Problems with fuzzball black hole creation and low energy physics

Question

The fuzzball proposal intrigues me, but I doubt its ability to match up with semi-classical low energy physics. A variation of my question was asked by the awkwardly titled post:"At the instant of fuzzball creation, do all strings instantly pop to its face?". And an answer was provided with a link to an article by Samir D. Mathur called "Spacetime has a ‘thickness’", arXiv:1705.06407 There, Mathur tries to reconcile fuzzballs with the scenario of a shell of matter falling through the smooth horizon of a black hole, where GR tells us nothing special happens.

While the article was relavent, I don't think it answered the original question about black hole creation. Mathur's article already assumes a horizon exists and examines the fate of matter falling towards it. I want to reformulate the question in a way that emphasizes the tension between fuzzballs and low energy physics.

Imagine you're in a spaceship floating in the center of an enormous spherical, homogenous cloud of a very clold, diffuse gas. The total mass of the cloud is about ~$3.9*10^9$ solar masses, and its initially radius is ~$1000 AU$, giving it a density of about ~$0.5 \frac{g}{m^3}$. Slowly, the cloud condenses around you. It eventually reaches a radius of ~$76 AU$, and a density of ~$1.25 \frac{kg}{m^3}$, or the average density of air on Earth.

Despite this mild density of ~$1.25 \frac{kg}{m^3}$, the cloud has collapsed into a black hole. Calculating the Schwarzschild radius

$$ R = \frac{2 G (3.9*10^9 M_☉)}{c^2} ≈ 76 AU$$

At the center of the cloud, you remain oblivious of your future demise, because the local physics around you is mild, and it will remain so for a while. But eventually the density will increase exponentially, and you'll get shredded by the singularity.

That's what would happen according to GR. But according to the fuzzball proposal, there is no interior to the black hole. There's only an extreme configuration of strings and D-branes at the location that is usually called the event horizon, and there's nothing beyond that.

But how would fuzzballs explain the scenario described above? According to GR, at the moment of the formation of the black hole, the local physics around an observer at the center of the gas cloud is simple low energy effective physics, it's table top stuff. But according to the fuzzballs proposal, the moment the cloud reaches the Schwarzschild radius then the entire interior instantly becomes a high energy mess of strings and D-branes that removes the interior from existence.

How does fuzzball proposal reconcile this?

Answer 1

I will list some intuitive points that should cover your question:

1. The collapse process is not described by semi-classical physics, but by something dubbed "Entropy-enhanced tunnelling". The intuitive argument is that even though the probability amplitude of a shell of matter with macroscopic mass M tunnelling into a fuzzball is extremely suppressed

$$P \propto e^{-GM^2}$$

there is an exponential number of fuzzballs to tunnel into

$$N \propto e^{GM^2}$$

making the total amplitude of order 1.

2. Low energy matter ($E \sim T_H$) and high energy matter ($E >> T_H$) behave differently, the former feeling the "thickness" of space. The thickness is related to the quantum fluctuations of the geometry, which are more intense in energy dense regions with respect to flat space. This phenomenon is also called Fuzzball Complementarity, and implies that fine-grained operators experience the details of the fuzzball microstate and coarse-grained operators (a human observer for instance) experience a black hole-like infall.

3. The tunnelling process is not instantaneous, but it can be shown that

$$t_{tunnel} << t_{evap}$$

which is an important sanity check to show that the process happen fast enough.

References:

• https://homes.psd.uchicago.edu/~ejmartin/GLSC2018/slides/Saturday/Mathur.pdf
• https://indico.cern.ch/event/37753/contributions/1802130/attachments/752935/1032913/mathur3.pdf
• https://arxiv.org/abs/1704.02123

Disclaimer: At this stage these are mainly toy models, dynamic fuzzballs formation, and fuzzball themselves, is still an active subject of research.

EDIT: (reply to the first two comments)

The first reference (see p.20 onwards) describes the case for an infalling shell of matter, but a similar reasoning can be done for your homogeneous cloud with the difference that the bubble fluctuations would start from the center too. Your 2 comment questions:

1. Not instantly, it takes some time to tunnel. Until enough matter has coalesced you have only virtual fuzzballs quantum fluctuations. Once enough matter is close to the would-be-horizon the tunnelling process is completed over the 76AU. If complementarity is correct, as a coarse-grained observer you would still not observe anything particular until being close to the time you would hit the would-be-singularity. This is assuming you would experience a black-hole-like experience resulting from the most typical condensate of fuzzballs geometries. If you experience a more general geometry (but this should be less likely in the ensemble), your actual fate seems much harder to describe, but it should be an interpolation betweeen GR black-hole and instant fuzz. I'm not advocating for this to be true by the way, just explaining how the theory seems to explain it.

2. Near the would-be-horizon. In principle there are even virtual quantum fluctuations of immense fuzzballs of mass say $M + M^{10}$, but those fluctuations don't have any actual mass-energy to put them on-shell.