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Hello guys,

I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here to share the statements hoping to improve the discussion.

Given that SU(2) is homomorphic to SO(3), we were discussing if this statements were True or not:

The first statement is true without any doubts but, now, in my viewpoint, that would (probably) imply that the spatial part of ##T_{\mu\nu}## is isotropic, in which case, anisotropic stresses would be zero but there would still be chances to have ##T_{i0} \neq 0## meaning that the second statement wouldn't be enough to ensure the third one.

The tough part is the logical relation between the first statement and the second one, my friend claims that the second one is right but he doesn't offer a clear explanation about it. What do you think guys?

I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here to share the statements hoping to improve the discussion.

Given that SU(2) is homomorphic to SO(3), we were discussing if this statements were True or not:

**[1]**A SU(2)-invariant matter lagrangian is also invariant under rotations**[2]**its energy-momentum tensor is isotropic**[3]**it has the same form as the one from a perfect fluid.The first statement is true without any doubts but, now, in my viewpoint, that would (probably) imply that the spatial part of ##T_{\mu\nu}## is isotropic, in which case, anisotropic stresses would be zero but there would still be chances to have ##T_{i0} \neq 0## meaning that the second statement wouldn't be enough to ensure the third one.

The tough part is the logical relation between the first statement and the second one, my friend claims that the second one is right but he doesn't offer a clear explanation about it. What do you think guys?

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