Non-rotational and rotational metric tensors

[Orion1]

General Relativity...

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

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

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

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

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

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

 

[PeterDonis]

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.

 

[Matterwave]

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.

 

[Ben Niehoff]

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

 

[sardar]

In General Relativity, the Einstein's field equations relate the curvature of spacetime to the distribution of matter and energy in the universe. The metric tensor is a mathematical tool used to describe the curvature of spacetime and is influenced by the distribution of matter and energy. In the case of a non-rotational spherically symmetric body of isotropic perfect fluid, the metric tensor and the stress-energy tensor are both diagonal and have zero values for off-diagonal elements, indicating no rotation.

However, if the metric has rotation, the off-diagonal elements become non-zero (J \neq 0) and this affects both the Einstein tensor and the Stress-Energy tensor. In this case, the Einstein tensor metric element functions g_{t\phi} become non-zero, indicating the presence of rotation in the spacetime. This in turn affects the corresponding Stress-Energy tensor metric element functions T_{t\phi}, which also become non-zero.

This is because the presence of rotation in the metric affects the distribution of matter and energy, which is described by the Stress-Energy tensor. Therefore, the Stress-Energy tensor also reflects the rotation in the spacetime.

In summary, the presence of rotation in the metric tensor affects both the Einstein tensor and the Stress-Energy tensor, and the corresponding metric element functions become non-zero. This is in accordance with the energy conditions of perfect fluids, which state that rotation can influence the distribution of energy and matter in the universe.