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Given, for instance, the perfect fluid energy-momentum tensor:

$$T_{\mu\nu} = (\rho+p)u_{\mu}u_{\nu} - pg_{\mu\nu}\tag{1}$$

We can put (due to diagonalization procedure) into the diagonal for as:

$$T_{\hat{\mu}\hat{\nu}} = Diag[\rho, \tau,p_{2},p_{3}] \tag{2}$$

On the other hand , if we specify a tetrad frame we write the very energy tensor into the same diagonal form.

Now, the Einstein tensor can be written in terms of tetrad frame as well:

$$G_{\hat{\mu}\hat{\nu}} = e_{\hat{\mu}}^{\mu}e_{\hat{\nu}}^{\nu}G_{\mu\nu} \tag{3}$$

My doubt is:

If we write the energy tensor in a tetrad frame, we need to express the Einstein tensor in a tetrad frame too?

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The Einstein field equations read $G_{\mu\nu}=8\pi T_{\mu\nu}$, so if we contract one side with $e^\mu_{\hat\mu}$ we have to do so to the other side as well. Hence, yes, both need to be in the orthonormal basis.

This is just a special case of the more general principle that indices should match on both sides of an equation (there are a few rare exceptions, to do with uncommon notational tricks).

In this case while you might have the same amount of indices on both sides, the point is they need to also be the same kind of indices.

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