## 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.

X[μν] = X[μν]
I don't know if that is significant.

1. I think point e is a 4-vector: each lambda is a component

1. 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.

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

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

Sorry 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