Menu Close

Contravariant and Covariant Four-vectors

Unlike Euclidean space, there are “two distinct four vectors” that can be defined on this four vector scheme (4D space-time manifold),

These are,

$x_{u}$ = (ct, $x$) = (ct, $x_{1}$, $x_{2}$, $x_{3}$)

$x^{u}$ = (ct, -$x$) = (ct, -$x_{1}$, -$x_{2}$, -$x_{3}$)

 

Relation between Contravariant and Covariant Four-vector

  • Contravariant

$x_{u}$ = $\eta_{uv}x^{v}$

 

  • Covariant

$x^{u}$ = $\eta^{uv}x_{v}$

Where $u$,$v$ are the indices (0,1,2,3)

And,

$\eta_{uv}$ = $\eta^{uv}$ =  $\begin{bmatrix}1 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{bmatrix}$

 

More Related Stuff