I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll. The blog contains answers to his exercises, commentaries, questions and more.
Saturday, 14 April 2018
Exercise 1.07 Tensors and Vectors
The Question was
To see my answer Click Here. The formulae are too complicated to put into this simple editor.
One thing that Sean Carroll had omitted to tell us was that an index can be raised or lowered by applying the metric tensor. So for example,
Xμν = ηνσ Xσμ
This is pretty vital for this exercise! The Wikipedia article is here.
No, no, no! Lambda is repeated so it is summed over. ##X_{\ \ \lambda}^\lambda=X_{\ \ 0}^0+X_{\ \ 1}^1+X_{\ \ 2}^2+X_{\ \ 3}^3## so it is just the sum of the diagonal components of ##X_{\ \ \nu}^\mu## which was the question (a). ##X_{\ \ \lambda}^\lambda## is a scalar.
I think point e is a 4-vector: each lambda is a component
ReplyDeleteNo, no, no! Lambda is repeated so it is summed over. ##X_{\ \ \lambda}^\lambda=X_{\ \ 0}^0+X_{\ \ 1}^1+X_{\ \ 2}^2+X_{\ \ 3}^3## so it is just the sum of the diagonal components of ##X_{\ \ \nu}^\mu## which was the question (a). ##X_{\ \ \lambda}^\lambda## is a scalar.
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