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16950 results

Page 23 of 848

  • 3 - Geodesics

  • Brian P. Dolan, National University of Ireland, Maynooth
  • Book:
    Einstein's General Theory of Relativity
    Published online:
    15 June 2023
    Print publication:
    29 June 2023, pp 44-84

  • Summary

    Geodesics are introduced and the geodesic equation analysed for the geometries introduced in chapter 2, using variation principles of classical mechanics. Geodesic motino on a sphere is described as well as the Coriolis effect and the Sagnac effect. Newtonian gravity is derived as the non-relativistic limit of geodesic motion in space-time. Geodesics in an expanding universe and heat death is described. Geodesics in Schwarzschild space-time are treated in detail: the precession of the perihelion of Mercury; the bending of light by the Sun; Shapiro time delay; black holes and the event horizon. Gravitational waves and gravitational lensing are also covered.

  • 1 - Introduction

  • Brian P. Dolan, National University of Ireland, Maynooth
  • Book:
    Einstein's General Theory of Relativity
    Published online:
    15 June 2023
    Print publication:
    29 June 2023, pp 1-13
    • Chapter
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  • Summary

    Newton's Universal Law of Gravitation is compared and contrasted to Coulomb’s Law and the differences highlighted. Tides are discussed, and the Equivalence Principle and how it leads to the notion of curved space-times is explained.

  • 5 - Einstein’s Field Equations

  • Brian P. Dolan, National University of Ireland, Maynooth
  • Book:
    Einstein's General Theory of Relativity
    Published online:
    15 June 2023
    Print publication:
    29 June 2023, pp 119-125

  • Summary

    Einsteins field equations are derived and discussed. It is argued that the Einstein tensor is proportional to the energy-momentum tensor and the constant of proportionality is derived by demanding that Newton’s Universal Law of gravitation be recovered in the non-relativistic limit. The modification of Einstein's equations when a cosmological constant is introduced is also presented.

  • 4 - The Geometry of Curved Spaces

  • Brian P. Dolan, National University of Ireland, Maynooth
  • Book:
    Einstein's General Theory of Relativity
    Published online:
    15 June 2023
    Print publication:
    29 June 2023, pp 85-118

  • Summary

    The mathematics required to analyse higher dimensional curved spaces and space-times is developed in this chapter. General coordinate transformations, tangent spaces, vectors and tensors are described. Lie derivatives and covariant derivatives are motivated and defined. The concepts of parallel transport and a connection is introduced and the relation between the Levi-Civita connection and geodesics is elucidated. Christoffel symbols the Riemann tensor are defined as well as the Ricci tensor, the Ricci scalar and the Einstein tensor, and their algebraic and differential properties are described (though technical details of the derivationa of the Rimeann tensor are let to an appendix).

  • 2 - Metrics

  • Brian P. Dolan, National University of Ireland, Maynooth
  • Book:
    Einstein's General Theory of Relativity
    Published online:
    15 June 2023
    Print publication:
    29 June 2023, pp 14-43

  • Summary

    The concept to the metric is introduced. Various geometries, both flat and curved, are described including Euclidean space; Minkowski space-time; spheres; hyperbolic planes and expanding space-times. Lorentz transformations and relativistic time dilation in flat space-time is discussed as well as gravitational red-shift and the Global Positioning System. Hubble expansion and the cosmological red-shift are also explained.

Einstein's General Theory of Relativity
  • A Concise Introduction
  • Brian P. Dolan
  • Published online:
    15 June 2023
    Print publication:
    29 June 2023
  • Einstein's general theory of relativity can be a notoriously difficult subject for students approaching it for the first time, with arcane mathematical concepts such as connection coefficients and tensors adorned with a forest of indices. This book is an elementary introduction to Einstein's theory and the physics of curved space-times that avoids these complications as much as possible. Its first half describes the physics of black holes, gravitational waves and the expanding Universe, without using tensors. Only in the second half are Einstein's field equations derived and used to explain the dynamical evolution of the early Universe and the creation of the first elements. Each chapter concludes with problem sets and technical mathematical details are given in the appendices. This short text is intended for undergraduate physics students who have taken courses in special relativity and advanced mechanics.

A General Relativity Coursebook
  • Ed Daw
  • Published online:
    03 May 2023
    Print publication:
    06 April 2023
  • General relativity is a subject that most undergraduates in physics are particularly curious about, but it has a reputation for being very difficult. This book provides as gentle an introduction to general relativity as possible, leading you through the necessary mathematics in order to arrive at important results. Of course, you cannot avoid the mathematics of general relativity altogether, but, using this book, you can gain an appreciation of tensors and differential geometry at a pace you can keep up with. Early chapters build up to a complete derivation of Einstein's Equations, while the final chapters cover the key applications on black holes, cosmology and gravitational waves. It is designed as a coursebook with just enough material to cover in a one-semester undergraduate class, but it is also accessible to any numerate readers who wish to appreciate the power and beauty of Einstein's creation for themselves.

Stellar Structure and Evolution
  • Marc Pinsonneault, Barbara Ryden
  • Published online:
    02 May 2023
    Print publication:
    06 April 2023
  • Stellar Structure and Evolution, the second volume in the Ohio State Astrophysics Series, takes advantage of our new era of stellar astrophysics, in which modern techniques allow us to map the interiors of stars in unprecedented detail. This textbook for upper-level undergraduate and graduate students aims to develop a broad physical understanding of the fundamental principles that dictate stellar properties. The study of stellar evolution focuses on the 'life cycle' of stars: how they are born, how they live, and how they die. As elements ejected by one generation of stars are incorporated into the next generation, stellar evolution is intertwined with the chemical evolution of our galaxy. Focusing on key physical processes without going into encyclopedic depth, the authors present stellar evolution in a contemporary context, including phenomena such as pulsations, mass loss, binary interactions, and rotation, which contribute to our understanding of stars.

Page 23 of 848