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2 results

  • Chapter 20 - A Little More Math

  • 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 419-444

  • Summary

    This last part of the book introduces the Einstein equation – the basic equation of general relativity, in much the same way that Maxwell’s equations are the basic equations of electromagnetism. Geometries such as the Schwarzschild geometry, or those of the FRW cosmological models, are particular solutions of the Einstein equation. Just three new mathematical ideas are needed to give an efficient and standard discussion of the Einstein equation: a more precise definition of vectors in terms of directional derivatives; the notion of dual vectors as a linear map from vectors to real numbers; and the covariant derivative of a vector field in curved spacetime. These mathematical concepts are introduced in this chapter.