Evaluating T(A,A) and Components of T(A,__) in Minkowski Spacetime

[Rasalhague]

Excercise Exercise 1.6 of Roger Blandford and Kip Thorne's online textbook Applications of Classical Physics:

"In Minkowski spacetime, in some inertial reference frame, the vector A and second rank
tensor T have as their only nonzero components A⁰ = 1, A¹ = 2, A² = A³ = 0. T⁰⁰ = 3, T⁰¹ = T¹⁰ = 2, T¹¹ = −1. Evaluate T(A, A) and the components of T(A,__) and A\otimesT."

http://www.pma.caltech.edu/Courses/ph136/yr2008/

The given answers to the first two questions are T(A, A) = -9, and T(A,__) = (1, -4, 0, 0). But I get 7, and (7, 0, 0, 0).

\left( \begin{matrix} 1 & 2 & 0 & 0 \end{matrix} \right) \left(\begin{matrix} 3 & 2 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right) = \left( \begin{matrix} 7 & 0 & 0 & 0 \end{matrix} \right)

\left( \begin{matrix} 7 & 0 & 0 & 0 \end{matrix} \right) \left( \begin{matrix} 1 \\ 2 \\ 0 \\ 0 \end{matrix} \right) = 7

 

[hamster143]

You forgot to lower the indices of A before doing the multiplication.

 

[Rasalhague]

Thanks! Yes, that's better:

\left( \begin{matrix} -1 & 2 & 0 & 0 \end{matrix} \right) \left(\begin{matrix} 3 & 2 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right) = \left( \begin{matrix} 1 & -4 & 0 & 0 \end{matrix} \right)

\left( \begin{matrix} 1 & -4 & 0 & 0 \end{matrix} \right) \left( \begin{matrix} -1 \\ 2 \\ 0 \\ 0 \end{matrix} \right) = -9