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,

X^{μ}_{ν} = η_{νσ} X^{σμ}

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.

DeleteI don't get why in f, we have $V_{\mu} V^{\mu}$=9 instead of 7. Do you introduce the metric implicitly?

ReplyDeleteStefano said "I don't get why in f, we have $##V_{\mu} V^{\mu}=9## instead of ##7##. Do you introduce the metric implicitly?"

DeleteSorry I made a mistake and the answer is 7 not 9. The metric does come into it. The answer would be 9 with a Euclidean metric. Thanks Stefano