Black holes and holography

TL;DR

The paper distinguishes two notions of holography in gravity—horizon-based holography tied to black-hole entropy and gauge/gravity holography with boundary-placed CFTs—and argues that resolving the information paradox requires real horizon-scale degrees of freedom. It advances the fuzzball program, showing that microstate geometries end outside the would-be horizon and emit information-carrying radiation via ergoregions, thereby preserving unitarity. By introducing approximate complementarity for expectation values and dynamics, the work provides a pragmatic bridge between fuzzball microstates and semiclassical descriptions, and discusses dualities within both gravity–gauge and gravity–gravity contexts. The framework connects microstate counting, entropy of Rindler/de Sitter spaces, and entropy-compatible holographic interpretations, offering a cohesive picture of how spacetime and information can be reconciled in quantum gravity.

Abstract

The idea of holography in gravity arose from the fact that the entropy of black holes is given by their surface area. The holography encountered in gauge/gravity duality has no such relation however; the boundary surface can be placed at an arbitrary location in AdS space and its area does not give the entropy of the bulk. The essential issues are also different between the two cases: in black holes we get Hawking radiation from the `holographic surface' which leads to the information issue, while in gauge/gravity duality there is no such radiation from the boundary surface. To resolve the information paradox we need to show that there are real degrees of freedom at the horizon of the hole; this is achieved by the fuzzball construction. While the fuzzball has no interior to the horizon, we argue that an auxiliary spacetime can be constructed to continue the collective dynamics of fuzzball for times of order the crossing time; this is an analogue of `complementarity'.

Figures (20)

• Figure 1: (a) Branes create a geometry that is AdS in the 'near' region; the dual CFT lives on a boundary placed anywhere in the AdS region, and describes gravity in all the region below it. (b) The singularity at $r=0$ can be avoided by moving to global AdS, where a 3-point function is computed by a simple path integral with no singularities. (c) If we have enough energy in global AdS, we make a black hole, and then we face the difficulties of Hawking's argument again. (The vertical direction is time, and the surface of the inner cylinder is the black hole horizon.)
• Figure 2: Eddington-Finkelstein coordinates for the Schwarzschild hole. Spacelike slices are $t=const$ outside the horizon and $r=const$ inside. Curvature length scale for a solar mass black hole is $\sim 3 ~km$ all over the region of evolution covered by the slices $S_i$.
• Figure 3: (a) The stretching of 'good slices' in the traditional black hole geometry leads to pair creation by the Hawking process and the consequent information problem. (b) If there are $Exp[S]$ fuzzball solutions, the wavefunction giving semiclassical geometry on the initial slice spreads over this vast phase space of solutions after some evolution, and we no longer get the traditional pair creation with growing entanglement.
• Figure 4: Schematic description of a microstate solution of Einstein's equations. There are 'local ergoregions' with rapidly changing direction of frame dragging near the horizon. The geometry closes off without having an interior horizon or singularity due to its peculiar topological structure.
• Figure 5: (a) The eternal hole, with two asymptotic infinities. (b) The boundary CFT consists of two entangled copies. (c) The state of each copy can be replaced by a gravitational solution, giving a pair of entangled solutions.