Covariant and contravariant basis

On a surface, basically, we must suppose

(1)
\begin{align} \boldsymbol{g}_i\cdot\boldsymbol{g}_j\neq \delta_{ij}. \end{align}

Hence, sometimes introducing another set of basis such that

(2)
\begin{align} \boldsymbol{g}_i\cdot\boldsymbol{g}^j=\delta_i^{\cdot j}, \end{align}

makes equations simpler.
These basis are referred to as dual basis. Additionally, $\boldsymbol{g}_i$ is called covariant basis, and $\boldsymbol{g}^i$ is called contravariant basis. Basically, $\delta_i^{\cdot j}$ is substantially same as Kronecker's delta $\delta_{ij}$, i.e., $\delta_i^{\cdot j}=0$ when $i=j$ and $\delta_i^{\cdot j}=1$ otherwise.
Given a tangent vector $\boldsymbol{a}$, to obtain sufficient information about this vector, it is sufficient to take inner product with two independent vectors. Of course, we chose basis as those vectors.
For example, we define

(3)
\begin{align} a_1=\boldsymbol{a}\cdot\boldsymbol{g}_1, \end{align}

and

(4)
\begin{align} a_2=\boldsymbol{a}\cdot\boldsymbol{g}_2, \end{align}

and call $a_1,a_2$ covariant components. On the other hand, we define

(5)
\begin{align} a^1=\boldsymbol{a}\cdot\boldsymbol{g}^1, \end{align}

and

(6)
\begin{align} a^2=\boldsymbol{a}\cdot\boldsymbol{g}^2, \end{align}

and call $a^1,a^2$ contravariant components.
Because there are two types of components, each tangent vector has two representations, i.e,

(7)
\begin{align} \boldsymbol{a}=a_1\boldsymbol{g}^1+a_2\boldsymbol{g}^2 \end{align}

and

(8)
\begin{align} \boldsymbol{a}=a^1\boldsymbol{g}_1+a^2\boldsymbol{g}_2. \end{align}

By taking inner product with Eq. (7) with $\boldsymbol{g}_i$, we obtain Eq. (5).
So as Eq. (8).
Contrary, by taking inner product with Eq. (7) with $\boldsymbol{g}^i$, we obtain

(9)
\begin{equation} a^i=a_1g^{1i}+a_2g^{2i} \end{equation}

and

(10)
\begin{equation} a_i=a^1g_{1i}+a^2g_{2i}. \end{equation}

Covariant and contravariant components can be converted as such. This translation can be performed with metric only. Using summation convention, we can write

(11)
\begin{align} a^\alpha=a_ig^{i\alpha} \end{align}

and

(12)
\begin{align} a_\alpha=a^ig_{i\alpha}. \end{align}

This is known as raising and lowering indices laws of tensors. Also, this is a generalization to an arbitrary dimension.
For contravaiant basis, we can define

(13)
\begin{align} g^{ij}=\boldsymbol{g}^i\cdot\boldsymbol{g}^j. \end{align}

This "the other metric" can be computed by

(14)
\begin{align} \left[\begin{array}{ccc}g^{11}g^{12}\\g^{12}g^{22} \end{array}\right]=\left[\begin{array}{ccc}g_{11}g_{12}\\g_{12}g_{22} \end{array}\right]^{-1}. \end{align}

Actually, in a most common manner, $g^{ij}$ is calculated first and then $\boldsymbol{g}^i$ are calculated second by using

(15)
\begin{align} \boldsymbol{g}^1=g^{11}\boldsymbol{g}_1+g^{12}\boldsymbol{g}_2 \end{align}

and

(16)
\begin{align} \boldsymbol{g}^2=g^{12}\boldsymbol{g}_1+g^{22}\boldsymbol{g}_2. \end{align}

Note that these satisfy Eq. (2). Using summation convention, we can write

(17)
\begin{align} \boldsymbol{g}^{j}=g^{ij}\boldsymbol{g}_i \end{align}

Because we have two different types of components, there are four ways to calculate inner products between tangent vectors and they are:

(18)
\begin{align} <\boldsymbol{a},\boldsymbol{b}>=a^ib^jg_{ij}=a_ib_jg^{ij}=a_ib^j=a^ib_j. \end{align}

They give the same scalar.

page revision: 7, last edited: 13 May 2015 04:22