002 : A Brief History of the Black Hole

A Curious Solution

Shortly after Einstein’s General theory of relativity was published, a curiosity was found by Karl Schwarzschild. A stable solution was found that seemed to describe a most peculiar object (one that was briefly thought about in the late eighteenth century). It was an object that warped the fabric of space and time in such a way that nothing could escape. Thus the “black hole” as we know it was conceived. This is the classic idea of a black hole; an object of fixed size, with a huge gravitational well. At its centre, the singularity, a point of infinite density, which itself is surrounded by the event horizon; the boundary of no return. We'll call this type of black hole a classical black hole.

While at the time thought too extreme to exist, the idea persisted, and many years later astronomers noticed that the motions of certain clusters of stars at the centre of galaxies were moving with incredible speeds around a seemingly invisible centre! That is, there was some massive object which wasn’t emitting any light, yet interacting with these nearby stars via gravitation. They determined that the mass required to cause the stars to orbit as they were was vastly more than even the most massive stars. This was a serious candidate for a black hole.

With more and more observations like this, the idea of black holes existing in our universe was becoming generally accepted by the scientific community. Few concepts have captured the imagination quite like these fantastic objects with such little observational facts. There is somewhat limited scientific evidence of these objects, nevertheless much research has been done on the mathematical objects which has modified our version of the classical black hole many times and in many ways.

Taking Quantum Leaps

With the advent of quantum field theory, certain aspects of the black hole needed thinking about. One surprising (almost paradoxical) result is that a black hole, an object where nothing can escape, should in fact radiate! This amazing feat of thought was developed by the late great Professor Stephen Hawking. The father of popularising these ideas and making them accessible to a general audience via his many best-selling popular science books.

To understand his revolutionary contribution, a little quantum field theory (QFT) is necessary. In QFT, the most fundamental physical entities are not point particles; they are fields. In this framework, the universe is filled with these quantum fields which interact with each other. Excitations of these fields can be represented as the familiar point particle theory. There is a specific aspect of quantum theory that is needed here, that is the notion of particle creation and annihilation. The hypothesis goes that particles can flit in and out of existence subject to certain conditions. One is that the particles that are created must be a pair such that one is a particle and the other is its antiparticle. They get created, live a short life, and annihilate with one another.

It is worth noting by the way that the marriage of general relativity and quantum field theory is the holy grail of theoretical physics. There have been many attempts; string theory, loop quantum gravity, to name but two. The difficulty arises from the fundamental differences between the two frameworks: In general relativity, spacetime is assumed to be a continuous manifold, whereas in QFT, the more we zoom in the more granular the universe appears to be, to the point that right at the lowest level it is hypothesised to be a quantum frenzy of madness. In this regime nothing of our usual aspects of the universe are familiar. The result of this is that when we incorporate the equations of quantum mechanics into general relativity, the equations “blow up”, that is they give nonsensical answers such as infinite probabilities. Thus, Hawking's attempt at a slight overlap of the quantum realm in a gravitational setting is a spectacular result.

Radiation Revelations

Now that we’re better equipped, let’s see what Hawking achieved. He placed the idea of the black hole in this quantum framework, one where our particle antiparticle pairs live. If we zoom into the boundary of the black hole, the event horizon, we have this sphere of no return, so if we have particles and antiparticles coming in and out of existence right at the boundary, antiparticles can fall into the black hole, while the particle (without it’s partner to annihilate with), can escape to infinity. Seen from far enough away, it would appear that the black hole is in fact radiating! This amazing result was named in Hawking’s honour; Hawking radiation. The temperatures were calculated to be quite less than the balmy 3 kelvin of the cosmic microwave background radiation, so measuring this is currently practically impossible. A consequence of Hawking radiation is that the antiparticle diet of the black hole causes it to shrink in size! So over time it would appear that the black hole evaporates. We have now moved away from the classical black hole to this quantum black hole.

Going back for a moment, the fundamental principle of quantum mechanics is that all the information of a system is embodied by its wave function; which cannot be destroyed. This has relevance to us here when speaking about black holes. Particularly where a black hole has evaporated to nothing. Is this possible? What happens to the wave functions of the matter that fell in through the event horizon? “The information paradox” as it is known has plagued physicists and cosmologists for decades. For thirty years Hawking firmly claimed that information can indeed be destroyed in a black hole, without any convincing opposing theories at demonstrating otherwise. Until in 2004, a potential solution came via the holographic principle. This forced Hawking to reconsider, and concede that black hole evaporation may preserve information. I could (and may) dedicate a whole article on the holographic principle, but the underlying postulate is that the information of an n-dimensional object can be encoded into an n-1-dimensional object. What this means is that the information falling into a black hole could be preserved and encoded on its boundary, the event horizon, thus potentially solving the information paradox of an evaporating black hole.

An Aside on Thermodynamics

When we start talking about “information” in physics, the concept of “entropy” is never far behind. Entropy has a few, equivalent definitions: It arose first in the thermodynamic “heat engines” of the industrial revolution as a measure of the unavailable work in a system. In statistical mechanics, it is related to the number of microscopic configurations that a thermodynamic system can have when in a state as specified by some macroscopic variables. It also comes up in information theory as the disorder of a system.

If we are comfortable with the idea that wave functions falling into black holes have their information encoded onto the boundary, then it makes sense to talk about a black hole’s entropy. “Entropy”, “radiation”, “evaporation” are all thermodynamic quantities, what are they doing in black hole physics? The information paradox itself arises from the second law of thermodynamics not being violated, and there are in fact a set of equations for black hole dynamics that are equivalent to the equations of thermodynamics!

Firewall Problems

It has been suggested by Polchinski et al. that information leaving a black hole during evaporation would cause a "firewall paradox”, that is, the process would produce a huge amount of energy creating a wall of plasma around the event horizon destroying any in-falling matter. This forces a contradiction of the principle of equivalence in general relativity, i.e. that there should be no distinction to an observer falling through the event horizon. A possible reconciliation proposed by Hawking is that of an “apparent horizon”. It is a hypothesized surface that traps light but which can also vary in shape due to quantum fluctuations. This is enough to allow the potential for light (and thus information) to escape.

Sir Roger Penrose previously used general relativity to show the equivalence between event horizons and apparent horizons. This is a good step, as classical phenomena can be distilled from quantum ones as the quantum numbers (usually Planck's constant h) approach zero. In a more recent paper, Penrose suggests that quantum mechanics might reveal the horizons to be different. Removal of the event horizon removes the firewall paradox. This would imply that information is lost. However, Hawking suggests that need not be the case. Hawking proposes that the structure of a black hole just below the horizon is chaotic. That is, the information is practically impossible to interpret, but importantly, not destroyed. There is a lot of skepticism with Hawking’s suggestion. His interestingly titled paper: "Information Preservation and Weather Forecasting for Black Holes", is quite sparse, short, and contains no calculations. Polchinski suggests that Hawking is replacing the firewall with a “chaos-wall”.

In January 2016, Hawking, Perry, and Strominger published a paper attempting to resolve the information paradox in a new way. Their suggestion is that empty space can perhaps carry information. More specifically, they propose that soft particles are at work. These particles can exist in a zero energy state, hence, particles falling into a black hole would leave information behind with them. Hawking et at. suggest a mechanism to do this but they have yet to explain how the information exchange would actually occur. The information transfer mechanism has been called “black hole (soft) hair”. This term they came up with to describe calculations that showed encoding data in quantum descriptions of the event horizon.

Recap & Conclusion

What started off as a curious solution to Einstein’s field equations, has lead us down the path of the truly bizarre: From hypothetical monsters, to convincing observations, to holograms and information paradoxes! Black holes are truly fantastic creatures that continue to capture the imagination of academics and amateurs alike. ▢