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3 results

  • 12 - The Einstein equations and the sources of a gravitational field

  • from Part II - The theory of gravitation
  • Jerzy Plebanski, National Polytechnic Institute of Mexico, Andrzej Krasinski, Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences
  • Book:
    An Introduction to General Relativity and Cosmology
    Published online:
    30 May 2024
    Print publication:
    06 June 2024, pp 123-155

  • Summary

    The derivation of the Einstein equations is presented following Einstein’s method. Hilbert’s derivation (from a variational principle) is also presented. The Newtonian limit of Einstein’s theory is discussed. A Bianchi type I solution of Einstein’s equations with a dust source is derived. A brief review of other theories of gravitation (Brans–Dicke, Bergmann–Wagoner, Einstein–Cartan and Rosen) is presented. The matching conditions for different metrics are derived. The weak-field approximation to general relativity is presented.

  • Chapter 22 - The Source of Curvature

  • from Part III - The Einstein Equation
  • James B. Hartle, University of California, Santa Barbara
  • Book:
    Gravity
    Published online:
    03 September 2021
    Print publication:
    24 June 2021, pp 471-490

  • Summary

    This chapter completes the description of the Einstein equation by finding the correct measure of energy density and the more general measure of spacetime curvature. A density is a quantity per unit spatial volume, such as rest-mass density, charge density, number density, energy density, and so on. The chapter begins by discussing how densities are represented in special and general relativity – for example, the densities of energy and momentum, and their conservation.

  • 41 - Nonperturbative Gravity: The Vacuum Schwarzschild Solution

  • from Part III - Other Spins or Statistics; General Relativity
  • Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
  • Book:
    Classical Field Theory
    Published online:
    04 March 2019
    Print publication:
    14 March 2019, pp 389-395

  • Summary

    The nonpeturbative Schwarzschild solution of the Einstein's equations in vacuum is found. We start with the Newtonian limit for gravity, and find the metric in terms of the Newton potential. We write an ansatz, and solve the equations of motion to find the solution. The Schwarzschild black hole and its event horizon are defined. Birkhoff's theorem for the uniqueness of the Schwarzschild solution is stated.