What the information paradox is {\it not}

TL;DR

Mathur reframes the black hole information paradox as a four-step argument and shows that many proposed resolutions fail because they do not address all steps A–D. He demonstrates that small corrections to Hawking's pair-creation process cannot restore purity, requiring a genuine change in near-horizon microstructure. The fuzzball proposal replaces the traditional horizon with a vast ensemble of horizonless microstate geometries, whose collective phase-space allows unitary evaporation and modifies collapse dynamics. The discussion also clarifies the roles of AdS/CFT, Euclidean saddles, and gravity entanglement, arguing that only a nonperturbative microstate structure can resolve the paradox in a consistent way.

Abstract

There still exist many confusions about the black hole information paradox and its resolution. We first give a precise formulation of the paradox, in four steps A-D. Then we examine several proposals for resolving the paradox. We note that in each case one of these four steps has been ignored, so that the proposal does not really target the essence of the paradox. Finally, we give a brief summary of the fuzzball construction and argue that it resolves the paradox in string theory. This resolution contains a deep lesson -- the phase space of quantum gravity is so large that the measure in the path integral can compete with the classical action for macroscopic objects undergoing gravitational collapse.

Figures (4)

• Figure 1: The Penrose diagram of a black hole formed by collapse of the 'infalling matter'. The spacelike slices satisfy all the niceness conditions N.
• Figure 2: A schematic set of coordinates for the Schwarzschild hole. Spacelike slices are $t=const$ outside the horizon and $r=const$ inside. Assuming a solar mass hole, the infalling matter is $\sim 10^{77}$ light years from the place where pairs are created, when we measure distances along the slice. Curvature length scale is $\sim 3 ~km$ all over the region of evolution covered by the slices $S_i$.
• Figure 3: On the initial spacelike slice we have depicted two fourier modes: the longer wavelength mode is drawn with a solid line and the shorter wavelength mode is drawn with a dotted line. The mode with longer wavelength distorts to a nonuniform shape first, and creates an entangled pair $b_1, c_1$. The mode with shorter wavelength evolves for some more time before suffering the same distortion, and then it creates the entangled pair $b_2, c_2$.
• Figure 4: The horizontal direction labels the value of $\lambda$, the eigenvalue. The horizontal line gives the level of the fermi sea of eigenvalues, obtained when the matrix model potential is the indicated 'upside-down harmonic oscillator' potential. (a) A small pulse representing a quantum scatters off the wall at $r=0$ (b) A pulse large enough to make a black hole in dilaton gravity spills over the wall instead of returning to the same asymptotic infinity.