Sean Carroll Lecture On General Relativity

[Arleen Jerdee]

Thisis a course on general relativity, given to Part III (i.e. masters level) students. It covers advanced material, but is designed to be understandable for students who haven't had a first course in the subject. Please do email me if you find any typos or mistakes.

2. Introducing Differential Geometry: PDF

Manifolds: Topological spaces, differentiable manifolds and maps between manifolds. Tangent Spaces: tangent vectors, vector fields, integral curves and the Lie derivative. Tensors, covectors and one-forms. Differential Forms: the exterior derivative, de Rahm cohomology, integration and Stokes' theorem. 3. Introducing Riemannian Geometry: PDF

The metric; Riemannian and Lorentzian manifolds, the volume form and the Hodge dual. The Maxwell action. Hodge theory. Connections and the covariant derivative, curvature and torsion, the Levi-Civita connection. The divergence theorem. Parallel transport, normal coordinates and the exponential map, holonomy, geodesic deviation. The Ricci tensor and Einstein tensor. Connection 1-forms and curvature 2-forms. 4. The Einstein Equations: PDF

The Einstein-Hilbert action, the cosmological constant; diffeomorphisms and the Bianchi identity; Minkowski, de Sitter and anti-de Sitter spacetimes; Symmetries and isometries, Killing vectors, conserved quantities; Asymptotics of spacetime, conformal transformations and Penrose diagrams; Coupling matter, the energy-momentum tensor, perfect fluids, spinors, energy conditions; Cosmology. 5. When Gravity is Weak: PDF

The Linearised theory, gauge symmetry, the Newtonian limit; Gravitational waves, de Donder gauge, transverse traceless gauge, LIGO; Gravitational wave production, binary systems, the quadrupole formula, gravitational wave sources. 6. Black Holes: PDF

The Schwarzschild solution, Birkhoff's theorem, Eddington-Finkelstein Coordinates, Kruskal diagrams and Penrose diagrams, weak cosmic censorship; The Reissner-Nordstrom solution, Cauchy horizons and strong cosmic censorship, Extremal black holes; The Kerr solution, global structure, the ergoregion, the Penrose process and superradiance, no hair theorems.

Problem SheetsJoo Melo has put together a preparatory worksheet, based on Chapter 1 of the lectures notes, to help refresh your understanding of geodesics before the course begins. It can be downloaded here.

General relativity is one of the towering achievements of modernphysics, the best answer we currently have to the question 'what isgravity?'. Here is where you get to grips with it - the maths can be abit gory, but by the end you should understand the Einstein equations!These enable us to describe how mass/energy curves spacetime, whichgives rise to the effects we used to call 'gravity'! After eachlecture I will link material to this page - so keep checking to seewhat is here (and hit reload to make sure you are getting the most upto date version!). These lecture notes are NOT a substitute forattending the lectures. But do look at them because I sometimes edit themAFTER the lecture, so I emphasise and try to find other ways ofexplaining any points which were obviously an issue in the lecture.

There is a highly recommended web sit of Sean Carroll's lecture notes on general relativity.I especially like hisNo-Nonsense Introduction to General Relativity. Only thing towatch is that he uses the opposite sign convention on his metric!His links are worth checking out as well.A very different approach (much more along the pure mathematics,differential geometry line) is anIntroduction to Differential Geometry and General Relativity.But its got some good pictures in it. And an excellent essay onfundamental meaning of GR (and quantum mechanics) I once did an experimental DU astrosoc talk onBlack holes - this was midway between a lecture and my usual'edutainment' approach to public talks.

There are also some fun relativity pages on the web

Popular science (non technical sites) include spacetime wrinkles.There are also some good visualisation sites like falling into ablack hole and a make your own orbits around a black hole (java appletsite).

Sean Michael Carroll (born October 5, 1966) is an American theoretical physicist and philosopher who specializes in quantum mechanics, cosmology, and philosophy of science. Formerly a research professor at the Walter Burke Institute for Theoretical Physics at the California Institute of Technology (Caltech) department of physics,[1] he is currently an external professor at the Santa Fe Institute,[2] and the Homewood Professor of Natural Philosophy at Johns Hopkins University.[3][4] He has been a contributor to the physics blog Cosmic Variance, and has published in scientific journals such as Nature as well as other publications, including The New York Times, Sky & Telescope and New Scientist. He is known for his atheism, his vocal critique of theism and defense of naturalism.[5][6][7][8] He is considered a prolific public speaker and science populariser.[8][9][10] In 2007, Carroll was named NSF Distinguished Lecturer by the National Science Foundation.[11]

He has appeared on the History Channel's The Universe, Science Channel's Through the Wormhole with Morgan Freeman, Closer to Truth (broadcast on PBS),[12] and Comedy Central's The Colbert Report. Carroll is the author of Spacetime And Geometry, a graduate-level textbook in general relativity, and has also recorded lectures for The Great Courses on cosmology, the physics of time and the Higgs boson.[13] He is also the author of four popular books: From Eternity to Here about the arrow of time, The Particle at the End of the Universe about the Higgs boson, The Big Picture: On the Origins of Life, Meaning, and the Universe Itself about ontology, and Something Deeply Hidden about the foundations of quantum mechanics. He began a podcast in 2018 called Mindscape, in which he interviews other experts and intellectuals coming from a variety of disciplines, including "[s]cience, society, philosophy, culture, arts and ideas" in general.[14] He has also published a YouTube video series entitled "The Biggest Ideas in the Universe" which provides physics instruction at a popular-science level but with equations and a mathematical basis, rather than mere analogy. The series has become the basis of a new book series with the installment, The Biggest Ideas in the Universe: Space, Time, and Motion, published in September 2022.[15]

Carroll received his PhD in astronomy in 1993 from Harvard University, where his advisor was George B. Field. His dissertation was entitled Cosmological Consequences of Topological and Geometric Phenomena in Field Theories. He worked as a postdoctoral researcher at the Massachusetts Institute of Technology (MIT) and Kavli Institute for Theoretical Physics at the University of California, Santa Barbara[16] and as an assistant professor at the University of Chicago until 2006 when he was denied tenure.[17] He is the Homewood Professor of Natural Philosophy at Johns Hopkins University, teaching in both the Department of Philosophy and The Department of Physics and Astronomy.

In 2010, Carroll was elected fellow of the American Physical Society for "contributions to a wide variety of subjects in cosmology, relativity and quantum field theory, especially ideas for cosmic acceleration, as well as contributions to undergraduate, graduate and public science education".[20] In 2014, he was awarded the Andrew Gemant Award by the American Institute of Physics for "significant contributions to the cultural, artistic or humanistic dimension of physics".[21] In 2015, he was awarded a Guggenheim Fellowship.[22]

He is also a very prolific public speaker, hosting the podcast series Mindscape,[23] which he describes as "Sean Carroll hosts conversations with the world's most interesting thinkers", and The Biggest Ideas in the Universe.[24] He also delivers public speeches as well as getting engaged in public debates in wide variety of topics.

Carroll has appeared on numerous television shows including The Colbert Report and Through the Wormhole.[25] He also worked as a consultant in several movies[26][27] like Avengers: Endgame[28] and Thor: The Dark World. Besides consulting, Carroll worked as a voice actor in Earth to Echo.[29]

Carroll has worked on a number of areas of theoretical cosmology, field theory and gravitation theory. His research papers include models of, and experimental constraints on, violations of Lorentz invariance; the appearance of closed timelike curves in general relativity; varieties of topological defects in field theory; and cosmological dynamics of extra spacetime dimensions. He has written extensively on models of dark energy and its interactions with ordinary matter and dark matter, as well as modifications of general relativity in cosmology. He has also worked on the foundations of quantum mechanics, especially the many-worlds interpretation, including a derivation of the Born rule for probabilities.

His most-cited work, "Is Cosmic Speed-Up Due To New Gravitational Physics?" (2003) was written with Vikram Duvvuri, Mark Trodden and Michael Turner. With over 1,900 citations, it helped pioneer the study of f(R) gravity in cosmology.[32][failed verification][third-party source needed]

Carroll has also worked on the arrow of time problem. He and Jennifer Chen posit that the Big Bang is not a unique occurrence as a result of all of the matter and energy in the universe originating in a singularity at the beginning of time, but rather one of many cosmic inflation events resulting from quantum fluctuations of vacuum energy in a cold de Sitter space. They claim that the universe is infinitely old but never reaches thermodynamic equilibrium as entropy increases continuously without limit due to the decreasing matter and energy density attributable to recurrent cosmic inflation. They assert that the universe is "statistically time-symmetric", insofar as it contains equal progressions of time "both forward and backward".[33][34][35] Some of his work has been on violations of fundamental symmetries, the physics of dark energy, modifications of general relativity and the arrow of time. Recently he started focusing on issues at the foundations of cosmology, statistical mechanics, quantum mechanics and complexity.

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