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Number of unknowns - Coordinate Transforms

  1. Jan 26, 2013 #1
    In general relativity, what are the total number of unknowns for a generic coordinate transform? Is it just 4 * 4 = 16? Is there a way to break those down into combinations of types, such as boosts, rotations, reflections (parity?), etc, or is it just left wide open from an interpretive standpoint? My feeling is the answer is in fact 16 unknown functions of space-time and that the actual interpretation can't really be broken out like we do in SR (Lorentz)...

    Your help is greatly appreciated...
     
  2. jcsd
  3. Jan 26, 2013 #2

    Dale

    Staff: Mentor

    It is an infinite number of unknowns. Even if you only had a 1D manifold, there are an infinite number of degrees of freedom.
     
  4. Jan 26, 2013 #3
    I made a mistake in my question. I know that functions have an unlimited number of degrees of freedom, I meant to ask how many functions are involved in a generic coordinate transform in R^4. My guess is 16, since there are two indexes in the transformation matrix, each running over 4 values, but wanted clarity since there might be symmetries that eliminate elements (though I'm guessing not).
     
  5. Jan 26, 2013 #4

    Dale

    Staff: Mentor

    I think it is only 4 unknown functions. You can always write it as:
    [itex]x'_0=f_0(x_0,x_1,x_2,x_3)[/itex]
    [itex]x'_1=f_1(x_0,x_1,x_2,x_3)[/itex]
    [itex]x'_2=f_2(x_0,x_1,x_2,x_3)[/itex]
    [itex]x'_3=f_3(x_0,x_1,x_2,x_3)[/itex]
     
  6. Jan 27, 2013 #5
    May be you are talking about gμv which seems to have 16 components.But it is symmetric,which will eliminate 6 so there will be only 10 independent components.
     
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