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Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry

1963; American Institute of Physics; Volume: 4; Issue: 6 Linguagem: Inglês

10.1063/1.1724316

1527-2427

F. K. Manasse, Charles W. Misner,

Relativity and Gravitational Theory

Fermi coordinates, where the metric is rectangular and has vanishing first derivatives at each point of a curve, are constructed in a particular way about a geodesic. This determines an expansion of the metric in powers of proper distance normal to the geodesic, of which the second-order terms are explicitly computed here in terms of the curvature tensor at the corresponding point on the base geodesic. These terms determine the lowest-order effects of a gravitational field which can be measured locally by a freely falling observer. An example is provided in the Schwarzschild metric. This discussion of Fermi Normal Coordinate provides numerous examples of the use of the modern, coordinate-free concept of a vector and of computations which are simplified by introducing a vector instead of its components. The ideas of contravariant vector and Lie Bracket, as well as the equation of geodesic deviation, are reviewed before being applied.