Hello everyone
I decided to start this blog page for those following my lecture series at ICTS, Bengaluru on Advanced General Relativity - A Centennial Tribute to Amal Kumar Raychaudhuri.
On this page you will find the lecture notes, exercises and comments/corrections on the course. You may also post your questions here and I will answer them as and when I can.
I advise those following the course on a regular basis to subscribe to notifications, that way you will receive an email every time there is a new post.
There is also an ICTS webpage where you will find the course details, Zoom link as well as links to the videos. These are handled by ICTS.
Information about pre-requisites and sources is available on the ICTS page but I'll also reproduce it here:
ADVANCED GENERAL RELATIVITY
Sunil Mukhi
Outline for an online course organised by ICTS, Bengaluru,
March 20 – April 30, 2023.
The goal of this course is to provide a brief, but hopefully precise, overview of topics relevant to current research on General Relativity. The course will focus exclusively on classical aspects of gravity and, towards the end, on the quantum behaviour of particles/fields in gravitational backgrounds.
List of topics:
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Brief review of basic General Relativity and relevant mathematics. [1]
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Geometry and topology of spacetimes: causality, globally hyperbolic space-times, Cauchy surfaces. [2]
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Geodesics, focusing and Raychaudhuri equations, Penrose and Hawking singularity theorems. [3]
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Review of Schwarzschild black holes, asymptotically deSitter and anti-deSitter space- times. [1]
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Black holes: causality properties, cosmic censorship. [2]
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Thermal properties of Rindler space and black holes, Hawking radiation, black hole entropy and thermodynamics. [3]
Total: 10-12 lectures of 90 minutes each. [ ] = tentative number of lectures assigned to each topic. The topics may evolve slightly as the course progresses.
Pre-requisites: The course should be accessible to anyone who has had a one-semester course on basic General Relativity as well as some exposure to basic Topology and Differential Geometry. In GR, familiarity will be assumed with space-time metrics, general coordinate invariance, Riemann curvature, Einstein equations, Einstein-Hilbert action, isometries, geodesics and some classical solutions. In TDG, familiarity will be assumed with topological spaces, separability, connectedness, compactness, homotopy, differentiable manifolds.
References for the course:
Edward Witten, “Light Rays, Singularities and All That”, arXiv:1901.03928. Robert Wald, “General Relativity” (textbook).
Tom Hartman “Lectures on Quantum Gravity and Black Holes” (Chapters 2 and 3 only!), downloadable at http://www.hartmanhep.net/topics2015.
Aron Wall, “Survey of Black Hole Thermodynamics’’, arXiv: and references therein, arXiv: 1804.10610.
References for the pre-requisites:
(i) Basic General Relativity:
• James Hartle, “Gravity – An Introduction to Einstein’s General Relativity”
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Sean Carroll, “Spacetime and Geometry: An Introduction to General Relativity”, Chapters 1—4.
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Steven Weinberg, “Gravitation and Cosmology”. Chapters 1—7, 12, 13.
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Sunil Mukhi, “Classical General Relativity” (22 video lectures),
https://youtu.be/f7LdeEKzIwY
(ii) Topology and Differential Geometry:
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Sunil Mukhi and N. Mukunda, “Lectures On Advanced Mathematical Methods For Physicists”, Chapters 1—4.
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Sunil Mukhi, “Topology and Differential Geometry for Physicists”, (11 video lectures) https://youtu.be/F_Ug6y54wJ4
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I.M. Singer and J.A. Thorpe, “Lecture Notes on Elementary Topology and Geometry”.
The first set of lecture notes will appear on this blog page, tomorrow after the lecture or on Saturday.
Respectful Sir The concept of Diffeomorphism is used extensively in GR...Sir we will be very thankful to you if you can accomodate the idea of diffeomorphism between 2 manifold or diffeomorphism etween 2 points in same spacetime manifold...Also Sir we will be very thankful if you can help us know diffeomorphism between points that are null seperated or spacelike seperated somewhere in the course...
ReplyDeleteThere is no such thing as "diffeomorphism between two points". A diffeomorphism of manifolds is a homeomorphic (topology-preserving) map from one to the other, which is differentiable and its inverse is also differentiable. Thus, a diffeomorphism is an equivalence that preserves the differentiable nature of a manifold.
DeleteOk Thankyou so much Sir :)...
DeleteRespectful Sir, Suppose we consider the Event Horizon of a Black Hole and a region just outside it(Which is timelike)...So that point P lies on Null Hypersurface or spacelike surface(if it is in the interior of Black Hole) and point Q in its neihbhourhood lies in a timelike hypersurface...Sir is it true that we can find Q to be Convex normal neighbourhood of P? Sir maybe we can find paths going from Q to P but is it possible to find paths from P to Q , so that Q is Convex normal neighbhourhood of P?
DeleteSir I am very sorry I will correct my question here...Sir given a point P(along timelike Hypersurface in suppose Schwarschild Spacetime) just outside the null hypersurface can we consider pair of points Q and R in its neighbhourhood so that Q lies in a spacelike hypersurface and R in timelike Hypersurface, is it possible that both Q and R lie in the convex neighbhourhood of P...Sir there can be paths from R to Q but I am a bit worried that there may not be paths from Q to R...
DeleteThe question does not seem to be well-posed. There is no meaning to statements like "given a point along a time-like hypersurface" or "Q lies in a space-like hypersurface". Any point in space-time lies in infinitely many space-like hypersurfaces, and also in infinitely many time-like hypersurfaces, and also in infinitely many null surfaces. Also, if there is a path from R to Q then the same path (with the parameter reversed) is a path from Q to R. You may want to look carefully at all the definitions again.
DeleteOk Sir , Sir I am worried that due to Causal structure of space-time, It may not be possible to have paths from a point which is in a Spacelike Hypersurface to a point which is in time like Hypersurface 🙏🏻🙏🏻🌼🌼...Sir please correct me if I am wrong 🙏🏻🙏🏻🌼🌼...Also Sir I could not understand that a point can be in a infinitely many Spacelike Hypersurface and infinitely many Time like Hypersurface
DeleteWhen one is confused one should look at 1+1 d Minkowski spacetime. Consider the origin (x,t)=(0,0). It lies in every straight line through the origin. Lines whose slope is >45 degrees are timelike, lines whose slope is <45 degrees are spacelike and lines whose slope is exactly 45 degrees are null. And these are only straight lines. There are also many curved lines that are spacelike or timelike and pass through the origin.
DeleteOk Sir I got it Thankyou so much Sir :)...
DeleteProfessor, is the nature of the manifold preserved if say a Rindler metric is induced for a non inertial frame?
ReplyDeleteYes, the manifold hasn't changed (although Rindler coordinates may cover only a part of the original manifold). What changes is the physical interpretation of what the observer sees.
ReplyDeleteProfessor, is taking infinite balls from P to boundary of P then repeating at the next P equivalent to foliation of the space from P to boundary?
ReplyDeleteI'm not sure of the context of this question, but it seems to be about the proof that sequences of curves from P to Q converge. In that case, I don't think this is a foliation. A foliation is a slicing of a manifold, while here we are trying to construct the convergence curve of the sequence. For this we take finite-size nested balls around a point P to construct a segment of the convergence curve, and then repeat from a different point to construct another segment and so on.
ReplyDelete