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Question about metric in SR

  1. Jan 15, 2014 #1
    Does [tex]\eta^{\alpha \beta}=\eta_{\alpha \beta}[/tex] in all coordinate systems or just inertial coordinate systems?
     
  2. jcsd
  3. Jan 16, 2014 #2

    stevendaryl

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    For non-inertial coordinate systems, the symbol [itex]g_{\alpha \beta}[/itex] is used instead of [itex]\eta_{\alpha \beta}[/itex]. And in general, [itex]g_{\alpha \beta}[/itex] is unequal to [itex]g^{\alpha \beta}[/itex]. [itex]g^{\alpha \beta}[/itex] is the inverse of [itex]g_{\alpha \beta}[/itex].

    Here's an example: In polar coordinates [itex]t, \rho, \phi, z[/itex],

    [itex]g_{tt} = 1[/itex]
    [itex]g_{zz} = -1[/itex]
    [itex]g_{\rho \rho} = -1[/itex]
    [itex]g_{\phi \phi} = -\rho^2[/itex]

    [itex]g^{tt} = 1[/itex]
    [itex]g^{zz} = -1[/itex]
    [itex]g^{\rho \rho} = -1[/itex]
    [itex]g^{\phi \phi} = -\frac{1}{\rho^2}[/itex]
     
  4. Jan 16, 2014 #3

    jcsd

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    [itex]g^{\alpha \beta}=g_{\alpha \beta}[/itex] means that the coordinate basis is orthonormal, which only corresponds to 'inertial coordinates' (Minkowski coordinates) in flat spacetime.
     
  5. Jan 16, 2014 #4
    Thanks guys.
     
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