# Sean carroll spacetime and geometry an introduction to general relativity pdf

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Providing an introduction to general relativity for advanced undergraduates and graduate students, this work leads readers from physics of flat spacetime, through the intricacies of differential geometry and Einstein's equations, and on to exciting applications such as black holes, gravitational radiation, and cosmology. Read more

## Lecture Notes on General Relativity

Albert Einstein - spacetime diagram for two black holes colliding to become one Einstein with Tagore General Introduction The purpose of this class: This class will provide an overview of the theory of general relativity, Einstein's theory of relativistic gravity, as well as some basic applications, including at least the solar-system tests of gravitational theories,some of the more interesting properties of black holes and gravitational waves, along with some surveys of cosmology.

This class will not completely prepare you for research in this area: it will be an overview with insufficient depth for that purpose. However, that is more likely than not exactly what you wanted anyway. The first third to half of the course will focus primarily on the basic structure of the theory, with relevant physical motivation and insight thrown in along the way, and also provide a reasonable introduction to the needed mathematics. You do NOT need to already know more physics and mathematics than is described in the Prerequisite section just below.

The major applications will come after that, although perhaps some discussion of motions around spherical stars, and weak gravitational waves will come in the earlier sections. Prerequisites: I assume you have a good foundation in standard undergraduate physics: classical mechanics, electromagnetism, and the usual junior-level special relativity. Also you should have a mathematics background in calculus, differential equations, and linear algebra. The mathematics of general relativity is differential geometry, but I am not assuming you have had any studies on that before: we will spend a good fraction of the first portion of the course learning the relevant differential geometry.

Only the basic ideas of spacetime, 4-vectors, Minkowski diagrams, etc. Textbooks and Syllabus: The general Syllabus for the course is at this link. It proceeds week by week, and should be seen as an ordered listing of the majority of things we want to talk about; however, you will see it has two different weeks scheduled as "catch-up" if and when we have fallen behind in the ordering. I will do this by fairly closely following one or more of my own handouts, listed below; therefore, I recommend you spend the time to read them carefully.

However, I do believe that it is useful to have a printed textbook as well, so I have asked the bookstore to provide two different books for your consideration, listed there as optional texts, and named below.

There are many textbooks on general relativity, and these two are reasonably different in their approach, so that they form a nice complimentary pair.

I will be referring to them from time to time; however, you do not necessarily need both but, rather, choose the one that best fits your own interests and approach. Also see comments below, concerning them. This book is a greatly-expanded version of Sean Carroll's earlier "Lecture Notes on General Relativity," which can still be acquired online, for free, from a page on his current website. He notes that about half of the book is newer than the lecture notes, that much of the material in the lecture notes was "polished and improved" when the book was published, and that there are many more worked-out examples, so that one should surely also purchase the book.

There is also quite an intriguing list of possible alternative ways to think about the universe and cosmology in a list of current research interests of Carroll's, given at this link. The text itself has especially good descriptions of Lie derivatives, and of Penrose diagrams, among other things, and his section explaining the concept of manifolds is so good I could only wish I had written it.

Press It has a nice, simple discussion of the Thomas precession in special relativity, and some approaches to the evolution of perturbations in the universe, which are the best I have seen anywhere. In addition, it also provides an understanding of what it means to say that gravity is an "emergent" theory, and some older introductions to quantizing gravity.

The emphasis is always based on trying to agree with, and explain, the observations. It is unfortunate that it does not have the very latest material on the Buchert equations concerning difficulties with averaging nonlinear processes relevant to the possibility of "dark energy. The first one is a listing of some books at a "popular" level , that concern ideas in general relativity.

The second is a listing of technical books, mostly not quite so recently-published , that are certainly still useful. As already stated, I will actually be following handouts of my own creation, presented as pdf-files on this website, which should be read, at least more or less, in the order listed below. Handouts to supplement the texts: parts of the course will follow these closely. Introductory comments about tidal gravitational forces, and geometry , 11 pages. A brief review of special relativity , along with some notational conventions, 25 pages, somewhat revised.

A useful summary of the Lorentz transformations of several useful physical quantities , 6 pages. In some sense all of the above handouts have dealt with review, or physical motivation. At this point we begin considering the mathematical needs for general relativity! Important notes on Covariant Derivatives and Curvature ; 73 pages. This contains the physical interpretations for the ideas named in the title, and is extremely important.

It also has the clearest discussion of non-holonomic tetrad basis sets, best for physical interpretations of components of vectors. Some useful notes on ways to view the curvature tensor, and its parts. Various very recent papers concerning the January, observations by LIGO of gravitational waves emitted as two black holes merged: The original Physical Review Letters paper.

From here on, the handouts consider various specific applications to physical systems. A summary of local properties of spherically symmetric, static spacetimes ; 9 pages; and also some notes on the Kruskal extensions. Discussion of observations made by a uniformly accelerating observer ; 15 pages. The Kerr metric , for rotating stellar objects: some rather brief listings of properties and equations; 4 pages. An example calculation showing an erroneous method for the determination of the affine connections for the Kerr metric.

Penrose conformal diagram for maximal extension of the Kerr manifold the important, original paper on rotating black holes: Rotating Black Holes: Locally Nonrotating frames, energy extraction, and scalar synchrotron radiation , by James M.

Bardeen, William H. Press, and Saul A. Teukolsky, The Astrophysical Journal, , A discussion of Lie derivatives and Killing vectors ; 15 pages All the notes below here concern many different opinions on the very intriguing subject of Cosmology : Notes on Robertson-Walker [or FLRW] Spacetimes: solutions of Einstein's equations where homogeneity and isotropy of the entire universe on average, over some scale are assumed, and, further, that the matter can be modelled by a perfect fluid with everywhere-defined flow lines.

Sometimes referred to as the "concordance" model, which, perhaps, means that this is the most well-agreed-upon model for the background space for the universe, from which perturbation theory should begin. Some background as to the controversy: Pedagogical discussion of the problems concerning horizons in FLRW Cosmologies , published in the American Journal of Physics, , in Excellent review article on the 83 years of General Relativity and Cosmology , 39 pages.

More recent, quite brief summary, posing questions to be answered A brief article by George Ellis, on probable need for using inhomogeneous cosmological, concerning "patched together" solutions, from An article from the Scientific American on difficulties with the idea of dark energy, A careful study of Inhomogeneity and the foundations of concordance cosmology , by Clarkson and Maartens, , from Classical and Quantum Gravity.

A detailed proposal, of the status, in , of dark energy and alternative explanations of the problems with the data. A more detailed list of references for some of the disparate current Work on Cosmologies alternative to the FLRW isotropic and homogeneous model. Quite a few different groups of workers noted. I will make no comments here about the proposal for inflation, a process involving quantum phenomena, which many believe probably occurred "before the current era. However, a particular interesting article is the one below: A particular explicit model for inflation: the emergent universe, Exams and Homework Assignments: There will be two examinations, currently scheduled for 21 March and 27 April, but no final examination.

In addition, there will be more or less weekly homework assignments , with solutions posted after they have been turned in. The grader for the course is Stephen Keith Sanders. If you wanted to speak with him, note that he usually attends class, but you could also email him for a meeting time and place. Solutions for HW 6 An Exam on Monday, 21 March, You may bring any personally-written material with you, or my handouts for this class.

Solutions are available here. HW 7, due Monday 28 March. Homework assignments and Solutions are pdf-files, except when occasionally there will be an html-file for a portion of the solutions. Solutions will be made available once the assignments have been turned in. Homework is DUE at the beginning of the class period on the due date! There are many modern software packages to perform tensor calculations.

I prefer the program grtensor , which is described in more detail in this linked webpage. After you have a reasonably-good understanding of how the process works , I see no reason why you shouldn't have an algebraic computing system do the work for you. Very interesting discussion and movies of both orbiting around and falling into a vacuum, Schwarzschild black hole.

Done by Andrew Hamilton, whose main home page has many other interesting links about special and general relativity and interesting things in the sky. Some interesting movies showing the fact that when one uses light rays to view very-fast-moving, 3-dimensional objects they appear to rotate, were made by Leo Brewin at Monash University in Australia. I have copied two of the movies that I liked the best, which may be found here: for a fast-moving Rubik's Cube moving toward you, and for a fast-moving steam locomotive moving past you.

They move fast; therefore, it is most interesting if you actually slowly "drag" the play button along while watching. An interesting history of the ideas in general relativity , beginning with Aristotle and Copernicus, along with many further links to biographical information on the researchers involved, can be found at this link, created by people at St. Andrews University in Scotland. A year-long course on General Relativity is taught every year at Cal.

This link takes you to the webpage for that class, created by Marc Kamionkowski. Finley's page of Other Interesting Links for Relativity From time to time, a student asks a question which is too complicated to fully discuss in class. If possible I will then create a webpage with a listing of articles appropriate to answering that question. At the beginning of the semester there were two such listings here. This paper seems to provide a correct answer to an old and somewhat controversial question.

Some earlier responses to this question are in this paper. Questions somewhat related to the radiation of charges are those concerning the backreaction of particles in a gravitational field. They are discussed very carefully and rigorously in this paper of Eric Poisson. Is the concept of a gravitonas a spin 2, massless object "something like" a photon" really an idea consistent with full, nonlinear general relativity?

This paper says no , although this is still a very controversial issue. This put here since there were quite a few questions about it at the last class, and there wasn't really time to discuss it more. However, the new book by Griffiths and Podolsky, listed in my online list of other books, is much more complete on this question.

Very interesting history of Gravity Probe B , As well, there is. COBE satellite data on cosmic microwave background Very interesting background primer on Cosmology and the astronomical measurements that allow us to make inferences about it. Click here to mail your comments and suggestions concerning the Homepage. Click here to go to Finley's own Home Page.

## Spacetime and geometry : an introduction to general relativity

These lecture notes are a lightly edited version of the ones I handed out while teaching Physics 8. Each of the chapters is available here as PDF. What is even more amazing, the notes have been translated into French by Jacques Fric. Je ne parle pas francais, mais cette traduction devrait etre bonne. Dates refer to the last nontrivial modification of the corresponding file fixing typos doesn't count. Note that, unlike the book, no real effort has been made to fix errata in these notes, so be sure to check your equations.

Sign in Create an account. Syntax Advanced Search. Sean M. Carroll California Institute of Technology. General Relativity in Philosophy of Physical Science.

Sean M. Carroll ductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three the spacetime interval — the metric — Lorentz transformations You may be concerned that this introduction to tensors has been.

## General Relativity - Fall 2006

An essential resource for learning about general relativity and much more, from four leading experts. Important and useful to every student of relativity, this book is a unique collection of some problems--with solutions--in the fields of special and general relativity, gravitation, relativistic astrophysics, and cosmology. The problems are expressed in broad physical terms to enhance their pertinence to readers with diverse backgrounds. In their solutions, the authors have attempted to convey a mode of approach to these kinds of problems, revealing procedures that can reduce the labor of calculations while avoiding the pitfall of too much or too powerful formalism. Although well suited for individual use, the volume may also be used with one of the modem textbooks in general relativity.

The course contents are:. A handout with this information will be distributed at the first lecture Monday January 7, and is also available in pdf format here. Lectures meet Mondays from to and on Thursdays from to in HH Attendance to the lectures is not compulsory, but if you come I ask you to pay attention and not disrupt the class with personal conversation.

This page collects any mistakes that people have been able to find in the book.

### General Relativity Autumn 2013

Albert Einstein - spacetime diagram for two black holes colliding to become one Einstein with Tagore General Introduction The purpose of this class: This class will provide an overview of the theory of general relativity, Einstein's theory of relativistic gravity, as well as some basic applications, including at least the solar-system tests of gravitational theories,some of the more interesting properties of black holes and gravitational waves, along with some surveys of cosmology. This class will not completely prepare you for research in this area: it will be an overview with insufficient depth for that purpose. However, that is more likely than not exactly what you wanted anyway. The first third to half of the course will focus primarily on the basic structure of the theory, with relevant physical motivation and insight thrown in along the way, and also provide a reasonable introduction to the needed mathematics.

Spacetime and Geometry is a graduate-level textbook on general relativity. It is exactly the same book , just with a different cover. Buy it: Amazon. A lonely, uncompensated, perhaps even impossible Task, yet some of us must ever be seeking, I suppose. In I taught a one-semester graduate course in general relativity at MIT. Along the way I typed up a detailed set of lecture notes. The book Spacetime and Geometry is a significantly revised and expanded version of these notes; about half of the finished book is completely new.

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The homework problem sets are not optional. You are encouraged to discuss the class material and homework problems with your classmates and to work in groups, but all submitted problems should represent your own work and understanding. The final exam will be held in Nat. Annex and will cover the complete course material. You must take the final exam to pass the course.

Очевидно, волнение отняло у него все силы. Его лицо залила мертвенная бледность. Беккер предпринял последнюю попытку: - Мистер Клушар, я хотел бы получить показания этого немца и его спутницы. Вы не скажете, где они могли остановиться. Клушар закрыл глаза, силы покинули. Он едва дышал.

Он. Беккер был уверен, что представляет собой отличную мишень, даже несмотря на то что находился среди огромного множества прихожан: его пиджак цвета хаки ярко выделялся на черном фоне. Вначале он хотел снять его, но белая оксфордская рубашка была бы ничуть ни лучше, поэтому он лишь пригнулся еще ниже.