulk of the Earth. The curvature of space-time in this case means that this horizon doesn’t actually exist: anything in the observable universe can reach you if it travels fast enough. On this view, since there’s no horizon, there’s nowhere for the Unruh/Hawking photons to be coming from, and hence you won’t experience them.

After writing this, I think the second of these possibilities seems the most plausible. Any thoughts?

3. I think that the first idea can’t work as you would have to extract the energy from a discrete amount of mass which cannot change without changing its form. Also conservation of momentum might be a problem eventually.
As for the second, I’m not sure that Hawking radiation is the same as Unruh (though saying that it may be). Also I’m not sure if it is relevant to the conversation but beyond the event horizon space and time flip around so for the external point of view the event horizon IS a special point in space even if it can’t be seen as such from someone moving through it. This may play a role in the production of Hawking radiation though again I’m not sure. As for the earth’s event horizon, it will occur at the earth’s radius minus about half a centimeter from the surface :)

As an aside, if considered as a purely energy based problem the actually event horizon (the point at which no return is possible) will be half the Schwarzchild radius. Bonus points if you can work out why

Jos

4. I think your comment doesn’t completely rule out the first option, as it could be indicative of new physics that means the matter’s form does indeed change. E.g. the particles could decay. But I agree that’s less parsimonious than the other option.

I agree that the event horizon is special from the point of view of an external observer. I don’t know about this coordinate flipping thing but would say the most special thing about the event horizon is that an external observer can never see anything move past it, in either direction. (I have a forthcoming post on this.)

I was puzzled by your answer to the location of the event horizon, until I realised that you mean the location of the event horizon that would form if the Earth were compressed into a black hole. I actually meant the location of the Rindler horizon in a space-time that’s linearly extrapolated from the local space-time of an object on the Earth’s surface. I’ve just worked out that this is given by $c^2/g$, about 0.97 light years away at its closest point. That’s where the horizon would be if you were accelerating at $g$ rather than sitting on the surface of a massive object.

Hmm, I’m not sure what you mean by “considered as a purely energy problem” – I think I’d need further clarification in order to collect those bonus points.

5. Consider that to escape a gravitational body you don’t ever have to reach the escape velocity, simply maintain a constant speed away from it. Thus the difference between escaping by an instantaneous impulse to a speed which decreases to zero at infinity and a constantly accelerating body which is accelerating at a rate slightly greater than the gravitational acceleration at every point. A body m will have a maximum of mc$latex^2$ energy, this can be (theoretically) converted to maintain this acceleration. You therefore run out of rest mas at infinity from a point which is half the Schwarzchild radius.
There may be other things stopping you doing this but it initially amused me thinking this way.