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2 results

  • 3 - The Riemann Tensor and Einstein’s Equation

  • Joel Franklin, Reed College, Oregon
  • Book:
    An Introduction to Gravity
    Published online:
    04 May 2024
    Print publication:
    11 April 2024, pp 102-135

  • Summary

    With the form of the target theory built up over the previous two chapters, we move to a geometric description of gravitational motion. By recasting the relative dynamics of a pair of falling objects as the deviation of nearby geodesic trajectories in a spacetime with a metric, Einstein’s equation is motivated. To describe geodesic deviation quantitatively, the Riemann tensor is introduced, and its role in characterizing spacetime structure is developed. With the full field equation of general relativity in place, the linearized limit is carefully developed and compared with the gravito-electro-magnetic theory from the first chapter.