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24897 results in Cosmology, Relativity and Gravitation

Page 51 of 1245

  • Preface

  • 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 xv-xvi
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  • 8 - The loop representation 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 188-208
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  • 6 - Causal structure

  • 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 180-225

  • Summary

    §6.1 deals with the question of the orientability of timelike and spacelike bases. In §6.2 basic causal relations are defined and the definition of a non-spacelike curve is extended from piecewise differentiable to continuous. The properties of the boundary of the future of a set are derived in §6.3. In §6.4a number of conditions which rule out violations or near violations of causality are discussed. The closely related concepts of Cauchy developments and global hyperbolicity are introduced in §6.5 and §6.6, and are used in §6.7 to prove the existence of non-spacelike geodesies of maximum length between certain pairs of points.

    In §6.8 we describe the construction of Geroch, Kronheimer, and Penrose for attaching a causal boundary to spacetime. A particular example of such a boundary is provided by a class of asymptotically flat spacetimes which are studied in § 6.9

  • 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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Page 51 of 1245