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Non-rotational and rotational metric tensors

  1. Nov 7, 2014 #1

    General Relativity...

    Non-rotational spherically symmetric body of isotropic perfect fluid Einstein tensor metric element functions:
    [tex]g_{\mu \nu} = \left( \begin{array}{ccccc} \; & dt & dr & d \theta & d\phi \\ dt & g_{tt} & 0 & 0 & 0 \\ dr & 0 & g_{rr} & 0 & 0 \\ d\theta & 0 & 0 & g_{\theta \theta} & 0 \\ d\phi & 0 & 0 & 0 & g_{\phi \phi} \end{array} \right) \; \; \; \; \; \; J = 0[/tex]

    Non-rotational spherically symmetric body of isotropic perfect fluid Stress-Energy tensor metric element functions:
    [tex]T_{\mu \nu} = \left( \begin{array}{ccccc} \; & dt & dr & d \theta & d\phi \\ dt & T_{tt} & 0 & 0 & 0 \\ dr & 0 & T_{rr} & 0 & 0 \\ d\theta & 0 & 0 & T_{\theta \theta} & 0 \\ d\phi & 0 & 0 & 0 & T_{\phi \phi} \end{array} \right) \; \; \; \; \; \; J = 0[/tex]

    If the metric has rotation [itex](J \neq 0)[/itex] and the Einstein tensor metric element functions become non-zero for [itex]g_{t \phi}[/itex], do the Stress-Energy tensors also become non-zero for the corresponding Stress-Energy tensors [itex]T_{t \phi}[/itex]?

    Rotational spherically symmetric body of isotropic perfect fluid Einstein tensor metric element functions:
    [tex]g_{\mu \nu} = \left( \begin{array}{ccccc} \; & dt & dr & d \theta & d\phi \\ dt & g_{tt} & 0 & 0 & g_{t \phi} \\ dr & 0 & g_{rr} & 0 & 0 \\ d\theta & 0 & 0 & g_{\theta \theta} & 0 \\ d\phi & g_{t \phi} & 0 & 0 & g_{\phi \phi} \end{array} \right) \; \; \; \; \; \; J \neq 0[/tex]

    Rotational spherically symmetric body of isotropic perfect fluid Stress-Energy tensor metric element functions:
    [tex]T_{\mu \nu} = \left( \begin{array}{ccccc} \; & dt & dr & d \theta & d\phi \\ dt & T_{tt} & 0 & 0 & T_{t \phi} \\ dr & 0 & T_{rr} & 0 & 0 \\ d\theta & 0 & 0 & T_{\theta \theta} & 0 \\ d\phi & T_{t \phi} & 0 & 0 & T_{\phi \phi} \end{array} \right) \; \; \; \; \; \; J \neq 0[/tex]

    Reference:
    http://en.wikipedia.org/wiki/General_relativity#Einstein.27s_equations - General Relativity and Einstein's equations
    http://en.wikipedia.org/wiki/Energy_conditions#Perfect_fluids - Energy Conditions of perfect fluids
     
    Last edited: Nov 7, 2014
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  3. Nov 7, 2014 #2

    PeterDonis

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    Staff: Mentor

    The Einstein Field Equation does not relate the metric tensor to the stress-energy tensor. It relates the Einstein tensor to the stress-energy tensor. The Wikipedia page describes (briefly) what the Einstein tensor is.
     
  4. Nov 8, 2014 #3

    Matterwave

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    I could not parse your post well enough to give you a good answer, OP, but it seems, like Peter is interpreting, that you think there is a direct relationship between the non-zero members of the metric tensor with the non-zero members of the stress-energy tensor. This is very false, since, for example, in a Schwarzschild space-time all members of the stress-energy tensor are 0 everywhere (except at the singularity) and yet the metric tensor is non-zero.
     
  5. Nov 8, 2014 #4

    Ben Niehoff

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    Also, it is possible to get off-diagonal terms in the Einstein tensor even if the metric tensor is diagonal.
     
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