Black Holes: The Key to Understanding the Universe

11

Spaghettified and Vaporised

‘In microphysics, however, the information does not sit out there. Instead, Nature in the small confronts us with a revolutionary pistol, “No question, no answer.” Complementarity rules.’

John Archibald Wheeler34

The ‘black holes are a bit like atoms in the way they emit radiation as a consequence of being tickled by the vacuum’ picture is a good one, but it disguises a major difference between the emission of light from everyday hot objects and the emission of Hawking radiation. The difference can be traced to the fact that it is gravitational effects that make the vacuum fluctuations real. This unique production mechanism gives rise to three properties of Hawking radiation that, taken together, appear nothing short of bamboozling.

Someone falling freely near to the horizon of a large black hole will not encounter any radiation.

Someone accelerating, so that they hover just above the horizon of a large black hole, will be vaporised by a flux of very hot radiation.

Someone far away from the black hole will experience a flux of cool radiation which appears to have been emitted by a glowing object at the Hawking temperature.

Let’s deal with each of these in turn.

That someone falling freely near the horizon of the large* black hole should not encounter any radiation is not hard to understand. This is Einstein’s Equivalence Principle. A freely falling observer feels as if they are at rest in plain old flat spacetime. From their perspective therefore, they experience the vacuum fluctuations (those particle–antiparticle pairs) in the same benign way as someone floating far away from the black hole. As a result, they float onwards unawares until, inexorably, they are spaghettified as they approach the singularity.

In contrast, someone hovering just above the horizon will encounter the positive energy part of the vacuum fluctuations. From their perspective they will be bombarded by a flux of real particles, separated from their partners by the geometry of spacetime. A very similar effect occurs even in flat spacetime, as was appreciated by Paul Davies and Bill Unruh in the mid-1970s.†35 In the Davies–Unruh effect, an accelerating rocket ship far from any gravitating objects will experience a thermal bath of particles and the temperature of the bath is proportional to the acceleration. The Equivalence Principle can be invoked to say that the same thing will happen to a rocket that accelerates to maintain a fixed position close to the horizon of a black hole. In that case, because spacetime in the immediate vicinity of the rocket is approximately flat, the rocket’s experience is just like that of a rocket accelerating in flat spacetime. And therefore, as a result of being immersed in a hot bath of particles, the rocket heats up. If it is close enough to the horizon, it will be vaporised.

A freely falling observer far from the hole will detect Hawking radiation, but the particles they receive will be benign, with energies characteristic of an object radiating at the Hawking temperature. This is as expected – it is what Hawking predicted. Another way to understand how Hawking radiation arises from this far away point of view is to think about the tidal gravity of the black hole. Tides on Earth are raised because the gravitational pull of the Moon varies across the Earth. The result is that the ocean surface, and to a very small extent the Earth itself, is distorted by the varying gravitational field. On Earth, tidal gravitational effects are only felt over very long distances – at widely separated points on the Earth’s surface – because they are caused by the difference in the gravitational pull of the Moon in two places. There is no measurable difference in gravitational pull over a couple of metres, which is why the Moon does not raise tides in your bath. Hawking radiation arises because the gravitational pull of the black hole varies across a vacuum fluctuation. For this to be a large enough effect to make the particles real, the vacuum fluctuation must be separated by approximately the size of the black hole. From the distant vantage point, an observer therefore sees a flux of long-wavelength (low-energy) particles.

Each of these three experiences are legitimate descriptions from different viewpoints, yet at first sight they appear to be mutually contradictory. Let us make the point even more starkly. Suppose you jump into a large black hole. From your perspective you are doomed to spaghettification as you approach the singularity, but you will cross the event horizon with no drama. Your friend in a rocket ship outside of the hole will never see you cross the horizon, but they will see you get closer and closer. That much we know from general relativity. They might decide to lower a thermometer down to measure the temperature at your position close to the horizon. The thermometer, hovering above the horizon, will experience the vacuum fluctuations as in case 2 above and will therefore be immersed in a hot bath of real radiation. From this vantage point, the near-horizon region is a scorching hot place. Your friend may conclude that you got burnt to a crisp outside the black hole and never made it through the horizon.

It seems there is little way out here. Should we abandon the Equivalence Principle, the very foundation of general relativity, and conclude that the horizon is a hot and dangerous place? Or should we insist that a freely falling observer must cross the horizon with no drama and conclude that there is something wrong with our quantum physics analysis of the vacuum? If we take one or other of these positions, we are required to ditch a core element of either general relativity or quantum theory.

There is a third way. A beautiful expression of centrism. It is possible that both perspectives are correct. From the outside perspective, the black hole has a scorching hot, impenetrable atmosphere which vaporises everything that approaches. Yet, according to in-fallers, the horizon is a complete non-entity and they pass through unharmed into the interior. A person can be both spaghettified and vaporised: Spaghettified from their own perspective and vaporised according to outsiders. This idea is known as ‘black hole complementarity’.36

From the outside, nothing is ever observed to fall into a black hole. We might say that the interior lies beyond the end of time for someone lurking outside. Stuff just falls into a hot atmosphere where it gets burnt up. From the outsiders’ point of view, it seems that a black hole is not so different to a hot, glowing coal.

According to the black hole complementarity paradigm, none of this is contradicted by the narrative of someone who falls into the hole, although the story they tell is different. They get to explore the interior and meet with the singularity. Crucially, however, they are not able to communicate any of this to the outsiders once they have crossed the horizon. Equally crucially, the outsiders are unable to inform the in-faller that they have been burnt up. A contradiction is avoided because the outsiders and in-fallers never get together to compare notes. This sounds like a proper bodge, to use our native vernacular, but it is a serious proposition.

You might immediately raise the following objection. What if the outsider collects some of the ashes of the in-faller and then jumps in after their friend to show them the evidence that they were incinerated? Being confronted with one’s own ashes would be a disconcerting experience, even for the most ardent advocate of black hole complementarity. The way out of this logical impossibility, according to complementarity, is that it takes time for the outsider to collect the evidence that the in-faller has burnt up. By the time the outsider has gathered the evidence and jumped in across the horizon, their friend has already been spaghettified and cannot therefore be presented with the evidence of their own demise. However bizarre it may seem, the picture we’ve just outlined appears to be essentially correct. The route to this realisation can be traced back to the 1980s and a very simple question.

The information paradox

The title of Stephen Hawking’s initial paper on Hawking radiation was ‘Black Hole Explosions?’ The reason for this eye-catching title is that the temperature of the black hole is predicted to rise as it shrinks, causing it to radiate more fiercely and shrink faster until it completely disappears in a flash of radiation.‡ ‘Faster’ is perhaps the wrong word to use, introducing an unwarranted sense of urgency to the process. You can pop a few numbers into Hawking’s formula to calculate the temperature of a typical stellar mass black hole – let’s say around five times the mass of the Sun. Doing so reveals that the temperature is ten billionths of a degree above absolute zero. That’s very much colder than the Universe today, which has cooled down to around 2.7 degrees above absolute zero in the 13.8 billion years since the Big Bang. At this moment in cosmic time, black holes are more like Wheeler’s iced teacups and, in accord with the Second Law of Thermodynamics, they are absorbing energy as they float in the relatively hot bath of the cosmos. There will come a time, however, as the Universe continues to expand and cool, when they become glowing hot spots in the ever-chilling sky and then they will begin to evaporate. A typical solar mass black hole will have a lifetime of approximately 1069 years, which is a very long time. At first sight, therefore, it might appear that we don’t need to worry about black hole explosions; their lifetimes are surely as close to infinite as makes no odds. If we stopped here, however, we would miss what is quite possibly the most important revolution in theoretical physics since Einstein’s time. The revolution was triggered by the following question: ‘Do black holes destroy information?’

Imagine that a book falls into a black hole. Over incomprehensible time scales, the black hole will gradually evaporate away as it emits Hawking radiation until it vanishes in a final burst of radiation. All that remains will be the Hawking radiation. Crucially, Hawking’s calculation makes a definite prediction about the nature of this radiation: it is thermal, which means that the radiation encodes no information at all. In other words, when the black hole has vanished, it is as if the book never existed. The information it contained has been erased from the Universe. In fact, all the information about everything that ever fell into the black hole, including the details of the collapsing star out of which it originally formed, will also be erased. Instead, all that remains is a bath of featureless, thermal radiation.

So what? In his book The Black Hole War, Leonard Susskind describes the moment in a small seminar room in an attic in San Francisco in 1983 when Hawking first made the claim that information is destroyed by black holes.37 The soon-to-be Nobel Prize winner Gerard ’t Hooft ‘stood glaring’ at the blackboard for an hour after Hawking’s talk. ‘I can still see the intense frown on Gerard’s face and the amused smile on Stephen’s.’ ’t Hooft was glaring because the laws of physics as we currently understand them preserve information. If we know the precise state of something at a given moment in time then, in principle, we can predict precisely what it will do in the future and know what it was doing in the past. This is determinism, the fundamental idea that the Universe evolves in a predictable way. All the known laws of physics deliver deterministic evolution. They take a system, be it a box of gas or star or galaxy, and evolve it uniquely into a single, well-defined configuration at some time in the future. And because things evolve in a unique and predictable way, the laws also allow us to calculate precisely what the system was like at any time in the past.§ But what of a Universe that contains black holes? Galaxies contain supermassive black holes at their cores, and those black holes swallow things. If the black hole subsequently disappears in a puff of information-free Hawking radiation, then it would be impossible in the far future to reconstruct any details about anything that had fallen in. It would in fact be impossible to deduce that the black hole ever existed at all, because it would have erased all trace of itself. This problem has become known as the black hole information paradox.

Figure 11.1 shows a Penrose diagram that illustrates the problem. It is obtained by sewing together two spacetime geometries: the Schwarzschild spacetime corresponding to the period that the black hole exists (Figure 8.3) and flat (Minkowski) spacetime when the black hole has gone. The singularity disappears with the black hole and the wavy line representing the end of time therefore ends before future timelike infinity at the upper tip of the diagram. We have no idea what happens at the rightmost point of the singularity, since this is the event where the black hole disappears and quantum gravity effects are important. But as we’ll see, and contrary to the expectations of many experts in the field until very recently, it turns out that we don’t need to know to solve the information paradox.

The shaded region of the diagram corresponds to times after the evaporation of the black hole, where spacetime is flat and there appears to be no trace of anything that crossed the event horizon. To see this, draw a light ray (a 45-degree line) from any point inside the event horizon and you’ll see that it ends on the singularity. The region behind the horizon remains a prison from which there is no escape because it is causally disconnected from the Universe outside; there is no spacetime directly above the singularity in Figure 11.1. The only thing that survives into the future when the black hole has disappeared is Hawking radiation. We’ve drawn a wiggly arrow to indicate the worldline of the last Hawking particle to be emitted as the black hole vanishes – it heads off happily to future lightlike infinity.

Figure 11.1. Penrose diagram for an evaporating black hole. The singularity disappears after the last Hawking particle (orange). The blue line is the worldline of a book thrown into the hole and in red is another Hawking particle. Both Hawking particles follow their respective dotted worldlines and end up at future lightlike infinity.

According to Hawking, the radiation is featureless and the black hole therefore erases all trace of everything that fell into it. Where did everything go? If the hole did not evaporate, we could at least offer the vague statement that ‘it fell into the singularity and we don’t really understand that place’. But after the hole evaporates, there isn’t any singularity anymore – there is no hiding place.

Black hole complementarity offers an apparently simple resolution to the paradox, at least from the point of view of an outside observer. Since nothing is ever seen to fall through the horizon, nothing is ever lost. Let’s think again about the book we tossed into the black hole, we have drawn its worldline in blue on the figure. From the perspective of the outside observer, according to complementarity, the book is incinerated on the horizon and its ashes are returned to the Universe as Hawking radiation (as illustrated by the red wiggly arrow). This is no different to burning a book on a campfire. If we burn a book, the information contained within can in principle be recovered if we make precise enough measurements of all the ashes and gases and embers that emerge. In practice, this is not possible, but practicality is not a word that concerns theoretical physicists. The point is that it is possible in principle. The information contained in the book got scrambled up during the burning process, but it wasn’t destroyed. We would claim that nothing has disappeared, it has just been converted from words on a page into particles in space. It’s clear from the Penrose diagram that as long as the book burns up before it crosses the horizon, it is possible to draw worldlines that link every atom in the book to future timelike or lightlike infinity.

Contrast this with the view from inside the black hole. The book is spaghettified as it approaches the singularity. Though we don’t know what happens at the singularity, the fact that it lies behind the horizon implies that the book is not able to get back out again. It is destroyed at the end of time inside the horizon. But that is OK because from the external viewpoint information is preserved. The stories in the book are written in the Hawking radiation and will always be there, in principle, for the super-beings of the future to read. If we accept complementarity, both the internal and external viewpoints are true.

What are the consequences for physical reality? We tend to think, based on our experience of the world, that large physical objects like books or astronauts can only be in one place at once and only a single fate can befall them. Quantum theory destroys this picture when we ask questions about the behaviour of sub-atomic particles, and complementarity appears to be an even more radical challenge to our intuition. It asks us to accept that there are two equally valid views of what happens to a big thing like an astronaut – you, for instance – freely falling towards a black hole. You are both spaghettified (inside) and vaporised (outside). This suggests that the relationship between the inside of a black hole and the outside is not the same as the relationship between ‘here’ and ‘over there’ according to our everyday experience. There is also a small but persistent fly in the ointment: Stephen Hawking’s calculation, which says that the Hawking radiation contains no information. In the history of the subject, attacking this sharp, well-defined problem in Hawking’s calculation turned out to be extremely fruitful in the quest to place complementarity on a rigorous footing, because the question is simple and easy to state: if the information is to come out, where did Stephen Hawking go wrong?

* Choosing a large black hole means that tidal effects are small at the horizon.

† Often called the Fulling–Davies–Unruh effect to acknowledge the earlier work in 1973 by Stephen Fulling.

‡ You can see this directly from Hawking’s formula – the temperature of a black hole is inversely proportional to its mass.

§ This is also true for the quantum evolution of the state of a system, even though the outcomes of individual experiments are not determined.