# Doubt on Tetrads, Energy-momentum tensors and Einstein's equations

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?

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.