/ Recent content on Suvrat Raju Hugo -- gohugo.io en © 2026 Mon, 01 Jun 2026 08:46:26 +0530 /courses/serc-school/notes-lecture-1/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/notes-lecture-1/ Lecture 1 covered the Schwarzschild Black Hole solution, which we derived assuming only spherical symmetry. We proved Birkhoff’s theorem. We also reviewed the gravitational redshift and the Kruskal extension, which shows that the horizon is locally a smooth region. Download PDF of Lecture Notes /courses/quantum-aspects/lecture-1/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-1/ This lecture offers a brief overview of the topics that we will cover in this course. Download PDF of Lecture Notes /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/serc-school/lecture-2/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/lecture-2/ In Lecture 2, we reviewed the Oppenheimer-Snyder solution that shows how black holes can be formed via the collapse of dust. We also considered other kinds of black hole solutions, including the Reissner-Nordstrom black hole, and charged Kerr black holes. Download PDF of Lecture Notes /courses/quantum-aspects/lecture-2-framework-qft/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-2-framework-qft/ We discussed the framework of quantum field theory in curved spacetime. This is an approximation where we take the background spacetime to be fixed, and then consider fluctuations of fields on this background. An important difference with QFT in flat space is the fact that the mode functions may not be plane waves. Moreover, not all observers agree on the vacuum. We also defined Bogoliubov coefficients and calculated what the vacuum state for one quantization looks like in terms of excited states in another quantization. /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/serc-school/lecture-3/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/lecture-3/ In Lecture 3, we considered the causal structure of the R-N black hole. At first sight, this leads to an exotic Penrose diagram. But after we looked at this diagram more closely and considered the instability of the inner horizon, the conclusion was that, qualitatively, for realistic collapse leading to a charged black hole or a charged rotating black hole, the causal structure of the spacetime is much like the causal structure of the Oppenheimer-Snyder solution. /courses/quantum-aspects/lecture-3-examples-qft-/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-3-examples-qft-/ In this lecture, we considered a simple example of QFT in curved space, by considering a massive scalar field in an expanding Universe. The expansion dynamically creates particles and we were able to calculate the exact Bogoliubov coefficients between the modes appropriate to an observer in the asymptotic future, and the modes appropriate to an observer in the asymptotic past. Then we introduced Rindler coordinates in Minkowski space. This is an important example since a lot of our intuition about the black hole derives from this setup. /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/serc-school/lecture-4/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/lecture-4/ In lecture 4, we complete our discussion of the Penrose process and frame the laws of Black Hole Thermodynamics. The rest of the lecture is an overview of what happens when we add quantum mechanics. We will review the old information paradox, which is a simple question of reversibility, and can be resolved rather easily. Then, we sketch more modern versions of the information paradox which are more subtle and may require us to accept that degrees of freedom in quantum gravity are (1) not exactly local and (2) state-dependent. /courses/quantum-aspects/lecture-4-rindler-space/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-4-rindler-space/ In this lecture, we discussed the transformation between the Minkowski and the Rindler vacua. We considered linear combinations of the Rindler modes that had the property that they could be written purely in terms of positive frequency Minkowski modes. Therefore, the Minkowski vacuum must be annihilated by these modes. By solving this equation, we established a very important result — the Minkowski vacuum looks like the thermofield doubled state from the point of view of a Rindler observer. /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/serc-school/lecture-5/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/lecture-5/ In Lecture 5, we discussed the basics of QFT in curved space. Since, in a general spacetime, there is no canonical choice of a time coordinate, the notion of “particles” becomes observer-dependent. We discussed how the physics of two different observers, using a different basis of mode functions, was related by Bogoliubov transformations. We then started to apply this formalism to understand how flat space appears for accelerated observers by quantizing a free-field in Rindler coordinates. /courses/quantum-aspects/lecture-5-more-rindler-/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-5-more-rindler-/ In this lecture, after deriving the explicit Bogoliubov coefficients between the Unruh and Minkowski modes, we turned to a consideration of Rindler space in d+1 dimensions. The modes here are considerably more complicated. So, instead of trying to derive the Bogoliubov transformations using their global properties, we started a local derivation of the Unruh. Our strategy was to analyze the near-light-cone properties of two-point correlation functions in the Minkowski vacuum, and then use this to derive the two-point function of Rindler modes in the Minkowski vacuum. /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/serc-school/lecture-6/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/lecture-6/ In lecture 6, we completed our discussion of the Unruh effect. The central result that for a set of observers in region I, using Rindler modes, even the Minkowski vacuum appears to be a thermal density matrix. We then applied this intuition to black holes. There we found that locally, the change of coordinates from Schwarzschild coordinates to Kruskal coordinates was very similar to the transformation between Rindler and Minkowski coordinates. /courses/quantum-aspects/lecture-6-more-rindler-/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-6-more-rindler-/ In our last lecture on QFT in curved spacetime, we completed our analysis from Lecture 5. We showed that unless the Rindler occupation numbers had the right thermal form, we would not be able to derive the short-distance behaviour of correlators in the Minkowski vacuum. This derivation is important since it just relies on local properties of correlation functions, and generalizes easily to the late-time collapsing black hole geometry and other situations, where a global analysis of the modes may be difficult. /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/serc-school/lecture-7/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/lecture-7/ In this lecture, we proceeded to derive the formula for Hawking radiation in two different ways. First, we examined Hawking’s original geometrical optics approach. This is useful because it gives us insight into where the radiation at future infinity originates from. Then, we started a more precise derivation using the properties of correlation functions in any smooth geometry. The geometry of the collapsing shell tells us that field correlators in position space have to have some universal properties. /courses/quantum-aspects/lecture-7-schwarzschild/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-7-schwarzschild/ In lecture 7, we moved to a discussion of the Schwarzschild black hole. We analyzed the red-shift between signals exchanged between observers who stay at different fixed values of $r$. We then analyzed geodesics, and arrived at the surprising conclusion that while an infalling observer crosses the horizon and reaches the singularity in finite time, the observer outside never sees this happen; instead the infaller vanishes after a while because any radiation that he emits get red-shifted below the IR-cutoff of the outside observer. /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/serc-school/lecture-8/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/lecture-8/ In lecture 8, we considered the “old information paradox”. This is the paradox, that appears to arise because Hawking radiation is thermal, and so this suggests that if we start with a pure state, that collapses to form a black hole, the process of Hawking radiation will convert it into a mixed state. Such evolution, from a pure to a mixed state would, of course, violate unitarity. But this is not really a paradox, because in large statistical systems, generic pure states look almost exactly like thermal states. /courses/quantum-aspects/lecture-8-more-schwarzs/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-8-more-schwarzs/ In this lecture, we completed our discussion of the Schwarzschild black hole by reviewing its Penrose diagram. This solution is particularly important because it is the unique vacuum solution with spherical symmetry; this fact is known as Birkhoff’s theorem. Schwarzschild black holes can also be found in higher dimensions, and we quickly reviewed these solutions. We then turned to black holes formed by collapse. Using techniques developed by B. Datt, Oppenheimer and Snyder were able to follow the collapse of dust into a singularity. /courses/quantum-aspects/lecture-10-charged-blac/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-10-charged-blac/ In lecture 10, we discussed the charged Schwarzschild black hole called the Reissner-Nordstrom black hole. In some ways, this black hole is similar to the Schwarzschild black hole, but it has some novel features. In particular, when we examine the structure of the maximal extension, we find that the black hole has both an outer and an inner-horizon. Moreover, at first sight, it appears possible to avoid the singularity inside the horizon, and emerge into another Universe. /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/serc-school/lecture-9/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/serc-school/lecture-9/ In the last lecture, we considered the modern information paradox. This starts with Page’s observation that although unitarity tells us that the final radiation will be pure, we can get a stronger result by adding the assumption that the black hole is a “generic state”. In this situation, we expect that the von Neumann entropy of the outgoing radiation will start to decrease after, what is called the “Page time” (after roughly “half” the black hole has evaporated) and continue to decline monotonically to zero (at which point the outgoing radiation is pure). /courses/quantum-aspects/lecture-11-kerr-black-h/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-11-kerr-black-h/ In Lecture 11, we discussed the charged Kerr black hole. This is the most general stationary black hole solution. The metric for the Kerr black hole is messier to deal with, but we examined the near-horizon geometry and worked out the coordinate transformations that allowed us to cross the horizon in a smooth coordinate system. The interesting new feature that appeared here, was the presence of the ergosphere. In this region, which is outside the horizon except at two points where it touches it, the vector (d/dt) becomes spacelike. /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/quantum-aspects/lecture-12-second-law-a/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-12-second-law-a/ In Lecture 12, we started to investigate the limits of energy extraction from black holes. We found that positivity of local energy in the ergosphere prevented processes that could decrease the area of the black hole. We then started a derivation of a more general version of the second law of thermodynamics. The main technical input required for this derivation is the “Raychaudhuri equation”, and we derived this equation. Download PDF of Lecture Notes /courses/quantum-aspects/lecture-13-area-theorem/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-13-area-theorem/ In the previous lecture, we derived the main technical input required for the myriad global theorems about black holes—the Raychaudhuri equation. The area theorem follows from simple reasoning about horizons and the application of this equation. First, we show that the horizon is generated by null geodesics that do not intersect. Then it follows that the expansion at each point must be positive, unless the horizon itself hits a singularity. Assuming that this does not happen, the area of the horizon must increase along its future-directed null coordinate. /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/quantum-aspects/lecture-14-black-hole-t/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-14-black-hole-t/ In this lecture, we rounded off our discussion of black hole thermodynamics by considering the first and zeroth law of black holes. To phrase these laws precisely, we needed to introduce the concept of surface gravity. The first law then simply states that: dM = k dA /(8 pi) + (work-terms) and the zeroth law states that the surface gravity is constant over the horizon of a stationary black hole. /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/quantum-aspects/lecture-15-hawking-radi/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-15-hawking-radi/ In this lecture, we reviewed Hawking’s original derivation of the Hawking radiation formula through ray-tracing. The main geometric insight that goes into this derivation is the behaviour of geodesics in the collapsing black hole-geometry. Light rays that embark from past null-infinity after a “last ray” get stuck inside the black hole. Light rays that embark from past null-infinity just before the “last ray” emerge to future null infinity. But an infinite region (in affine-parameter space) on future null infinity is generated by a very small region just before the last-ray on past-null infinity. /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/quantum-aspects/lecture-16-local-deriva/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-16-local-deriva/ The ray-tracing derivation of Hawking radiation suffers from a trans-Planckian problem. The frequencies that we see at late times originate from very blueshifted frequencies in the past. To sidestep this issue, we derived the formula for Hawking radiation using only local assumptions. Our main assumption was that the future horizon remains smooth. This is very reasonable — it is simply the statement that quantum corrections are small and do not affect the geometry at leading order. /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/quantum-aspects/lecture-17-naive-inform/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-17-naive-inform/ In this lecture, we first considered entanglement across the horizon. By using techniques from Lecture 16, we derived the two point function for modes across the horizon. Modes across the horizon are independent degrees of freedom but they are nevertheless correlated in a specific manner if the horizon is smooth. We then turned to the simplest version of the information paradox. If Hawking radiation is universal with no knowledge of the initial state, then it appears that the final state at future null infinity is universal regardless of which initial state we started from. /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/quantum-aspects/lecture-18-resolution-n/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-18-resolution-n/ The naive information paradox is easy to resolve. We can show that expectation values of an operator in a typical pure state are exponentially close to expectation values in a mixed state. So, unless we can measure correlators to an accuracy of e^{-S/2}, we cannot differentiate between a pure final state and a mixed final state. This means thatHawking’s computation is not precise enough to lead to a paradox. Furthermore, the relevant coupling constant when we take the back-reaction of Hawking radiation into account is 1/S. /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/quantum-aspects/lecture-19-page-curve/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-19-page-curve/ In previous lectures, we formulated and resolved the naive information paradox. As a prelude to refining the information paradox, in this lecture we analysed the rate at which information must emerge from a black hole. This requires an assumption that is stronger than just unitarity. But, using insights from generic finite-dimensional systems in random states, we argued that the von Neumann entropy of the emitted Hawking radiation obeyed the “Page curve”. /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/quantum-aspects/lecture-20-nice-slices-/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-20-nice-slices-/ After reviewing the Page curve, we discussed the construction of “nice slices.” These are Cauchy slices that stay away from the singularity in all regions, and can also be designed to have small extrinsic curvature at all points. Nevertheless, we can use these nice slices to capture the evolution of the black hole geometry all the way from the formation of the black hole to the point where it has substantially evaporated (but remains large in Planck units). /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/quantum-aspects/lecture-21-strong-subad/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-21-strong-subad/ In this lecture, we first reviewed the strong subadditivity of the von Neumann entropy. This is the statement that, given three systems, A, B, C, we have:S_{AB} + S_{BC} \geq S_{A} + S_{C}. We can apply this to black hole evaporation to obtain a paradox as follows. Consider an old black hole, past its Page time. Divide the geometry into three regions: a near-horizon region outside the black hole, B, an analogous region inside the black hole, C, and the rest of the exterior geometry, A. /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/quantum-aspects/lecture-22-black-hole-c/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-22-black-hole-c/ Both the cloning and the strong subadditivity paradox can be resolved by black hole complementarity. The phrase “complementarity” is used by different people to mean different things! However, broadly speaking, there are two perspectives on the issue: The degrees of freedom inside the black hole are “scrambled versions” of degrees of freedom outside the black hole. So the Hilbert space of the theory does not factorize into an interior and exterior. /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/quantum-aspects/lecture-23-toy-model-bl/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-23-toy-model-bl/ In this lecture, we discuss a simple and precise model of black hole complementarity based on arXiv:1603.02812. The formula that we wish to establish is: f(x) = P(f(y_1) … f(y_N)) where f(x) is a simple “local” field operator, and the points y_1, … y_N are all spacelike separated from x and P is a “very complicated” polynomial.Nothing about this formula is special to black-holes, and the simple example we consider to establish the formula is empty anti-de Sitter space. /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/quantum-aspects/lecture-24-paradoxes-la/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-24-paradoxes-la/ In this lecture, we turned our attention to the information paradox for large AdS black holes. Large AdS black holes don’t evaporate, so there is no information paradox in the usual sense. However, there is still an interesting question — which arises from a conflict between unitarity and effective field theory — which is as follows: “do generic AdS black holes have an empty interior that can be described holographically?” /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/quantum-aspects/lecture-25-state-depend/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects/lecture-25-state-depend/ In this final lecture, we discussed a state-dependent construction of the black hole interior. State-dependence is the idea that the observables used by the infalling observer depend on the state of the black hole. So, a seemingly well-defined observable like “what does the infalling observer see once he has spent 2 seconds in free fall past the black hole horizon” may secretly be an ambiguous observable. It requires more information to specify this observable precisely, and that additional information is the state of the system. /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 /blog/2026-06-01-measuring-cross-lingual-gaps-in-llms/ Mon, 01 Jun 2026 08:46:26 +0530 /blog/2026-06-01-measuring-cross-lingual-gaps-in-llms/ On the arXiv today, with Purvam Jain, Preethi Jyothi and Vihari Piratla. Modern LLMs have become increasingly good at a variety of languages. How does one detect cross lingual gaps in their abilities? We propose a set… /blog/2026-03-14-imperial-overreach-in-iran/ Sat, 14 Mar 2026 09:09:54 +0530 /blog/2026-03-14-imperial-overreach-in-iran/ This is an article I wrote 7 years ago analyzing the allegations against the Iranian nuclear program and the history of the conflict between the US and Iran. Reading it again, I think it is still useful to understand… /blog/2026-02-09-seeing-the-page-curve-with-blinders-on/ Mon, 09 Feb 2026 08:16:19 +0530 /blog/2026-02-09-seeing-the-page-curve-with-blinders-on/ Today, with Hao, Andreas, Mark, Lisa and Carlos: We tried to ensure that our main points are clear from the intro, even for a non-expert. I encourage you to click through and read at least the first few pages. Here is… /blog/2026-01-23-public-events-in-hyderabad/ Fri, 23 Jan 2026 18:57:26 +0530 /blog/2026-01-23-public-events-in-hyderabad/ If you are in Hyderabad over the weekend, here are a couple of fun events that I’ll be part of on Saturday. A panel discussion at the Hyderabad Lit. Fest and then a public outreach lecture at Lamakaan. Both events are… /blog/2026-01-22-a-model-of-errors-for-llms/ Thu, 22 Jan 2026 22:37:30 +0530 /blog/2026-01-22-a-model-of-errors-for-llms/ Here is something different, with Praneeth Netrapalli. LLMs have made spectacular progress over the past few years. Yet they still make errors on relatively simple tasks and we don’t have a good understanding of why… /blog/2025-09-10-the-debate-over-israel-s-suspension-from-the-ioaa/ Wed, 10 Sep 2025 11:48:58 +0530 /blog/2025-09-10-the-debate-over-israel-s-suspension-from-the-ioaa/ There has been debate on the suspension of Israel from the IOAA. Alok and I wrote this oped for the Hindu today, which can be read here: I would also recommend the article by Madhu and Aditi in the Scroll that appeared… /blog/2025-08-26-review-of-the-ai-con/ Tue, 26 Aug 2025 21:11:54 +0530 /blog/2025-08-26-review-of-the-ai-con/ I wanted to think through various issues at the interface of science and society in relation to recent progress in AI/ML. So I reviewed “The AI Con” by Emily Bender and Alex Hanna for The India Forum. If people get a… /blog/2025-08-20-israel-suspended-from-the-ioaa/ Wed, 20 Aug 2025 15:30:08 +0530 /blog/2025-08-20-israel-suspended-from-the-ioaa/ Israel has been suspended from the International Olympiad on Astrononomy and Astrophysics (IOAA). The IOAA is part of a network of science Olympiads that seek to identify the most talented high-school students in each… /blog/2025-08-01-summer-school-with-scientists-for-palestine/ Fri, 01 Aug 2025 02:05:17 +0530 /blog/2025-08-01-summer-school-with-scientists-for-palestine/ Very privileged to have attended the Advanced Summer School on Quantum Physics organized by Scientists for Palestine. I learned a lot from the courageous and generous Palestinian students who attended the event despite… /blog/2025-06-10-indiapakistan-academics-dialogue/ Tue, 10 Jun 2025 21:56:19 +0530 /blog/2025-06-10-indiapakistan-academics-dialogue/ Please join us for an interaction between academics from India and Pakistan on Thursday evening. See below for details and Zoom link. This meeting is open to everyone. 𝐈𝐧𝐝𝐢𝐚-𝐏𝐚𝐤𝐢𝐬𝐭𝐚𝐧 𝐚𝐜𝐚𝐝𝐞𝐦𝐢𝐜𝐬 𝐝𝐢𝐚𝐥𝐨𝐠𝐮𝐞 When nations are… /blog/2025-04-05-discussion-on-the-ugc-draft-regulations/ Sat, 05 Apr 2025 17:44:06 +0530 /blog/2025-04-05-discussion-on-the-ugc-draft-regulations/ Academics for Democratic Rights ( and India Academic Freedom Network ( are organizing a virtual discussion on the UGC Draft Regulations tomorrow: Sunday, 6 April at 3 pm. The discussion will be led by Dr. Furqan Qamar… /blog/2025-04-03-academic-freedom-at-harvard/ Thu, 03 Apr 2025 22:34:23 +0530 /blog/2025-04-03-academic-freedom-at-harvard/ I am surprised at how rapidly elite U.S. Universities have crumbled under pressure from the Trump administration and are cracking down on pro-Palestinian activism. Here is an email I sent to Harvard’s president today… /blog/2025-02-13-nuclear-liability-and-dangerous-concessions-to-reactor-vendors/ Thu, 13 Feb 2025 17:05:58 +0530 /blog/2025-02-13-nuclear-liability-and-dangerous-concessions-to-reactor-vendors/ In her budget speech earlier this month, the Finance Minister slipped in a seemingly unrelated remark: the government plans to amend the Indian nuclear liability act. This was deliberately timed so that the Prime… /blog/2025-01-24-introductory-quantum-field-theory-lectures/ Fri, 24 Jan 2025 19:31:42 +0530 /blog/2025-01-24-introductory-quantum-field-theory-lectures/ Lectures from my introductory QFT course last year: This is a fairly standard course that we offer at ICTS annually. I don’t think I said much that was new. We keep the Moodle open for anyone to join and view. A few… /blog/2024-12-31-locality-and-observers-in-quantum-gravity/ Tue, 31 Dec 2024 09:57:19 +0530 /blog/2024-12-31-locality-and-observers-in-quantum-gravity/ New paper to end the year with Kristan and Antony. Last year, with Jonathan, Antony and Kristan studied quantum information measures for subregions in gravitational theories. Some of their results might appear puzzling… /blog/2024-12-17-scholasticide-and-genocide-in-palestine/ Tue, 17 Dec 2024 15:11:42 +0530 /blog/2024-12-17-scholasticide-and-genocide-in-palestine/ Last month, as part of Academics for Democratic Rights, we organized a talk by Sundos Hammad and Ahmad Sabobeh who are part of the Right to Education campaign at Birzeit University in Palestine. The talk was titled… /blog/2024-11-17-achin-vanaik-in-bengaluru-palestine-and-the-crisis-in-west-asia/ Sun, 17 Nov 2024 15:53:04 +0530 /blog/2024-11-17-achin-vanaik-in-bengaluru-palestine-and-the-crisis-in-west-asia/ Achin Vanaik will be in Bengaluru to talk about the genocide in Palestine and the crisis in West Asia. We are having a talk at ICTS on Tuesday, 19 November 3:30-5 pm as part of our general talk series. Then, later in… /blog/2024-09-22-opposing-the-indiaisrael-business-summit-at-iisc/ Sun, 22 Sep 2024 18:17:05 +0530 /blog/2024-09-22-opposing-the-indiaisrael-business-summit-at-iisc/ IISc seems to think that this is the right time to promote an India-Israel business summit. More than 1500 students and faculty members from across the country, and other members of civil society have written to the… /blog/2024-08-13-phymaths-podcast-a-conversation-with-hassaan-saleem/ Tue, 13 Aug 2024 22:10:25 +0530 /blog/2024-08-13-phymaths-podcast-a-conversation-with-hassaan-saleem/ Fun podcast discussion with Hassaan Saleem. We discussed physics and life and some politics and some other questions! /blog/2024-05-09-an-email-to-harvard-on-student-protests-and-gaza/ Thu, 09 May 2024 20:52:17 +0530 /blog/2024-05-09-an-email-to-harvard-on-student-protests-and-gaza/ An email that I sent to Harvard’s interim president, Alan Garber, today. /blog/2024-04-25-letter-to-columbia-university-in-support-of-student-protesters/ Thu, 25 Apr 2024 08:54:20 +0530 /blog/2024-04-25-letter-to-columbia-university-in-support-of-student-protesters/ Some of us have drafted a letter to the President of Columbia University. The letter is available here: If you would like to endorse this letter, please use this Google Form: We plan to send it off to Shafik by the… /blog/2024-03-09-are-page-curves-relevant-for-real-black-holes/ Sat, 09 Mar 2024 17:11:43 +0530 /blog/2024-03-09-are-page-curves-relevant-for-real-black-holes/ Vasudev Shyam got Geoff Penington and me to be part of a podcast discussion today morning. We discussed whether islands and the Page curve are relevant for realistic black holes. As Geoff pointed out at the end, it is… /blog/2024-01-22-what-does-same-place-mean-over-time/ Mon, 22 Jan 2024 10:55:17 +0530 /blog/2024-01-22-what-does-same-place-mean-over-time/ Since I spend some of my time thinking about how to localize observations in gravity, I couldn’t help thinking about this question today. How do we decide that a structure we build today is at the same spatial location… /blog/2023-11-16-theoretical-physicists-statement-on-palestine/ Thu, 16 Nov 2023 21:19:01 +0530 /blog/2023-11-16-theoretical-physicists-statement-on-palestine/ A statement issued by an international group of theoretical physicists on the crisis in Palestine: /blog/2023-10-30-a-candlelight-call-for-palestine-in-bengaluru/ Mon, 30 Oct 2023 22:29:57 +0530 /blog/2023-10-30-a-candlelight-call-for-palestine-in-bengaluru/ The Karnataka government continues its appalling policy of preventing any public events in solidarity with Palestine. In Bengaluru, police shows up at venues before organizers do and even film screenings have been… /blog/2023-10-28-karnataka-s-ban-on-pro-palestine-protests/ Sat, 28 Oct 2023 08:23:10 +0530 /blog/2023-10-28-karnataka-s-ban-on-pro-palestine-protests/ The Karnataka government has clamped down completely on protests for Palestine. In any case, in Bengaluru, space for protests is sharply restricted and protests are allowed only in Freedom park. For those who haven’t… /blog/2023-08-06-oppenheimer-hiroshima-and-the-use-of-the-atomic-bomb/ Sun, 06 Aug 2023 13:55:33 +0530 /blog/2023-08-06-oppenheimer-hiroshima-and-the-use-of-the-atomic-bomb/ Today is Hiroshima day. I was troubled by the discourse around “Oppenheimer” as we approached this anniversary. Here is my dissenting take on the movie. The movie glosses over the fact that, during the second world… /blog/2023-07-14-scientists-need-the-oxygen-of-free-speech/ Fri, 14 Jul 2023 19:24:28 +0530 /blog/2023-07-14-scientists-need-the-oxygen-of-free-speech/ an article on the recent events in IISc and IISER Mohali. /blog/2023-07-03-academics-protest-iisc-s-cancellation-of-the-uapa-discussion/ Mon, 03 Jul 2023 19:19:00 +0530 /blog/2023-07-03-academics-protest-iisc-s-cancellation-of-the-uapa-discussion/ A letter from members of the academic community to the director of IISc protesting the events that I described in my last post. This was signed by more than 500 academics and scientists, including several members of… /blog/2023-06-28-iisc-cancels-a-discussion-on-the-uapa/ Wed, 28 Jun 2023 22:39:18 +0530 /blog/2023-06-28-iisc-cancels-a-discussion-on-the-uapa/ An update on the talks on the UAPA by Natasha and Devangana I posted about a few days ago. We had a talk at ICTS on Monday without incident. A talk was supposed to take place in IISc today but, at the last minute, the… /blog/2023-06-22-talks-by-natasha-narwal-and-devangana-kalita-on-the-uapa/ Thu, 22 Jun 2023 18:48:02 +0530 /blog/2023-06-22-talks-by-natasha-narwal-and-devangana-kalita-on-the-uapa/ Next week, we are planning to have three talks by Natasha and Devangana who will discuss the UAPA and its misuse through the lens of their own experience. Two of the talks are aimed at students in ICTS and IISc—a… /blog/2023-06-02-some-thoughts-about-reviewing/ Fri, 02 Jun 2023 07:45:09 +0530 /blog/2023-06-02-some-thoughts-about-reviewing/ I have always found the Nature publishing model distasteful. Fortunately, in high-energy theory we have rid ourselves of these scientific gatekeepers and publication is largely via the arXiv, with peer-review being… /blog/2023-03-30-holography-of-information-in-de-sitter-space/ Thu, 30 Mar 2023 13:09:41 +0530 /blog/2023-03-30-holography-of-information-in-de-sitter-space/ And here is the second one (again with Tuneer, Joydeep, Victor, and Priyadarshi. ) The question we ask and answer here is: how does the holography of information work in de Sitter space. The background to this is the… /blog/2023-03-30-the-hilbert-space-of-de-sitter-quantum-gravity/ Thu, 30 Mar 2023 13:07:38 +0530 /blog/2023-03-30-the-hilbert-space-of-de-sitter-quantum-gravity/ I am quite excited about a pair of papers that we put out today with Tuneer, Joydeep, Victor, and Priyadarshi. Here is the first one: The question we ask and answer here is: “What is the right Hilbert space for quantum… /blog/2023-02-03-academic-statement-against-blocking-the-bbc-modi-documentary/ Fri, 03 Feb 2023 15:04:22 +0530 /blog/2023-02-03-academic-statement-against-blocking-the-bbc-modi-documentary/ Our recent statement expressing our dismay at the government’s decision to block the BBC Documentary, India: the Modi question. Some news coverage is here: and here: and a few other places. /blog/2022-11-13-recent-black-hole-information-debates/ Sun, 13 Nov 2022 21:56:12 +0530 /blog/2022-11-13-recent-black-hole-information-debates/ nice non-technical summary by Nirmalya Kajuri of some of the recent debates on black-hole information. /blog/2022-10-06-magic-cards/ Thu, 06 Oct 2022 10:08:01 +0530 /blog/2022-10-06-magic-cards/ I’m often struck by how differently children think. My 9.5 year old son is fond of playing the following “magic trick” on his friends. The magician asks the subject to think of a number between, say, 1 and 63. Then the… /blog/2022-09-16-a-short-video-on-black-hole-information/ Fri, 16 Sep 2022 17:22:54 +0530 /blog/2022-09-16-a-short-video-on-black-hole-information/ A small outreach video on black hole information that we prepared recently. Many thanks to Anupam Ghosh and the ICTS outreach team for putting this together. Let me know if you have comments. (Even if they are… /blog/2022-07-01-excess-mortality-during-india-s-covid-19-pandemic/ Fri, 01 Jul 2022 09:44:21 +0530 /blog/2022-07-01-excess-mortality-during-india-s-covid-19-pandemic/ Recorded video of an important discussion hosted by the Indian Scientists Response to COVID-19. The panel had Rukmini S, Shankar and Prabhat Jha and discussed estimates of excess mortality during the pandemic. I… /blog/2022-03-25-recent-popular-media-coverage-of-black-hole-information/ Fri, 25 Mar 2022 11:10:09 +0530 /blog/2022-03-25-recent-popular-media-coverage-of-black-hole-information/ This is a post about the recent popular media coverage of the black hole information paradox. It is a post that I would have preferred not to make. However, I care about public outreach of science and I think that when… /blog/2022-03-12-eleven-years-after-fukushima/ Sat, 12 Mar 2022 08:35:46 +0530 /blog/2022-03-12-eleven-years-after-fukushima/ With Mv Ramana in The Hindu today on the eleventh anniversary of the Fukushima disaster. /blog/2022-03-08-the-sci-hub-case-and-the-future-of-scientific-publishing/ Tue, 08 Mar 2022 11:16:22 +0530 /blog/2022-03-08-the-sci-hub-case-and-the-future-of-scientific-publishing/ As many of you know, some time back Elsevier, Wiley and the American Chemical Society filed a suit against Sci-Hub in the Delhi High Court. A group of scientists, of which I am a part, filed an intervention petition in… /blog/2021-12-09-ias-talk-split-property-and-the-information-paradox/ Thu, 09 Dec 2021 16:14:15 +0530 /blog/2021-12-09-ias-talk-split-property-and-the-information-paradox/ Another talk on the split property and the information paradox at the Institute for Advanced Study workshop on Quantum Information and Spacetime. A little shorter than previous talks, so perhaps easier to follow. /blog/2021-11-19-pi-colloquium-split-property-and-the-information-paradox/ Fri, 19 Nov 2021 12:22:33 +0530 /blog/2021-11-19-pi-colloquium-split-property-and-the-information-paradox/ A colloquium I gave at the Perimeter Institute a few days ago. The idea was to target a broad theoretical physics audience and so much of it is relatively non-technical. The abstract below is hopefully self… /blog/2021-10-12-failure-of-the-split-property-in-gravity/ Tue, 12 Oct 2021 13:15:37 +0530 /blog/2021-10-12-failure-of-the-split-property-in-gravity/ The CQG journal is bringing out a special issue on the recent developments on the Page curve and the editors invited me to contribute something. So I took the excuse to write a “perspective essay” on the topic that… /blog/2021-09-26-cndp-statement-on-aukus-and-the-quad/ Sun, 26 Sep 2021 15:58:10 +0530 /blog/2021-09-26-cndp-statement-on-aukus-and-the-quad/ Our statement, as part of the Coalition for Nuclear Disarmament and Peace (CNDP), on AUKUS and the Quad dialogue. The full text is available here: /blog/2021-09-23-questions-about-pandemic-models/ Thu, 23 Sep 2021 12:01:38 +0530 /blog/2021-09-23-questions-about-pandemic-models/ If you are around today at 7 pm, please come for this discussion hosted by BSS Karnataka. We will discuss the public discourse around mathematical models that have been used for the pandemic in India. The talk will be… /blog/2021-08-02-holography-from-the-wheelerdewitt-equation/ Mon, 02 Aug 2021 07:32:48 +0530 /blog/2021-08-02-holography-from-the-wheelerdewitt-equation/ In a paper today, with Chandramouli Chowdhury, Victor Godet and Olga Papadoulaki, we establish a link between two famous but seemingly different approaches to quantum gravity. On one hand, we have the Wheeler-DeWitt… /blog/2021-07-09-inconsistency-of-islands-in-long-range-gravity/ Fri, 09 Jul 2021 09:44:43 +0530 /blog/2021-07-09-inconsistency-of-islands-in-long-range-gravity/ We report a surprising result today with Hao Geng, Andreas Karch, Carlos Pérez, Lisa Randall, Marcos Riojas and Sanjit Shashi. We show that the island rule leads to inconsistent results in standard theories of long… /blog/2021-04-18-black-hole-information-paradox-course-archive/ Sun, 18 Apr 2021 17:51:56 +0530 /blog/2021-04-18-black-hole-information-paradox-course-archive/ We have archived our recent ICTS course on the black hole information paradox. This was a research-level topical course covering several recent developments. The lecture archive is here: and some more resources… /blog/2021-04-03-the-black-hole-wars/ Sat, 03 Apr 2021 20:08:50 +0530 /blog/2021-04-03-the-black-hole-wars/ I was part of a “podcast with video” series on the “Black Hole Wars” that the students at DAMTP organized as part of the 2021 Cambridge Festival. This was the concluding episode in a series exploring the information… /blog/2021-02-08-caste-and-skin-colour-in-the-mahabharata/ Mon, 08 Feb 2021 18:59:49 +0530 /blog/2021-02-08-caste-and-skin-colour-in-the-mahabharata/ This is pretty far afield from the topics that I usually write about but after reading Rudrangshu Mukherjee’s article on “Dharma and Caste in the Mahabharata” in the India Forum I felt that perhaps a brief response was… /blog/2021-01-15-the-citizenship-amendment-act-and-academic-freedom/ Fri, 15 Jan 2021 19:24:20 +0530 /blog/2021-01-15-the-citizenship-amendment-act-and-academic-freedom/ An opinion piece I wrote for the Scientific American describing our response, as members of the Indian scientific community, to the Citizenship Amendment Act. Members of the scientific community have also spoken out on… /blog/2020-12-22-black-holes-and-the-reversibility-of-time/ Tue, 22 Dec 2020 19:57:22 +0530 /blog/2020-12-22-black-holes-and-the-reversibility-of-time/ A popular-level talk on Black Holes and the Reversibility of Time that I gave today to start off the new ICTS outreach series called the Vigyan Adda. I even talked about the principle of holography of information at… /blog/2020-12-11-lessons-from-the-information-paradox/ Fri, 11 Dec 2020 19:48:34 +0530 /blog/2020-12-11-lessons-from-the-information-paradox/ Here is another article that appeared today, reviewing some of the lessons that I think the information paradox teaches us about broader aspects of quantum gravity. I started this article with the plan of writing up… /blog/2020-12-10-information-transfer-with-a-gravitating-bath/ Thu, 10 Dec 2020 21:32:37 +0530 /blog/2020-12-10-information-transfer-with-a-gravitating-bath/ We have (in my unbiased opinion :-) ) an interesting paper out today: with Hao Geng, Andreas Karch, Carlos Perez-Pardavila, Lisa Randall, Marcos Riojas, and Sanjit Shashi. One of the significant problems we managed to… /blog/2020-11-04-bhi-talk/ Wed, 04 Nov 2020 07:55:34 +0530 /blog/2020-11-04-bhi-talk/ I’ve never physically been in the U.S. on the day of a presidential election, but at least I got to be there virtually yesterday at the Black Hole Initiative at Harvard! /blog/2020-10-07-holography-of-information-talks-at-stanford-and-perimeter/ Wed, 07 Oct 2020 12:51:50 +0530 /blog/2020-10-07-holography-of-information-talks-at-stanford-and-perimeter/ Here are some talks I gave at Stanford and at Perimeter in the past couple of weeks on the “Holography of Information from semiclassical gravity.” The talks cover somewhat similar ground, but the emphasis is slightly… /blog/2020-09-03-eradicate-nuclear-weapons-before-they-eradicate-us/ Thu, 03 Sep 2020 22:33:11 +0530 /blog/2020-09-03-eradicate-nuclear-weapons-before-they-eradicate-us/ Tomorrow (4 Sept), I’m going to speak at a webinar titled “Lets eradicate nuclear weapons before they eradicate us.” This is jointly organized by Rotary clubs in Cochin, Karachi and Michigan. I think there are a number… /blog/2020-09-02-qastm-seminar/ Wed, 02 Sep 2020 20:38:36 +0530 /blog/2020-09-02-qastm-seminar/ I’m going to describe some of our recent work in the QASTM seminar hosted by Sayantan tomorrow. I’ll try and keep the discussion as simple as possible, so please come and ask questions. (update: recorded video here /blog/2020-08-16-recording-asking-questions-about-mathematical-models/ Sun, 16 Aug 2020 19:27:59 +0530 /blog/2020-08-16-recording-asking-questions-about-mathematical-models/ The recorded video for the Chai and Why discussion today on “Asking questions about mathematical models” is available here: Thanks very much to Chai and Why? and to March For Science Mumbai for organizing the event… /blog/2020-08-15-asking-questions-about-mathematical-models/ Sat, 15 Aug 2020 09:56:18 +0530 /blog/2020-08-15-asking-questions-about-mathematical-models/ I’m going to lead a virtual discussion tomorrow (Sunday, 16 Aug, 11 AM) as part of the TIFR Chai and Why series titled: “Asking questions about mathematical models”. I promise to keep the discussion very simple. So if… /blog/2020-08-07-extracting-bulk-information-from-the-ads-boundary/ Fri, 07 Aug 2020 08:06:21 +0530 /blog/2020-08-07-extracting-bulk-information-from-the-ads-boundary/ With Chandramouli and Olga: “A physical protocol for observers near the boundary to obtain bulk information in quantum gravity”. Think of observers who live near the boundary of global AdS and can only explore the… /blog/2020-08-05-hiroshima-day-discussions-on-nuclear-disarmament/ Wed, 05 Aug 2020 21:41:26 +0530 /blog/2020-08-05-hiroshima-day-discussions-on-nuclear-disarmament/ I will be part of some panel discussions tomorrow (6 August). 1) The first is organized by the All India Peace and Solidarity Organization (MP). The topic is “75वें हिरोशिमा दिवस पर वेब-चर्चा” and I will speak about… /blog/2020-07-07-black-hole-information-at-strings-2020/ Tue, 07 Jul 2020 21:55:55 +0530 /blog/2020-07-07-black-hole-information-at-strings-2020/ A talk that I gave at the Strings 2020 conference: Since the Strings-conference audience is very broad, it was intended for people who are not directly thinking about the same questions, and provides a quick summary of… /blog/2020-06-27-did-india-s-lockdown-avert-covid-19-deaths/ Sat, 27 Jun 2020 08:19:43 +0530 /blog/2020-06-27-did-india-s-lockdown-avert-covid-19-deaths/ About a month ago, the government conducted a press-conference reporting the results of various “studies”, each claiming that the lockdown had saved thousands of lives. A few days later, some independent scientists… /blog/2020-05-29-india-s-lockdown-has-failed/ Fri, 29 May 2020 16:49:24 +0530 /blog/2020-05-29-india-s-lockdown-has-failed/ The government would like to claim that the lockdown has “saved lives”. But in this article for the Wire, I tried to explain why it makes no sense to gauge the efficacy of the lockdown by blindly comparing the current… /blog/2020-05-22-the-pandemic-hegemony-and-disarmament/ Fri, 22 May 2020 07:23:28 +0530 /blog/2020-05-22-the-pandemic-hegemony-and-disarmament/ Abolition 2000, which is a global network of nuclear disarmament groups asked 9 disarmament-campaigners across the world to record short videos in advance of its AGM that takes place on the weekend. The question we… /blog/2020-04-28-accountability-during-a-crisis/ Tue, 28 Apr 2020 06:17:10 +0530 /blog/2020-04-28-accountability-during-a-crisis/ An op-ed I wrote for the Hindu on the importance of accountability during a crisis. /blog/2020-04-25-virtual-seminars-and-a-talk-at-the-albert-einstein-institute/ Sat, 25 Apr 2020 14:58:52 +0530 /blog/2020-04-25-virtual-seminars-and-a-talk-at-the-albert-einstein-institute/ A virtual seminar I gave at the Albert Einstein Institute at Potsdam a few days ago A few comments on the topic of virtual seminars, which have become the norm over the past few weeks. Michal Heller invited me to give… /blog/2020-04-02-misleading-mathematical-models/ Thu, 02 Apr 2020 17:40:05 +0530 /blog/2020-04-02-misleading-mathematical-models/ This is an unusual post. I was reluctant to make it, but it seems to be necessary. A few days ago, the paper appeared, with various suggestions for controlling the epidemic in India: a lockdown of 49 days or a lockdown… /blog/2020-03-31-scientists-statement-on-the-lockdown/ Tue, 31 Mar 2020 10:31:29 +0530 /blog/2020-03-31-scientists-statement-on-the-lockdown/ Some of us have prepared a statement on the current lockdown and Covid-19 epidemic. It states that the lockdown, by itself, is not a permanent cure and so urges the government to prepare a post-lockdown plan. It also… /blog/2020-03-03-why-india-should-not-import-u-s-nuclear-reactors/ Tue, 03 Mar 2020 07:22:23 +0530 /blog/2020-03-03-why-india-should-not-import-u-s-nuclear-reactors/ Op-ed with Mv Ramana for the Hindu today: The idea that India should import reactors from the United States is a zombie idea that should have been dead long ago but keeps rearing its head from time to time. This… /blog/2020-02-07-the-holographic-nature-of-null-infinity/ Fri, 07 Feb 2020 09:58:30 +0530 /blog/2020-02-07-the-holographic-nature-of-null-infinity/ The holographic nature of null infinity! With Alok, Siddharth and Pushkal. I think that we have some exciting and interesting results. We clarify an important aspect of how holography should work in flat space — the… /blog/2020-01-25-anti-caa-protests/ Sat, 25 Jan 2020 08:04:09 +0530 /blog/2020-01-25-anti-caa-protests/ An interview that I did with rediff.com a few days ago: Its mostly just a set of simple questions and answers but some aspects might still be of interest. /blog/2020-01-07-iisc-student-march/ Tue, 07 Jan 2020 21:45:05 +0530 /blog/2020-01-07-iisc-student-march/ An absolutely incredible march organized by IISc students yesterday night, in which at least several hundred students marched through the campus. The video is very poor because I’m a poor videographer and because the… /blog/2019-10-09-a-quantum-test-for-smooth-horizons/ Wed, 09 Oct 2019 09:11:01 +0530 /blog/2019-10-09-a-quantum-test-for-smooth-horizons/ A simple quantum test for smooth horizons, with Kyriakos and Pushkal. We use the short distance behaviour of the two-point function to set out some very general necessary conditions for a quantum state of the fields to… /blog/2019-09-22-academics-speak-out-on-kashmir/ Sun, 22 Sep 2019 09:01:10 +0530 /blog/2019-09-22-academics-speak-out-on-kashmir/ Many of us have been disturbed by what is happening in Kashmir, and have felt helpless at our inability to do anything about it. Some of us felt that it would be very useful to have a statement from members of the… /blog/2019-07-05-talking-sanctions-endangering-peace/ Fri, 05 Jul 2019 03:04:11 +0530 /blog/2019-07-05-talking-sanctions-endangering-peace/ This is an op-ed I wrote for the Hindu on the U.S.-Iran conflict. /blog/2019-06-27-understanding-the-u-s-iran-conflict/ Thu, 27 Jun 2019 22:43:50 +0530 /blog/2019-06-27-understanding-the-u-s-iran-conflict/ My views on what is happening in Iran (for the blog and next issue of Aspects of India’s Economy). This piece is a little long — 6500 words — but I’ll try and write a shorter summary soon. (Update: see Hindu op-ed… /blog/2019-03-28-is-holography-implicit-in-canonical-gravity/ Thu, 28 Mar 2019 08:21:58 +0530 /blog/2019-03-28-is-holography-implicit-in-canonical-gravity/ New essay asking “Is Holography Implicit in Canonical Gravity?” The precise conjecture is that if two solutions of the Wheeler-deWitt equation coincide near the boundary of anti-de Sitter space at one point of time… /blog/2019-03-22-foxing-stockfirst-solution/ Fri, 22 Mar 2019 19:48:51 +0530 /blog/2019-03-22-foxing-stockfirst-solution/ After the chess puzzle in my previous post, there was a lot of discussion about chess programs, but no one seems to have attempted the puzzle itself :-) Its a difficult puzzle, but the solution is also really beautiful… /blog/2019-03-17-foxing-stockfish/ Sun, 17 Mar 2019 20:02:44 +0530 /blog/2019-03-17-foxing-stockfish/ At my son’s children’s chess tournaments, I sometimes meet a gentleman who rails against the recent advances in artificial intelligence. “Machines lie”, he roars. “Even in chess, they have no positional understanding."… /blog/2019-03-09-a-talk-on-a-toy-model-for-the-information-paradox/ Sat, 09 Mar 2019 08:32:32 +0530 /blog/2019-03-09-a-talk-on-a-toy-model-for-the-information-paradox/ Following the last post, I got a number of additional questions about the paper presenting a toy model for the information paradox. This is a talk I gave at a workshop in Princeton some time ago. The organizers at the… /blog/2019-03-05-bell-correlators-and-the-amps-paradox/ Tue, 05 Mar 2019 07:05:35 +0530 /blog/2019-03-05-bell-correlators-and-the-amps-paradox/ Not a new paper but, in this version, I’ve added some comments on the difference between gauge theories and gravity and the relevance to the AMPS paradox, about which I got a number a questions. It seems to have been… /blog/2019-03-01-a-conversation-in-delhi/ Fri, 01 Mar 2019 07:07:12 +0530 /blog/2019-03-01-a-conversation-in-delhi/ I had a sad and startling conversation with a taxi driver the day before yesterday. I had gone to Ashoka University to give a talk, and the ride to Delhi airport is almost a couple of hours. I hadn’t seen the news in… /blog/2019-02-20-bounds-on-thermal-relativistic-correlators/ Wed, 20 Feb 2019 08:29:05 +0530 /blog/2019-02-20-bounds-on-thermal-relativistic-correlators/ We considered relativistic correlation functions at finite temperature in the limit where the momenta of the operators become spacelike and large. Surprisingly, these correlators are bounded by a beautiful geometric… /blog/2019-02-17-the-crisis-in-venezuela/ Sun, 17 Feb 2019 17:06:03 +0530 /blog/2019-02-17-the-crisis-in-venezuela/ some thoughts on the situation in Venezuela. /courses/quantum-aspects-black-holes-and-information-paradox/ Mon, 01 Jan 0001 00:00:00 +0000 /courses/quantum-aspects-black-holes-and-information-paradox/ Starting with a review of the classical aspects of black holes, this course continues on to quantum aspects of black holes. We review the classical aspects of black hole thermodynamics, the derivation of Hawking radiation, and examine different formulations of the information paradox. Towards the latter half of the course, we discuss some advanced topics, including black hole complementarity and the information paradox in AdS/CFT. Course Materials # Advanced Topics (Archival) # Lecture 25: State-dependence in the black hole interior Lecture 24: Paradoxes for Large AdS Black Holes Lecture 23: A Toy Model of Black Hole Complementarity Lecture 22: Black Hole Complementarity Lecture 21: The Strong Subadditivity Paradox Lecture 20: Nice slices and the Cloning Paradox Lecture 19: The Page Curve Core Lectures # Lecture 18: Resolution to the naive Information Paradox (Content available in latest BH Course) Lecture 17: The Naive Information Paradox Lecture 16: A Local Derivation of Hawking Radiation Lecture 15: Hawking Radiation Through Ray-Tracing Lecture 14: Black Hole Thermodynamics Lecture 13: The Area theorem Lecture 12: The second law and the Raychaudhuri Equation Lecture 11: The Kerr Black Hole Lecture 10: Charged Black Holes Lecture 8: More on the Schwarzschild black hole and dust collapse Mathematica: Geodesics in Oppenheimer-Snyder Lecture 7: The Schwarzschild Black Hole Mathematica: Schwarzschild Solution Mathematica: Animation for light source falling into BH Lecture 6: More on Rindler Space and Unruh detectors Lecture 5: More on Rindler Space Lecture 4: Rindler Space Lecture 3: Examples of QFT in curved space Lecture 2: Framework of QFT in Curved Space Lecture 1: Overview