/courses/black-hole-information-paradox/ Recent content in The Black Hole Information Paradox on Suvrat Raju Hugo -- gohugo.io en © 2026 /courses/black-hole-information-paradox/lecture-1-introduction-and-two-point-qft-cor/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-1-introduction-and-two-point-qft-cor/ We started with a broad overview of what we plan to cover in these lectures, and with some general discussion about different ways of understanding Hawking radiation. We then moved to a systematic exploration of short-distance two point correlation functions in a quantum field theory in a curved spacetime. Download PDF of Lecture Notes /courses/black-hole-information-paradox/lecture-2-entangled-modes-across-null-surfac/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-2-entangled-modes-across-null-surfac/ We showed that in a quantum field theory in curved spacetime, in the vicinity of any null surface, it is possible to extract degrees of freedom that are entangled in a very specific manner. These degrees of freedom are obtained by integrating the field with a carefully chosen smearing function. Although the intermediate steps required careful calculations, we obtained a simple and beautiful final answer. In a spacetime with spherical symmetry, it is possible to generalize this construction and obtained entangled modes that are labeled by a spherical harmonic rather than a point. /courses/black-hole-information-paradox/assignment-1/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/assignment-1/ This assignment is primarily related to material from the first two lectures. But it also includes some simple fun calculations that provide a sense of the kind of physical numbers that appear in quantum gravity. Please submit the assignment by emailing it to directly to Chandramouli Choudhury. If you have questions about the assignment, or submission, please contact Chandramouli. Download Assignment 1 /courses/black-hole-information-paradox/lecture-3-quantum-fields-black-hole-backgrou/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-3-quantum-fields-black-hole-backgrou/ We started with a brief introduction to the geometry of a black hole formed from collapse in asymptotically flat space. We then analysed late-time solutions to the wave-equation in this background. We used this to write down a mode expansion of the field. This expansion is valid both in the black-hole exterior and in the black-hole interior, provided we consider a region that is well-separated from both the infalling matter and the singularity. /courses/black-hole-information-paradox/lecture-4-hawking-radiation/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-4-hawking-radiation/ In this lecture, we derived the form of Hawking radiation both in asymptotically flat space and in AdS. This result follows immediately when one puts together the results of previous lectures on (a) the universal entanglement of local degrees of freedom across a null surface and on (b) the mode-expansion of a quantum field in the black-hole geometry. Physically, the phenomenon of Hawking radiation follows from a combination of short-distance physics — which implies that certain local-degrees of freedom near the black-hole horizon are thermally occupied — and the late-time isometry of the geometry that allows us to relate these near-horizon degrees of freedom to global modes. /courses/black-hole-information-paradox/assignment-2/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/assignment-2/ This assignment explores aspects of the wave-equation in black-hole backgrounds. You are first asked to parse this equation so as to put it in a convenient form. This form allows an analysis of the so-called “greybody factors” of black holes. A third question introduces a commonly-used method of solving this equation via an infinite series. The last problem is meant to give you a sense of the kind of physical time-scales that are relevant for black-hole evaporation. /courses/black-hole-information-paradox/lecture-5-hawkings-original-paradox/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-5-hawkings-original-paradox/ In this lecture, we examined Hawking’s original version of the information paradox by reading key parts of Hawking’s paper “Breakdown of predictability in gravitational collapse”. Hawking argued that this computation of black-hole radiation suggested that the final state on future null infinity would be thermal. Moreover, he argued that this was to be expected since an observer at future null infinity would be forced to adopt a “principle of ignorance” about the degrees of freedom that fell into the black hole. /courses/black-hole-information-paradox/lecture-6-how-close-are-pure-and-mixed-state/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-6-how-close-are-pure-and-mixed-state/ In this lecture, we explored a simple question in quantum statistical mechanics. From the point of view of physical observations, how closely do typical pure states resemble mixed states. We found that typical superpositions of states from a given energy band are exponentially close to the microcanonical density matrix. This simple result, when applied to black-hole evaporation, tells us that Hawking’s original argument is not precise enough to lead to a paradox. /courses/black-hole-information-paradox/lecture-7-resolving-hawkings-original-parado/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-7-resolving-hawkings-original-parado/ We concluded our discussion of Hawking’s original paradox. Hawking original argument had two aspects. The first was a concrete computation of low-point correlators at future null infinity. We already explained why this was insufficient to conclude that the final state would be thermal. But, in addition, Hawking argued that the external observer would have to adopt a “principle of ignorance” about the black hole interior and this is why the external observer would see a mixed state for the final radiation. /courses/black-hole-information-paradox/lecture-8-some-results-quantum-information/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-8-some-results-quantum-information/ In this lecture, we discussed some simple results from quantum information.We showed how entanglement could be quantified using a simple set of two-point correlators called CHSH correlators. These correlators also allow for a beautiful formulation of the “monogamy of entanglement.” We then went on to discuss the average entanglement between subsystems of a large system. We showed that, in a typical state, for a pair of pseudospin operators in the smaller system, it is possible to find a set of operators in the larger system so that the two pairs are maximally entangled. /courses/black-hole-information-paradox/assignment-3/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/assignment-3/ This assignment contains some exercises to help you understand how pure states can mimic mixed states to very high accuracy. The last question explores some properties of CHSH correlators as a measure of entanglement. Download assignment3.pdf Download Assignment 3 /courses/black-hole-information-paradox/lecture-9-monogamy-paradox/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-9-monogamy-paradox/ The monogamy paradox can be stated very simply. If the black hole horizon is smooth, degrees of freedom across the horizon must be entangled with each other. On the other hand, general results from quantum information imply that, for an old black hole, modes near the horizon must be entangled with the early radiation emitted by the black hole. This is a paradox because it violates the monogamy of entanglement. /courses/black-hole-information-paradox/lecture-10-fuzzballs-firewalls-and-complemen/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-10-fuzzballs-firewalls-and-complemen/ We discussed three possible resolutions to the monogamy paradox: the fuzzball proposal, the firewall proposal and a set of resolutions that go by the name of “A=R_b, complementarity, or ER=EPR” The fuzzball and firewall proposals state that the interior does not exist and so they resolve the monogamy paradox by getting rid of the entanglement between modes across the horizon. The spirit of the resolution using complementarity is that the degrees of freedom inside the black hole can be written as scrambled versions of degrees of freedom from the early radiation. /courses/black-hole-information-paradox/lecture-11-aspects-ads-hilbert-space/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-11-aspects-ads-hilbert-space/ In this lecture, we considered some aspects of the Hilbert space of QFTs and gravity in AdS. We started by considering a minimally coupled scalar in AdS. We found that the Hilbert space could be constructed by exciting the vacuum with asymptotic operators smeared over boundary regions. We then found that this description continues to hold wheninteractions are turned on, and even in theories of gravity. This description of the Hilbert space is overcomplete but it automatically includes black holes formed from collapse and other states that we usually think of as being “nonperturbative”. /courses/black-hole-information-paradox/lecture-12-holography-information-ads/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-12-holography-information-ads/ In this lecture, we first discussed some additional aspects of the Reeh-Schlieder theorem from Lecture 11. A Mathematica file with some simple numerical examples that demonstrate our result can be downloaded here. Download the Mathematica demonstration of how to generate states from a small time band In this lecture, we proved the principle of holography of information for theories of gravity in asymptotically AdS spacetimes. Gravity is different from other quantum field theories because the projector on the vacuum belongs to the algebra of asymptotic operators; in no other theory can one use an operator from the asymptotic algebra to project onto a unique state. /courses/black-hole-information-paradox/lecture-13-low-energy-tests-holography-infor/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-13-low-energy-tests-holography-infor/ We started this lecture by considering a fun thought experiment that illustrates some of the physics of the principle of holography of information. Consider a set of astrophysicists with sensitive equipment who live near the boundary of AdS. Their equipment can only be used between time $0$ and $\epsilon$. We showed how these astrophysicists could entirely identify a low-energy bulk state using gravitational effects without ever actually exploring the bulk! This shows that, even at low-energies, information in a theory of quantum gravity can be localized very differently from local quantum field theories. /courses/black-hole-information-paradox/lecture-14-holography-information-flat-space/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-14-holography-information-flat-space/ In this lecture, we first reviewed aspects of the Hilbert space of massless excitations in four-dimensional flat space. We then discussed how the principle of holography of information manifests itself in flat space. The result is that, in a theory of quantum gravity, under weak assumptions, a state of massless excitations can be completely identified using observations in a small neighbourhood of the past boundary of future null infinity. This is contrast to local quantum field theories, where observations on all of future null infinity would be required to identify the state. /courses/black-hole-information-paradox/lecture-15-low-energy-tests-and-perspective-/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-15-low-energy-tests-and-perspective-/ In this lecture, we started by illustrating the physics of the principle of holography of information in flat space using a simple thought experiment. Consider a pulse that arrives at scri-+ near u = 0. We showed that using gravitational effects, and measuring correlations of fluctuations of the metric and dynamical fields, it is possible to completely characterize the pulse through observations near u -> -\infty before the pulse actually reaches scri-+! /courses/black-hole-information-paradox/lecture-16-entropy-null-infinity/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-16-entropy-null-infinity/ In this lecture, we examined the von Neumann entropy of a segment (-\infty, u_0) of scri-plus in a theory of gravity. We showed that if one considers the fine-grained entropy, then the principle of holography of information tells us that this entropy is a constant, independent of u_0. This formalizes the statement that “information is always outside” in a theory of gravity. We discussed the physics of this claim and contrasted black holes with ordinary objects, like burning coal. /courses/black-hole-information-paradox/assignment-4/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/assignment-4/ This assignment explores aspect of the Reeh-Schlieder theorem. It attempts to (1) demystify the result by relating it to the familiar Fourier series (2) check its consistency with locality and (3) demonstrate a slight generalization of the result that we used in the lectures. For the purposes of this assignment, you can forget about gravity. Each question has some explanatory text that is not part of the question itself. Please don’t get intimidated by this! /courses/black-hole-information-paradox/assignment-5/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/assignment-5/ This assignment asks you to (1) show how two-point functions accessible near the boundary in a theory of gravity in AdS can be used to characterize a bulk excitation (2) compute a two-point function at null infinity (3) check some technical details of the toy-version of the monogamy paradox. Each question has some explanatory comments that are not part of the question itself but may help to provide perspective. Download Assignment 5 /courses/black-hole-information-paradox/lecture-17-entanglement-entropy-holographic-/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-17-entanglement-entropy-holographic-/ In this lecture, we discussed the prescription for computing entanglement entropy in holographic CFTs. We worked out a simple example of a minimal area surface in AdS3 and found an answer that matches the known answer for the entropy of an interval in a 1+1 dimensional CFT. We also discussed subregion duality, which is the idea that a region on the boundary has information about its entanglement wedge in the bulk. /courses/black-hole-information-paradox/lecture-18-introduction-islands/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-18-introduction-islands/ We completed our discussion of simple entanglement wedges in AdS/CFT. We reviewed the replica trick for computing the entanglement entropy and indicated how it can be used to obtain the holographic prescription for entanglement entropy. We then reviewed the setup, where a holographic theory is coupled to a nongravitational bath, that has been used in recent analyses of the Page curve. We wrote down the “island formula” for the entanglement entropy of a part of the nongravitational bath and briefly sketched how the formula leads to a Page curve. /courses/black-hole-information-paradox/lecture-19-islands-branes/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-19-islands-branes/ Islands can be understood most simply in a doubly holographic setup. Here, we consider a CFT in d-dimensions that lives on a half space with a boundary that hosts a (d-1)-dimensional CFT. The entire system has a bulk dual as a theory of gravity in (d+1)-dimensional AdS that is terminated by a brane.The entropy of regions of the nongravitational boundary can be computed by the standard RT prescription except that RT surfaces are allowed to end on the brane. /courses/black-hole-information-paradox/lecture-20-gravity-bath/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-20-gravity-bath/ In this lecture, we completed our study of an example of a phase transition between RT surfaces that leads to the emergence of a Page curve in holographic systems coupled to nongravitational baths. We then discussed how the physics changes when we turn on gravity in the bath. Within the framework of braneworlds, one example is obtained by introducing two branes in a black string geometry. The only extremal surface that extends between the two branes, and is also extremal with respect to its endpoints is the horizon. /courses/black-hole-information-paradox/lecture-21-paradoxes-large-ads-black-holes/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-21-paradoxes-large-ads-black-holes/ In this lecture, we turned our attention to large black holes in asymptotically AdS spacetimes. These black holes are different from the evaporating black holes we have considered so far because they dominate the microcanonical ensemble above a given energy. This leads to a new set of paradoxes. We showed that if one assumes that degrees of freedom in the black hole interior are described by state-independent operators then this leads to a contradiction with the idea that typical states have a smooth interior. /courses/black-hole-information-paradox/lecture-22-paradoxes-eternal-black-hole/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-22-paradoxes-eternal-black-hole/ In this lecture, we showed that if one assumes that (a) degrees of freedom in the black hole interior are described by the same operators in the thermofield doubled state and states related to it by simple Hamiltonian evolution (b) disentangled states are not connected by a wormhole, then one can extend the paradoxes that we found for large black holes to the eternal black hole. So even the seemingly well-understood duality between the eternal black hole and the thermofield doubled state has a subtlety when we try and map bulk degrees of freedom in the black-hole interior to boundary degrees of freedom. /courses/black-hole-information-paradox/lecture-23-interior-reconstruction-adscft/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-23-interior-reconstruction-adscft/ We discussed the construction of “mirror operators”. These are boundary operators that are dual to degrees of freedom in the interior of a large black hole. In order to perform this construction, we introduced the notion of the little Hilbert space, and of equilibrium states. We then checked that the mirror operators, as defined, had the right correlators with ordinary operators. The commutator of interior operators with simple operators outside the black hole does not vanish as an operator. /courses/black-hole-information-paradox/lecture-24-state-dependence/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-24-state-dependence/ In this lecture, we completed our discussion of the construction of mirror operators. We described how the construction accounts for the nontrivial commutator of interior operators with the Hamiltonian. We also described how the construction accounts for near-equilibrium states and thereby avoids the “frozen vacuum” problem. By adding mirror operators constructed in different little Hilbert spaces, one obtains operators that can be used to describe the black hole interior for any state within the direct sum of these spaces. /courses/black-hole-information-paradox/assignment-6/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/assignment-6/ The first question in this assignment asks you to compute the holographic entanglement entropy of a “belt” on the boundary. The second question explores some simple properties of islands when the bath is nongravitating. The final question is about gravitating baths, and leads you through the (simple) calculation that is used to support the conclusion that the Page curve is trivial when gravity is dynamical everywhere. Download Assignment 6 /courses/black-hole-information-paradox/assignment-7/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/assignment-7/ This assignment explores various aspects of the mirror-operator construction. The first problem asks you to show that the mirror of the adjoint of an operator is the adjoint of the mirror of the operator. Problems 2-4 lead you through the Tomita-Takesaki approach to mirror operators. In problem 5, you are asked to numerically construct mirror operators in a simple spin-chain model. Download Assignment 7 /courses/black-hole-information-paradox/lecture-25-more-state-dependence/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-25-more-state-dependence/ In this lecture, we showed how the use of state-dependent mirror operators resolves all paradoxes associated both with large AdS black holes and the eternal black hole. All such paradoxes tacitly invoke a trace over the Hilbert space. But if the operator in the trace varies from one basis element to another, then the paradoxes disappear. On the other hand, state-dependent gives rise to its own puzzle since it suggests that black holes are anomalously sensitive to low-energy perturbations. /courses/black-hole-information-paradox/lecture-26-discussion/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/lecture-26-discussion/ In this concluding lecture, we discussed a number of open-ended questions. We summarized the status of state dependence and also discussed the question of whether typical black hole microstates are expected to have structure at their horizons. We explained why classical solutions cannot be used to make inferences about typical states. Download PDF of Lecture Notes /courses/black-hole-information-paradox/summary/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/black-hole-information-paradox/summary/ This summary provides a quick overview of the topics that we discussed in this course, and some perspective on how they fit together. Download slides Download PDF