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Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry

J. Math. Phys. 4, 735 (1963); doi:10.1063/1.1724316

Issue Date: June 1963

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F. K. Manasse and C. W. Misner
Palmer Physical Laboratory, Princeton University, Princeton, New Jersey
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. ©1963 American Institute of Physics
History: Received 31 May 1962
Permalink: http://link.aip.org/link/?JMAPAQ/4/735/1
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0022-2488 (print)   1089-7658 (online)
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REFERENCES (22)
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  1. E. Fermi, Atti Acad. Naz. Lincei Rend. Cl. Sci. Fiz. Mat. Nat. 31, 21, 51 (1922).
  2. L. P. Eisenhart, Non-Riemannian Geometry, (American Mathematical Society Colloquium Publications, New York, New York, 1927), Sec. 25. The Fermi normal coordinates developed in the present paper are also defined in (symmetric) affine spaces, and all our results which can be stated in affine spaces are valid there. The proofs are obtained by replacing every set of orthonormal vectors by a set of linearly independent vectors.
  3. L. O'Raifeartaigh, Proc. Roy. Irish Acad. A59, 2 (1958).
  4. J. L. Synge, Relativity, The General Theory (North-Holland Publishing Company, Amsterdam, 1960).
  5. See, for example, L. P. Eisenhart, Riemannian Geometry, (Princeton University Press, Princeton, New Jersey, 1926).
  6. Here etaµnu = diag(−1,1,1,1) is the Lorentz metric. We shall use Greek indices for space-time (µ,nu,etc. = 0,1,2,3), while Latin indices give components along spatial axes (i,j,etc. = 1,2,3). Our sign conventions for the curvature tensor are

    [dformula R[sub mu[sup nu] alpha beta] = [partial-derivative][sub alpha] Gamma[sub mu[sup nu] beta] - [partial-derivative][sub beta] Gamma[sub mu[sup nu] alpha] - (Gamma[sub mu[sup sigma] alpha] Gamma[sub sigma[sup nu] beta] - Gamma[sub mu[sup sigma] beta] Gamma[sub sigma[sup nu] alpha])]

    , and

    [dformula R[sub mu nu] = R[sub mu[sup alpha] alpha nu]]

    . The Riemann tensor convention corresponds to Cartan's definition (reference 13) of the curvature forms Oµnu = (1/2)Rµnualphabetadxalphadxbeta in terms of the connection forms

    [dformula omega[sub mu[sup nu]] = Gamma[sub mu[sup nu] alpha]dx[sup alpha],theta[sub mu[sup nu]] = d omega[sub mu[sup nu]] - alpha[sub mu[sup sigma]] omega[sub sigma[sup nu]]]

    , which definition is also valid in orthogonal (or other non-holonomic) frames.

  7. T. Levi-Civita, Math. Ann. 97, 291 (1926).
  8. F. K. Manasse, J. Math. Phys. 4, 746 (1963) (following paper).
  9. J. L. Synge and A. Schild, Tensor Calculus (University of Toronto Press, Toronto, 1952), Sec. 1.3.
  10. T. Y. Thomas, The Differential Invariants of Generalized Spaces (Cambridge University Press, New York, 1934,) p. 30.
  11. N. Steenrod, The Topology of Fibre Bundles (Princeton University Press, Princeton, New Jersey, 1951), Sec. 6.4.
  12. O. Veblen and J. H. C. Whitehead, The Foundations of Differential Geometry, (Cambridge University Press, New York, 1932, reprinted 1953), Sec. 16.
  13. E. Cartan, Leçons sur la géométrie des espaces de Riemann (Gauthier-Villars, Paris, 1951).
  14. C. Chevalley, Theory of Lie Groups. (Princeton University Press, Princeton, New Jersey, 1946), p. 77.
    15. A definition suitable for differentiable, rather than analytic, manifolds can be found in reference 15 or in H. Flanders, Trans. Am. Math. Soc. 75, 311 (1953).
    A definition of differentiable manifold which parallels Chevalley's for the analytic case is given by de Rham, Variétes Différentiables (Hermann et Cie., Paris, 1955).
  15. T. J. Willmore, An Introduction to Differential Geometry (Clarendon Press, Oxford, England, 1959), Chap. 6, Sec. 2.
  16. Since we think of the metric or any other tensor as an object which is independent of our choice of coordinate system, we prefer that the indication of the particular coordinate system to which a set of tensor components gµnu refers be placed on the component (index) part of the symbol rather than on the tensor part. Thus gµnu and gµ[prime]nu[prime] are components of the same metric tensor in two coordinate systems, while, should the occasion arise, gµnu and g[prime]µnu might represent two different metrics in a single coordinate system. See also the transformation laws of Eqs. (40) and (76).
  17. A vector tµ given only at a point, or along a curve, etc., can always and in many ways be considered part of a vector field by arbitrarily defining tµ(yalpha) at other points.
  18. A single vector [partial-derivative]/[partial-derivative]t differs from a partial derivative by the possibility of vanishing; e.g., the tangent to a constant curve P(t) = P0 is the zero vector ([partial-derivative]/[partial-derivative]t = 0, since [partial-derivative]f/[partial-derivative]t = df(P0)/dt = 0 for all functions f. However, in regions where [partial-derivative]/[partial-derivative]t[not-equal]0, coordinates can be introduced so that [partial-derivative]/[partial-derivative]t = [partial-derivative]/[partial-derivative]y0 is a conventional partial derivative.
  19. To see that the Lie Bracket does not always vanish, an example suffices. For the unit vectors etheta = [partial-derivative]/[partial-derivative]theta and e[cursive phi] = (sin theta)−1[partial-derivative]/[partial-derivative][cursive phi] on the unit sphere, compute from Eq. (20)

    [dformula [[bold e][sub theta],[bold e][sub [cursive phi]]] = - cot theta [bold e][sub [cursive phi]] [not-equal] 0]

    .

  20. See, for example, F. J. Murray and K. S. Miller, Existence Theorems, (New York University Press, New York, 1954), Chap. 2, Theorems 1, 3; Chap. 3, Theorem 2; Chap. 5, Theorem 6. A discussion of the properties of geodesies from which we have borrowed much is found in H. Seifert and W. Threlfall, Variationsrechnung im Grossen (B. G. Teubner, Leipzig, 1938), footnote 20, p. 97.
  21. For physical applications see F. A. E. Pirani, Acta Phys. Polon. 15, 389 (1956);
    22. Phys. Rev. 105, 1089 (1957);
    and J. Weber, General Relativity and Gravitational Waves (Interscience Publishers, Inc., New York, 1961), Chap. 8.
  22. The entire derivation may be regarded as a process of evaluating the commutators which relate Eq. (49) in the curious from

    [dformula ((delta)/(delta n))((delta)/(delta s))(((delta)/(delta s))) = 0]

    to an equation whose leading term is

    [dformula ((delta[sup 2]n)/(delta s[sup 2])) = ((delta)/(delta s))((delta)/(delta s))(((delta)/(delta n)))]

    .

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