4.2 The Fuzzball Theory

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Black hole thermodynamics

The second law of thermodynamics says that entropy, which is the amount of disorder in a system, always increases.
Jacob Bekenstein noticed that while the entropy of a black hole is totally unobservable, black holes increase in size when new material (and their associated entropy) is thrown inside. From this, he reasoned that we can infer the entropy of a black hole by the size of the event horizon surface area. Stephen Hawking later fixed the proportionality constant and derived the formula: $$S_{bh} = \frac{Ac^{3}}{4G \hbar}$$
Not only does this formula provide a measure of a black hole’s entropy, it also preserves the second law of thermodynamics. When an object falls into a black hole its entropy is not lost; the black hole grows in size and therefore its entropy grows too.
This raised another puzzle. According to Ludwig Boltzmann, entropy is also a counting measure – the more possible configurations ($N$) the microconstituents of a system can occupy, the higher its entropy. Boltzmann entropy is calculated by taking the logarithm of the total number of possible configurations: $$S = \ln(N)$$
A black hole of consists of empty space and a singularity, so by definition it has only one state, which corresponds to 0 entropy. Yet Bekenstein and Hawking’s calculations show that black hole entropy is enormous: the numerator has the speed of light cubed which is a huge number, and the denominator has both Newton’s gravitational constant and Planck’s constant, both of which are tiny numbers. There seems to be a troubling contradiction: how can a black hole simultaneously have a huge entropy and no entropy at all?

The surface topography of the fuzzball contains the entropy of the black hole

The fine microstructure of the fuzzball surface, which can take on many possible states, is where the entropy of a black hole exists.
Just like gas particles in a box can move around and change positions, the surface of a fuzzball can twist and curve and change as well. The total number of possible fuzzball surface states reproduces the Bekenstein entropy of a black hole.
The probability of a star fluctuating into any one fuzzball configuration is incredibly low. However, there are so many possible final fuzzball states that their individually low probabilities are cancelled by the massive number of potential outcomes. Therefore, the probability that a collapsing star transforms into a fuzzball is nearly 1.
In the new fuzzball view, black holes are firmly in the domain of quantum theory, their evolution dominated by their enormous entropy.

Fuzzball complementarity

The fuzzball theory has the interesting result of eliminating the interior of the black hole. This suggests that rather than crossing the event horizon, which classically was just empty space, any infalling particle would collide into the surface.
Objects in quantum mechanics exhibit complementarity, that is, a type of duality. For example, depending on the method of observation, they can appear to be waves or particles.
If fuzzballs exhibit complementarity, it is possible that the experience of colliding into the surface could mimic the experience of falling into a classical black hole.
In quantum theory every state is characterized by an energy, which can in turn be related to frequency ($\nu$) by: $E=h\nu$
Every system can therefore be defined by a set of frequencies. If two systems have the same frequencies, they are physically indistinguishable. This situation is called a duality.
It is possible that when an observer collides into a fuzzball, the frequencies of the surface oscillations match the frequencies inside a classical black hole, meaning that colliding into the surface of a fuzzball is in fact indistinguishable from falling into a classical black hole.