Critique of the fuzzball program

We explore the viability of fuzzballs as candidate microstate geometries for the black hole and their possible role in resolutions of the information paradox. We argue that if fuzzballs provide a description of black-hole microstates then the typical fuzzball geometry can only differ significantly from the conventional black-hole geometry at a Planck-scale distance from the horizon. However, precisely in this region, quantum fluctuations in the fuzzball geometry become large, and the fuzzball geometry becomes unreliable. We verify these expectations through a detailed calculation of quantum expectation values and quantum fluctuations in the two-charge fuzzball geometries. We then examine some of the solutions discovered in the work by Bena et al. [Phys. Rev. Lett. 117, 201601 (2016)]. We show, based on a calculation of a probe two-point function in this background, that these solutions, and others in their class, violate robust expectations about the gap in energies between successive energy eigenstates and differ too much from the conventional black hole to represent viable microstates. We conclude that, while fuzzballs are interesting starlike solutions in string theory, they do not appear to be relevant for resolving the information paradox and cannot be used to make valid inferences about black-hole interiors.

Figure 1

Logical flowchart of the paper. Statistical mechanics results are in magenta rectangles. The physical expectation on which they rely is referred to in brackets. Implications for the fuzzball are in orange ovals. Major conclusions are in brown boxes. Sections in which a conclusion is verified are given in brackets.

Figure 2

The subspace, $H_E$, is a compact manifold. Most pure states in the space are very close to the maximally mixed state. An exponentially small volume of states (displayed in blue) can be a typical.

Figure 3

An unusual possibility for the dynamics of fuzzball microstates. Probes of one microstate only excite other microstates in a single tower (solid black lines), and transitions between towers (dashed red lines) are disallowed.

Figure 4

A schematic representation of what a typical fuzzball geometry must look like, if fuzzballs represent black-hole microstates. The geometry must closely resemble the black-hole geometry away from the horizon (unshaded region) and then suddenly deviate away to cause some extra dimension to pinch off when we reach within the Planck length of the horizon (blue region).

Figure 5

A plot of the difference parameter, ${\mathsf{d}}_5$, and the “quantumness parameter”, ${\mathsf{q}}_5$, for the harmonic functions $f_5$ and $f_1$. The plot shows that in the average fuzzball geometry, precisely in the region where $f_5$ and $f_1$ differ significantly from their conventional value, quantum fluctuations becomes large and the fuzzball geometry becomes unreliable.

Figure 6

A plot of the difference parameter, ${\mathsf{d}}_W$, and the quantumness parameter, ${\mathsf{q}}_W$, for the 1-form $W_i$. The difference parameter is uniformly 1, since this function vanishes in the conventional black hole. But the value it ostensibly takes in the average fuzzball solution is always unreliable since quantum fluctuations are the same size as its expectation value.

Figure 7

A graph of $V(\xi )$ vs $\xi$ with $\gamma =10$, $\omega =0$, $\kappa =4$, and different values of $n$.

Figure 8

Comparison between a numerical calculation (dots) of the gap between the first two allowed frequencies and the formula (4.16) (solid curve). Other parameters are $\gamma =100$, $n=2$. In its regime of validity (large $\kappa$), the formula (4.16) shows excellent agreement with the numerical results.

Figure 9

Comparison between a numerical calculation (dots) of the asymptotic value $C$ with the formula (4.15) (solid curves) for different values of $n$ and $\gamma$. Here, $\kappa =5$ is held fixed. In the appropriate regime of validity of the analytic formula (large $\gamma$), it agrees very well with the numerical results.