In the context of general relativity, I am working with the energy-momentum tensor $T$, which is a rank-2 tensor whose components are usually denoted by $T^\mu_{\ \ \ \ \nu}$. However, I am unsure of how to write the fact that this tensor is symmetric. Which of the following two ways is correct?
$$T^\mu_{\ \ \ \ \nu}=T^\nu_{\ \ \ \mu}\ \ \ \ \ \text{or}\ \ \ \ \ T^\mu_{\ \ \ \ \nu} =T_\nu^{\ \ \mu}$$
I am thoroughly confused by this staggered indices notation, and would greatly appreciate any help.
An abstract tensor is a machine that eats vectors and covectors and returns a scalar. Lets consider a mixed $(1,1)$-tensor that takes a covector in its first slot and a vector in its second slot: $$\tag{1} T(U^*,V)\,. $$ When we have basis vector fields $\partial_\mu$ and $dx^\nu$ we can write this as $$\tag{2} T={T^\mu}_\nu\,\partial_\mu\otimes dx^\nu $$ where ${T^\mu}_\nu$ are $T$'s components. For this tensor it does not really make sense to speak of symmetry because it does not know what to do with $$\tag{3} T(V,U^*)\,. $$ How about the components ${T^\mu}_\nu\,?$ When we have a metric tensor $g^{\mu\nu}$ that we use to raise indices and which is symmetric then, as we know, $$\tag{4} {T^\mu}_\nu=g^{\mu\rho}T_{\rho\nu}\,. $$ The components $T_{\rho\nu}$ are those of a tensor $$\tag{5} T_{\rho\nu}\,\ dx^\rho\otimes dx^\nu $$ that eats two vectors $V,U$ and can therefore be symmetric or not: Unlike $T(U^*,V)$ it knows what to do with the vectors when we flip them.
Lemma. If the tensor (5) is symmetric, and if $g^{\mu\nu}$ is symmetric then $$\tag{6} {T^\mu}_\nu={T_\nu}^\mu\,. $$ Proof. $$\tag{7} {T^\mu}_\nu=g^{\mu\rho}T_{\rho\nu}=g^{\rho\mu}T_{\nu\rho} ={T_\nu}^\mu\,. $$ $$\tag*{$\Box$} \quad $$