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Riemannian manifold and general relativity

  1. Sep 8, 2007 #1
    1. The problem statement, all variables and given/known data
    My book ("General Relativity" by Hobson on page 32) says that an N-dimensional manifold has 1/2 * N * (N-1) independent metric functions. I am confused about why there is a limit at all to the number of independent metric functions [tex] g_{\mu \nu} [/tex]. It probably has to do with the symmetric property of the metric.

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 8, 2007 #2


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    I think you're confused about what they mean by "indpendent functions". If we ignored symmetry considerations, the metric would have N2 indepedent functions: g11(x), g12(x), ..., gNN(x). But the metric must be symmetric, gmn(x)=gnm(x). The number of indepednet functions is then the number of elements on or above the main diagonal in an NxN matrix; this number is (1/2)N(N+1). [Not, as you wrote, (1/2)N(N-1); your book has the correct result.]
  4. Sep 8, 2007 #3
    Thank you. And you're right it is 1/2 * (N) * (N+1)
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