In an exercise on classical field theories, I'm trying to derive the general formula of the Energy-momentum tensor. According to the formula in the lecture notes, this tensor includes a term of minus the Lagrangian density along the diagonal. However, this term does not appear in my derivation unless I assume that the Lagrangian density does not depend explicitly on space-time.
If I do make this assumption, the term appears as a result of partial integration of the term which is the product of the Lagrangian density and the derivative of the variation parameter with respect to a component of space-time. Doing the partial integration gives us the partial derivative of the Lagrangian density with respect to this space-time component times the variation parameter. But, since we assumed that the Lagrangian density does not depend explicitly on space-time, we might as well ignore this term, and again I do not obtain the diagonal Lagrangian density term.
In other words, if we assume that the Lagrangian density does not depend on space-time explicitly, this term doesn't actually contribute to the conservation law, so we might as well ignore it.
So the questions are:
Can the Lagrangian density depend on space-time explicitly, and
Why does the Energy-momentum tensor have a diagonal Lagrangian term?