- #1

spaghetti3451

- 1,344

- 33

## Homework Statement

Imagine we have a tensor ##X^{\mu\nu}## and a vector ##V^{\mu}##, with components

##

X^{\mu\nu}=\left( \begin{array}{cccc}

2 & 0 & 1 & -1 \\

-1 & 0 & 3 & 2 \\

-1 & 1 & 0 & 0 \\

-2 & 1 & 1 & -2 \end{array} \right), \qquad V^{\mu} = (-1,2,0,-2).

##

Find the components of:

(a) ##{X^{\mu}}_{\nu}##

(b) ##{X_{\mu}}^{\nu}##

(c) ##X^{(\mu\nu)}##

(d) ##X_{[\mu\nu]}##

(e) ##{X^{\lambda}}_{\lambda}##

(f) ##V^{\mu}V_{\mu}##

(g) ##V_{\mu}X^{\mu\nu}##

## Homework Equations

## The Attempt at a Solution

(a) ##{X^{\mu}}_{\nu}=X^{\mu\rho}\eta_{\rho\nu}=\left( \begin{array}{cccc}

2 & 0 & 1 & -1 \\

-1 & 0 & 3 & 2 \\

-1 & 1 & 0 & 0 \\

-2 & 1 & 1 & -2 \end{array} \right)

\left( \begin{array}{cccc}

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \end{array} \right)=\left( \begin{array}{cccc}

-2 & 0 & 1 & -1 \\

1 & 0 & 3 & 2 \\

1 & 1 & 0 & 0 \\

2 & 1 & 1 & -2 \end{array} \right)

##,

where the rows of the left matrix are multiplied by the columns of the right matrix because the summation is over the second index of ##X^{\mu\rho}## and the first index of ##\eta_{\rho\nu}##.

(b) ##{X_{\mu}}^{\nu}=\eta_{\mu\rho}X^{\rho\nu}=

\left( \begin{array}{cccc}

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \end{array} \right)

\left( \begin{array}{cccc}

2 & 0 & 1 & -1 \\

-1 & 0 & 3 & 2 \\

-1 & 1 & 0 & 0 \\

-2 & 1 & 1 & -2 \end{array} \right)

=\left( \begin{array}{cccc}

-2 & 0 & -1 & 1 \\

-1 & 0 & 3 & 2 \\

-1 & 1 & 0 & 0 \\

-2 & 1 & 1 & -2 \end{array} \right)

##,

where the rows of the left matrix are multiplied by the columns of the right matrix because the summation is over the second index of ##\eta_{\mu\rho}## and the first index of ##X^{\rho\nu}##.

(c) ##X^{(\mu\nu)}=\frac{1}{2}(X^{\mu\nu}+X^{\nu\mu})=\frac{1}{2}\Bigg[\left( \begin{array}{cccc}

2 & 0 & 1 & -1 \\

-1 & 0 & 3 & 2 \\

-1 & 1 & 0 & 0 \\

-2 & 1 & 1 & -2 \end{array} \right)+\left( \begin{array}{cccc}

2 & -1 & -1 & -2 \\

0 & 0 & 1 & 1 \\

1 & 3 & 0 & 1 \\

-1 & 2 & 0 & -2 \end{array} \right)

\Bigg]=\left( \begin{array}{cccc}

2 & -0.5 & 0 & -1.5 \\

-0.5 & 0 & 2 & 1.5 \\

0 & 2 & 0 & 0.5 \\

-1.5 & 1.5 & 0.5 & -2 \end{array} \right)

##

(d) ##X_{[\mu\nu]}=\frac{1}{2}(X_{\mu\nu}-X_{\nu\mu})=\frac{1}{2}(\eta_{\mu\rho}X^{\rho\sigma}\eta_{\sigma\nu}-\eta_{\nu\sigma}X^{\sigma\rho}\eta_{\rho\mu})##

Are my answers to (a), (b) and (c) correct?

With part (d), I'm not sure if I should take the original matrix to ##X^{\rho\sigma}## or the transposed matrix to ##X^{\rho\sigma}##? Does it make a difference anyway?