The black hole information paradox

Instructor: Samir D. Mathur (Ohio State University)

Sep 14-18, SISSA, Trieste (Italy)

Welcome to this set of lectures on the black hole information paradox. This paradox was first noted by Hawking around 1974, when he looked at the nature of radiation emitted by black holes. It has raised one of the most serious problems for the foundations of theoretical physics, since it says that general relativity and quantum theory cannot be compatible: general relativity predicts black holes will form, and then the explicit computation of vacuum fluctuations around the black hole background shows that the emitted radiation does not carry the information of the matter that made the hole.

Numerous efforts have been made to resolve the paradox. In these lectures a principal goal will be to understand what the paradox is, and why it is so robust. We will see that if we assume that quantum gravity effects are confined to within a bounded range like planck length or string length, then Hawking's argument is a 'theorem'; information loss will happen. We will end with seeing how string theory has now provided us with a resolution of the paradox: quantum gravity effects actually spread the information of the hole not all over the interior of a horizon shaped ball, so that the black hole does not have an `information-free horizon'. It remains true that the only length scale formed by the fundamental constants c, h, G is planck length, but a black hole is made of a large number of quanta N, and we need to ask if quantum gravity effects extend over distances of order planck length or some power of N times planck length. Explicit construction of black hole microstates indicates that the latter is true, so that we end up generating a macroscopic distance - horizon radius - for these quantum effects. It appears plausible that this counterintuitive behavior of quantum gravity can be traced to the very large entropy of the black hole; Exp[S] orthogonal wavefunctions of the black hole cannot be fitted into a radius smaller than horizon radius.

Note: There will be 5 days of lectures, with 2 hours of lectures and 2 hours of tutorials each day. The outline of the lectures are given below, and the reading material for each lecture is given as pdf files. I strongly urge the students to start working through this material now, and solving the assignments given with the material before the course starts; since the course lasts only 5 days, you will get much more out of the course if you are already familiar with the content and are at a stage where you are clarifying difficulties.

Lecture 1.1: The information paradox arises because of the creation of Hawking radiation. This radiation arises because the black hole has a horizon, inside which the time coordinate t actually becomes spacelike. Thus we begin by working out the spacetime geometry of the black hole: Starting from the Schwarzschild metric, finding Kruskal coordinates to cover the spacetime, and making a Penrose diagram to describe the causal structure created by a collapsing shell. While most of the lectures will not need a detailed knowledge of general relativity, it is nevertheless helpful to have a good grounding in the metric of the black hole before proceeding. This material is in the notes

The black hole geometry

Assignment 1.1: The problems are embedded in the text of the above notes.

Lecture 1.2: We now come to a crucial step: foliating the black hole spacetime with spacelike slices. There are two important points about this foliation: (a) The spacelike slices will necessarily have to be time dependent, which forces particle creation (b) The slices can be chosen to to avoid the singularity and lie entirely in a domain of low curvature, so that we are entitled to expect 'normal low energy physics' all through the evolution. We mark out the regions in the spacetime where we see the infalling matter, the Hawking radiation, and negative energy partners of the Hawking quanta. This material is in

What exactly is the information paradox? (Section 4)

CERN lectures(Section 3)

Assignment 1.2: Show how to foliate the spacetime of a black hole formed by the collapse of a shell, such that all slices are `good' everywhere, and we capture both the infalling matter and the Hawking radiation on the slices.

Lecture 2.1: We first discuss the creation of particles in curved spacetime. Next we discuss, in pictures, where particle creation happens in the black hole geometry. We end by writing the state of the created Hawking pairs. The qualitative discussion is in

What exactly is the information paradox? (Section 5)

The detailed expression for the Hawking quanta will not be derived in the lecture (it may be covered in the tutorial if there is enough time), but the dedicated student should work through the derivation in

Quantum Emission from Two-Dimensional Black Holes (Giddings and Nelson)

Lecture 2.2: We discuss the entangled nature of the Hawking state. We discuss how information is lost when the black hole evaporates, and what alternative scenarios are possible. This material is covered in

CERN lectures(Section 4)

Assignment 2: Write down a detailed explanation of how the Hawking process occurs, and leads to information loss.

Lecture 3.1: We now come to the most crucial part of the ifnormation paradox. Hawking's computation has been opposed on the grounds that it was a leading order calculation: taking into account small quantum corrections might introduce subtle correlations among the outgoing photons,and thus carry the information out in the Hawking radiation. In that case, of course. there would be really no paradox. We show that such is NOT the case, by showing that if the black hole has a traditional horizon, then information CANNOT be carried by such correlations. To do this we use standard arguments from quantum information theory to derive 3 lemmas, and then use them to prove a theorem: If the evolution of low energy modes is corrected by only a small fraction epsilon, then the creation of each new Hawking pair necessarily INCREASES the entanglement entropy between the inside of the hole and the outside by (1-2 epsilon) ln 2. This material is in

CERN lectures(Section 5)

Lecture 3.2: We use the above theorem to put Hawking;s argument in a rigorous form: If we assume that a `traditional black hole' forms in the theory, then we WILL have information loss. We define the `traditional black hole' in a precise fashion, incorporating the assumption that quantum gravity effects are small on macroscopic scales so the horizon is close to `empty space'. Putting Hawking's argument in this version is useful because it tells us what is the least we will have to do if we want information to indeed come out of the hole: the evolution of low energy modes at the horizon will have to be corrected by order UNITY, not by a small amount. This material is in

CERN lectures(Section 6)

Assignment 3: Write out a detailed proof of the three lemmas and the theorem they lead to. Then use these to prove the rigorous form of Hawking's argument.

Lecture 4.1 and 4.2: We will spend these lectures in debating 10 leading misconceptions about black holes and the information paradox. It is remarkable that this 30 year old problem is so poorly understood. The main difficulties stem from confusions about black holes, thermodynamics, quantum unitarity, encoding of information in correlations, the role of AdS/CFT etc. Due to these confusions we fail to appreciate the enormous power of Hawking's paradox, and thus fail to realize that we will have to make a basic change in our understanding of how quantum gravity should behave for macroscopic black holes. Thus understanding these puzzles will help us draw the essential lessons from the resolution of the paradox, and allow us to apply these lessons in situations like the early Universe.

What exactly is the information paradox? (Section 6)

CERN lectures(Section 8)

and other will be posted in the notes

Common misconceptions about the information paradox

Assignment 4: This assignment (to be posted) will ask the students to pinpoint the fallacies in several commonly made statements about the information paradox.

Lecture 5.1: In this lecture we turn to the resolution of the information problem in string theory. We explicitly construct the microstates of the simplest black holes, and note that they do not have a traditional horizon. The material is given in

The fuzzball proposal for black holes:an elementary review (Sections 4, 5)

Lecture 5.2: We will discuss the qualitative reason why quantum gravity effects can succeed in modifying the distribution of information througout the black hole interior: the large entropy of the black hole implies a very large phase space of solutions, which gives avery large number of orthogonal quantum states. Such a large number of orthogonal states cannot fit in too small a region, and lead to a spreading of data throughout a horizon sized 'fuzzball'. The material for this discusiion is in the papers

Black hole size and phase space volumes

Fuzzballs and the information paradox: a summary and conjectures

How fast can a black hole release its information?

Here are some review articles which have additional information:

Black Holes, Black Rings and their Microstates (Bena and Warner)

Black Holes as Effective Geometries (Balasubramanian, de Boer, El-Showk and Messamah)