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"Semi" Synchronous coordinates

  1. Jul 13, 2015 #1
    I understand that one can always construct a set Synchronous coordinates (or Gaussian normal coordinates) on a neighborhood of a point in spacetime.

    My question is:
    Does one can construct a metric with only $g_{0i}=0$ such that
    $dS^2=g_{00}dt^2 + g_{ij}dx^{i} dx^{j}$ (where $i=1,,,,D$ and $j=1,,,,D$) not just in a neighborhood of a point but to all spacetime in general?

    In this case one should use
     
  2. jcsd
  3. Jul 13, 2015 #2
    Yes...
    One can use

    g'_{0i}=\frac{\partial x^{\alpha}}{\partial x'^{0}}\frac{\partial x^{\\beta}}{\partial x'^{i}}g_{\alpha\beta}=0

    so it seems we have enough PDE...
     
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