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Page 33 of 848

  • 4 - Maxwell theory

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp 88-112
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  • Dedication

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp vii-viii
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  • Contents

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp ix-xii
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  • Foreword

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp xiii-xiv
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  • 10 - Knot theory and physical states of quantum gravity

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp 238-274
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  • 7 - Quantum gravity

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp 161-187
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  • 2 - Differential geometry

  • Stephen W. Hawking, University of Cambridge, George F. R. Ellis, University of Cape Town
  • Book:
    The Large Scale Structure of Space-Time
    Published online:
    17 February 2023
    Print publication:
    16 February 2023, pp 10-55

  • Summary

    The spacetime structure is that of a manifold with a Lorentz metric and associated affine connection.

    We introduce in §2.1 the concept of a manifold and in §2.2 vectors and tensors, which are the natural geometric objects defined on the manifold. A discussion of maps of manifolds in §2.3 leads to the definitions of the induced maps of tensors, and of sub-manifolds. The derivative of the induced maps defined by a vector field gives the Lie derivative defined in §2.4; another differential operation which depends only on the manifold structure is exterior differentiation, also defined in that section. This operation occurs in the generalized form of Stokes’ theorem.

    The connection is introduced in §2.5, defining the covariant derivative and the curvature tensor. The connection is related to the metric on the manifold in §2.6; the curvature tensor is decomposed into the Weyl tensor and Ricci tensor, which are related to each other by the Bianchi identities.

    The induced metric and connection on a hypersurface are discussed in §2.7, and the Gauss–Codacci relations are derived. The volume element defined by the metric is introduced in §2.8, and used to prove Gauss’ theorem.

  • 1 - Holonomies and the group of loops

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp 1-28
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  • 3 - The loop representation

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp 52-87
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  • Index

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp 319-324
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  • Frontmatter

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp i-vi
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  • 5 - Yang-Mills theories

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp 113-130
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  • 3 - General relativity

  • Stephen W. Hawking, University of Cambridge, George F. R. Ellis, University of Cape Town
  • Book:
    The Large Scale Structure of Space-Time
    Published online:
    17 February 2023
    Print publication:
    16 February 2023, pp 56-77

  • Summary

    Wepresent the theory as a number of postulates about a mathematical model for spacetime.

    In §3.1 we introduce the mathematical model and in §3.2 the first two postulates, local causality and local energy conservation. These postulates are common to both special and general relativity, and thus may be regarded as tested by the many experiments that have been performed to check the former. In §3.3 we derive the equations of the matter fields and obtain the energy–momentum tensor from a Lagrangian.

    The third postulate, the field equations, is given in §3.4. This is not so well established experimentally as the first two postulates, but we shall see that any alternative equations would seem to have one or more undesirable properties, or else require the existence of extra fields which have not yet been detected experimentally.

    The mathematical model we shall use for spacetime, i.e. the collection of all events, is a pair (ℳ, g) where ℳ is a connected four-dimensional Hausdorff C∞ manifold and g is a Lorentz metric (i.e. a metric of signature + 2) on ℳ.

  • 2 - Loop coordinates and the extended group of loops

  • Rodolfo Gambini, Universidad de la República, Uruguay, Jorge Pullin, Pennsylvania State University
  • Foreword by Abhay Ashtekar, Pennsylvania State University
  • Book:
    Loops, Knots, Gauge Theories and Quantum Gravity
    Published online:
    10 November 2022
    Print publication:
    16 February 2023, pp 29-51
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  • Index

  • Juan A. Valiente Kroon, Queen Mary University of London
  • Book:
    Conformal Methods in General Relativity
    Published online:
    04 November 2022
    Print publication:
    09 February 2023, pp 587-594
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Page 33 of 848