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Chapter 7: The Description of Curved Spacetime

Chapter 7: The Description of Curved Spacetime

pp. 135-168

Authors

, University of California, Santa Barbara
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Summary

This chapter (and the next one) covers some basic mathematics needed to describe four-dimensional curved spacetime geometry. Much of this is a generalization of the concepts introduced in Chapter 5 for flat spacetime. Coordinates are a systematic way of labeling the points of spacetime. The choice of coordinates is arbitrary as long as they supply a unique set of labels for each point in the region they cover, but for a particular problem, one coordinate system may be more useful than another. We then define the metric for a general geometry and explain common conventions. We show how to compute lengths of curves, areas, three-volumes, and four-volumes for a given metric. Concepts such as wormholes, extra dimensions, the Lorentz hyperboloid, and null spaces are introduced.

Keywords

  • coordinates
  • Penrose diagram
  • defining the metric
  • summation convention
  • local inertial frames
  • light cones
  • world lines
  • embedding diagrams
  • wormholes
  • vector fields
  • Lorentz hyperboloid

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