Difference between arbitrary basis and coordinate basis and Abstract Index Notation

Question

I am teaching myself GR reading Harvey Reall's notes on GR (referring to pages 22 and 24 - not of the pdf but of the page labelling) I encountered the concept of an arbitrary and coordinate basis. What exactly is the difference between two? And what is this abstract index notation all about? We use Latin Indices to denote Vectors and Greek indices to denote components of Vectors. But why then can one replace the Latin Indices sometimes with Greek ones?

Answer 1

One frequently encounters quantities which depend on positional coordinates, but whose numerical values are independent of the choice of the coordinate system. Examples of this kind are represented by lengths of vectors, the density of a material or the magnitude of the gradient of a temperature field.

The most important property of a tensorial quantity is the ability to form such invariant expressions that evaluate to the same value in all coordinate systems.

Let us illustrate this by an example. In a Cartesian coordinate system of dimension $n=2$ there lives a vector $\bar{a}=4\bar{e}_1+3\bar{e}_2$ at point $P=(\bar{x}_1, \bar{x}_2)=(\sqrt{2},\sqrt{2})$. The components of this vector at $P$ is $\bar{a}^\rho=(4,3)$.

If we instead use a polar coordinate system at the same point $P=(2,\pi/4)$ the components of the same vector is $a^{\nu}=\frac{1}{\sqrt{2}}(7,-1/2)$.

What is the length of $a^b$? Surely it must be the same in both coordinate systems. To measure this we obviously need an impartial (invariant) expression that is valid in any coordinate system.

Unsurprisingly there is such an expression. Using abstract index notation (ain) the length of a vector is $||a^b||^2=g_{hk}a^h a^k$. We just need to transform this into the corresponding expression for the components in each coordinate system. For the polar coordinate system we have $a^\mu =\frac{1}{\sqrt{2}}(7,-\frac{1}{2}),\, a_\nu=(\frac{7}{\sqrt{2}},-\sqrt{2})$ and

$$g_{\mu\nu}=\left( \begin{array}{cc} 1 & 0 \\ 0 & r^2 \\ \end{array} \right),\quad ||a^b||^2=g_{\mu\nu}a^\mu a^\nu=a^\nu a_\nu=25$$

For the Cartesian coordinate system we have $\bar{a}^\rho=(4,3)$ and

$$\bar{g}_{\rho\sigma}=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right),\quad ||a^b||^2=\bar{g}_{\rho\sigma}\bar{a}^\rho \bar{a}^\sigma=\bar{a}^\sigma \bar{a}_\sigma=25$$

The abstract index notation (ain) is a convenient way to express tensorial quantities and form invariants without reference to a specific coordinate system. With some care you can convert an expression from regular index notation to ain when you are sure the expression is indeed tensorial. Notice the difference between the components $\bar{a}^\mu$ and $a^\nu$, yet they express the same vector $a^b$.