/courses/quantum-aspects/ Recent content in Quantum Aspects of Black Holes on Suvrat Raju Hugo -- gohugo.io en © 2026 /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/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/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/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/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/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/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/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/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/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/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/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/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/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/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/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/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/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/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/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/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/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.