The information paradox: A pedagogical introduction

TL;DR

This paper provides a rigorous pedagogical treatment of the black hole information paradox, formalizing Hawking's argument within a solar-system limit that enforces locality and a set of niceness conditions.It proves that small corrections to the leading-order Hawking state cannot restore unitarity, leading to either mixed radiation or remnants under a traditional horizon.The discussion surveys string-theory insights, notably fuzzball microstates and fractionation, showing that horizon-free microstate geometries can emit unitary radiation carrying information, thereby resolving the paradox without violating quantum mechanics.Overall, the work argues that resolving the information paradox requires nonlocal or highly nonperturbative quantum-gravity effects that extend to horizon scales, challenging the traditional semiclassical picture of black holes.

Abstract

The black hole information paradox is a very poorly understood problem. It is often believed that Hawking's argument is not precisely formulated, and a more careful accounting of naturally occurring quantum corrections will allow the radiation process to become unitary. We show that such is not the case, by proving that small corrections to the leading order Hawking computation cannot remove the entanglement between the radiation and the hole. We formulate Hawking's argument as a `theorem': assuming `traditional' physics at the horizon and usual assumptions of locality we will be forced into mixed states or remnants. We also argue that one cannot explain away the problem by invoking AdS/CFT duality. We conclude with recent results on the quantum physics of black holes which show the the interior of black holes have a `fuzzball' structure. This nontrivial structure of microstates resolves the information paradox, and gives a qualitative picture of how classical intuition can break down in black hole physics.

Figures (7)

• Figure 1: (a) Spacelike slices in an evolution; the intrinsic geometry of the slice distorts in the region between the right and left sides (b) Particle pairs are created in the region of distortion (c) There is matter far away from the region of distortion; locality would say that the state of this matter is only weakly correlated with the state of the pair.
• Figure 2: A schematic set of coordinates for the Schwarzschild hole. Spacelike slices are $t=const$ outside the horizon and $r=const$ inside. Infalling matter is very far from the place where pairs are created ($\sim 10^{77}$ light years) 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: The Penrose diagram of a black hole formed by collapse of the 'infalling matter'. The spacelike slices satisfy all the niceness conditions N.
• Figure 4: The creation of Hawking pairs. The new quanta $c_{n+1}, b_{n+1}$ are not created by interaction with either the matter $|\psi\rangle_M$ (represented by the black square) or with the earlier created pairs. Rather the creation is by a Schwinger process which moves $|\psi\rangle_M$ further away from the place of pair creation, and also moves the earlier created $c, b$ quanta away from the place of pair creation. The new pairs are created in a state which to leading order is entangled between the new $b,c$ quanta but not entangled with anything else. Small corrections to this leading order state does not change this entanglement significantly, so the entanglement keeps growing all through the radiation process, unlike the case of radiation from normal hot bodies.
• Figure 5: A toy model of radiation from a normal hot body, showing the evolutions given in eqs.(\ref{['first']}),(\ref{['second']}). The left side of the vertical bar shows atoms in the body, the right side shows radiated photons with the arrow depicting their spin. If the atom near the boundary is in state $1$ (unfilled circle) then we get the linear combination of states on the right, and if the atom is in state $2$ (filled circle), then we get an orthogonal different linear combination.