I have a question regarding the definition of the energy-momentum tensor. I've seen it defined as a (2,0) tensor, so it has 2 upper indices $T^{ab}$, but many times it is written as a (0,2) tensor with lower indices (just like in Einstein's equations). I know that you can raise and lower indices using a metric, so $g_{ak}g_{bl}T^{ab}=T_{kl}$, but the textbooks I am using (mainly Carroll's Lecture notes on general relativity) often refer to the energy-momentum tensor as a (0,2) tensor, without mentioning the use of a metric. Is the metric always implied when writing $T_{ab}$?
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$\begingroup$ Sometimes given the context there is no need to worry about upper and lower indices. In that case, an author may write write everything in upper or lower indices. In contexts where upper and lower indices have a specified meaning, the metric is “always implied.” $\endgroup$– Jbag1212Apr 26, 2022 at 15:24
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The metric $g_{\mu\nu}$ can be used to raise and lower indices. In this case you are lowering the indices on the SEM tensor, that is converting it from contravariant to covariant. So the answer is yes. This can be true for any tensor. However, if you are looking for a general definition of the SEM Tensor, we have that $$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta S_M}{\delta g^{\mu\nu}}$$, where $S_M$ is the matter action and the $\delta$ is the functional derivative with respect to the metric. Furthermore, the SEM tensor is normally given in its covariant form but it's just a matter of convention.
