Symmetries and Curvature Structure in General Relativity


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Introduction

This is an advanced graduate course which develops the math and physics of general relativity from scratch up to the highest level. The going will sometimes be steep but I'll try to be always careful. The purpose is to prepare students for studies in quantum gravity, relativistic quantum information, black hole physics and cosmology. Quick summary of the contents:.

General Relativity Topic 17: Symmetries, Killing Vectors and Maximally Symmetric Spaces

Differentiable manifolds. Lecture 2: Algebra of function germs. Algebraic definition of tangent space. Lecture 3: Physical and geometric definitions of tangent space. Tangent bundle and its sections.


  1. Understanding gravity—warps and ripples in space and time!
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  4. Introduction.

Lecture 4: Algebra of differential forms. Anti- derivations. Exterior derivative.


  • Newton and the laws of gravity;
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  • Problems inside an empty hole.
  • Lie derivative on forms and on general tensors. Infinitesimal diffeomorphisms. Lecture 6: Orientability and volume. Stokes', Green's and Gauss' theorems. Klein Gordon and Maxwell equations. The grades will be based on an essay about the general relativity part of this course and a project on the cosmology part of this course, in equal parts.

    The essay topic is taken from the first half of the course. Your task is to review the various mathematical methods that are being used to describe the "shape" of a Riemannian or Lorentzian manifold. Remember, there are methods to describe the shape of a manifold by using the fact that a nontrivial shape implies violations of Pythagoras' law. Then there are methods using deficiency angles or causal dynamical triangulations look it up , and methods that use the nontriviality of parallel transport or the affine connection, the tetrad fields that diagonalize the metric-tensor, the overtone spectrum and spectral geometry for compact Riemannian manifolds , the Weyl and Ricci curvature decomposition, the causal structure plus a conformal function description, and there are probably many others.

    Try to cover several ideas and approaches and feel free to include new ideas if some come to mind. The point is to explain some of those approaches and how they do or may relate to each other in your own words - possibly with a discussion of the possible pros and cons of these methods, i. Focus on the main ideas and concepts rather than getting lost in details. You can choose to write much more on one or two approaches than on others.

    But in any case your essay should contain an overview over and a discussion of a range of methods. The project topic is chosen from the second half of the course in particular the lectures 15, 16, 17 : "Numerical explorations of the dynamics of FRW cosmologies". Your task is:. Main task: Study the past and future evolution of the night sky.

    Namely, explore the behavior of the flat FRW solutions numerically, i. Here is a suggestion for how you may want to structure your explorations you are free to structure your explorations differently :.

    The “true magic hidden inside General Relativity”

    For practice, and as a consistency check, I would suggest to first try to recover numerically the behavior in each cosmological epoch that we solve for analytically in class. Namely, make an assumption for a fixed value of the equation of state parameter, separately for the epochs e. Then solve the dynamical equations and plot the results for each epoch and compare with the analytically predicted behavior.

    Then, make the equation of state parameter w vary in a prescribed way. To this end, assume that w is a given function w rho of the energy density, that you choose so that w rho runs successively between the values for radiation, dust and vacuum energy dominance as the universe gets diluted, i.

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    Notice that w rho should not drop uniformly but should drop so as to reflect the presence of more or less clearly defined epochs connected by transitional periods. One question you may want to explore whether the behavior of the dynamics is sensitive to how slowly or how abruptly w changes. For example, w may changes very quickly at a time when the matter in the universe undergoes a phase transition.

    However, several experiments, including recent experimental data. Don't forget to state all your sources clearly, also for the project. This also includes the software that you use. If you use a specialized software package for GR, mention this. In this case the expectations are somewhat higher of course. An essay should be a review of existing literature on a given topic. The sources can be textbooks, lecture notes or review articles or original articles or some of each. All and everything that is used needs to be cited.

    Chapter 6 The Lorentz Group. Chapter 7 SpaceTimes and Algebraic Classification. Chapter 8 Holonomy Groups and General Relativity.

    About the Book

    Chapter 11 Conformal Symmetry in Spacetimes. Chapter 12 Projective Symmetry in Spacetimes. Chapter 13 Curvature Collineations. Gravitational collapse and black holes are discussed in part VI, which covers such topics as the Kruskal extension of the Schwarzschild metric, the critical mass of a star, the gravitational collapse of a sphere of dust and a brief discussion of the properties of the Kerr metric. Also included in this new edition is a short but useful section on the problem of quantization, of quantum field theory in a given curved spacetime and on the Hawking effect, and on the conformal structure of infinity leading to a definition of asymptotic flatness.

    Dafermos : The evolution problem in general relativity

    In part VII on cosmology, after showing how the cosmological principle leads to the Robertson-Walker metric, the various Friedmann models are derived and discussed. The final sections discuss a Bianchi type I model and the Goedel universe. For its size at about four hundred pages the book covers a great deal of material, and it does this very clearly and well. There are some misprints, but almost all of these are of an obvious kind and easily corrected. At the end of most sections there are some well chosen exercises perhaps there could have been more of these and useful suggestions for further reading.

    Students who work through the book carefully, completing the details for those calculations and proofs where these are outlined in the text, will learn a great deal about relativity and be well placed for further study in the subject. While it is probably too advanced in its scope for the average final year UK undergraduate course, I would recommend it as a reference book for further reading at this level or as a text for study by a beginning graduate student.

    Current Developments in Mathematics

    DOI: Country of publication. United Kingdom. Descriptors DEI. Descriptors DEC. Publication Year Publication Year. Language Language. Reference Number Reference Number.

    Symmetries and Curvature Structure in General Relativity Symmetries and Curvature Structure in General Relativity
    Symmetries and Curvature Structure in General Relativity Symmetries and Curvature Structure in General Relativity
    Symmetries and Curvature Structure in General Relativity Symmetries and Curvature Structure in General Relativity
    Symmetries and Curvature Structure in General Relativity Symmetries and Curvature Structure in General Relativity
    Symmetries and Curvature Structure in General Relativity Symmetries and Curvature Structure in General Relativity
    Symmetries and Curvature Structure in General Relativity Symmetries and Curvature Structure in General Relativity
    Symmetries and Curvature Structure in General Relativity Symmetries and Curvature Structure in General Relativity

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