- #1
JasonJo
- 429
- 2
I'm doing a reading of Carroll's General Relativity text, and I am a bit confused about one bit of tensors. Particularly the trace of a tensor.
For a (1, 1) tensor we just sum up the diagonal elements, but for other tensors this does not work. For the flat Euclidean metric for Minkowski space the trace is 4, not 2.
So how exactly do you calculate the trace of an arbitrary tensor? I.e., let's say we wanted to take the trace of a (2, 0) tensor [tex]X^{\mu \nu}[/tex]?
Carroll mentions that if we lower or raise an index, we are taking the trace of a different tensor. In the case of a (2, 0) we could lower an index and get a (1, 1) tensor, but that trace is not the trace of a (2, 0) tensor.
What exactly is the proper procedure?
For a (1, 1) tensor we just sum up the diagonal elements, but for other tensors this does not work. For the flat Euclidean metric for Minkowski space the trace is 4, not 2.
So how exactly do you calculate the trace of an arbitrary tensor? I.e., let's say we wanted to take the trace of a (2, 0) tensor [tex]X^{\mu \nu}[/tex]?
Carroll mentions that if we lower or raise an index, we are taking the trace of a different tensor. In the case of a (2, 0) we could lower an index and get a (1, 1) tensor, but that trace is not the trace of a (2, 0) tensor.
What exactly is the proper procedure?