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I Static, Isotropic metric

  1. Jun 12, 2017 #1
    In Weinberg's book it is said that a Static, Isotropic metric should depend on ##x## and ##dx## only through the "rotational invariants" ##dx^2, x \cdot dx, x^2## and functions of ##r \equiv (x \cdot x)^{1/2}##. It's clear from the definition of ##r## that ##x \cdot dx## and ##x^2## don't depend on the angular displacement. What I don't understand is why ##dx^2## is invariant under rotations, since it's the "pure" metric when written in spherical coordinates, and so it depends on the usual angles ##\theta## and ##\varphi##.
     
  2. jcsd
  3. Jun 12, 2017 #2

    PeterDonis

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    2016 Award

    Staff: Mentor

    Because it's a distance (an infinitesimal one, but still a distance), and distances are invariant under rotations.

    Formally, yes, but if you do a rotation of the coordinates, you will find that ##\theta## and ##\varphi## change in concert in such a way as to leave ##dx^2## invariant.
     
  4. Jun 12, 2017 #3
    Thanks!
     
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