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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

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# Non-rotational and rotational metric tensors

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