Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 6; Problem 6.3.

The *trace* of a rank-2 tensor is given by the contraction . In matrix terminology, it is the sum of the diagonal elements. If we start with a contravariant tensor , then we can calculate the trace as follows:

That is, we first lower the second index, then contract the top and bottom indices.

Since the trace contains no free index, it should be a scalar, which means it should be invariant. We can prove this by doing the transformation.

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guogood! the concept of the trace of metric(tensor) is good.