Jtdylktuy

Assume usual polar coordinates equations:

$$ x=rcos(theta) $$
$$ y=rsin(theta) $$

In what I think is called covariant we have a Jacobian of:

$$
J=begin{pmatrix}
frac{partial{x}}{partial{r}} & frac{partial{x}}{partial{theta}} \
frac{partial{y}}{partial{r}} & frac{partial{y}}{partial{theta}} \
end{pmatrix}=begin{pmatrix}
cos(theta) & -rsin(theta) \
sin(theta) & rcos(theta) \
end{pmatrix}
$$

and a metric tensor of:
$$
g_{ij}=J^T J=begin{pmatrix}
1 & 0 \
0 & r^2 \
end{pmatrix}
$$

My problems appears when I try the contravariant part:

$$
J=begin{pmatrix}
frac{partial{r}}{partial{x}} & frac{partial{r}}{partial{y}} \
frac{partial{theta}}{partial{x}} & frac{partial{theta}}{partial{y}} \
end{pmatrix}=begin{pmatrix}
cos(theta) & sin(theta) \
-frac{sin(theta)}{r} & frac{cos(theta)}{r} \
end{pmatrix}
$$

because the metric tensor:
$$
g^{ij}=J^T J=begin{pmatrix}
cos^2(theta)+frac{sin^2(theta)}{r} & ... \
... & ... \
end{pmatrix}
$$

results different of the expected one:
$$begin{pmatrix}
1 & 0 \
0 & frac{1}{r^2} \
end{pmatrix}
$$

why the expected result is not the one I obtain?

( I obtain the expected only if I change the product from $J^T J$ to $J J^T$ ).

Maybe a bit of a preamble will be useful here. If $xi$ are some coordinates defining the local metric $eta$ then, under the transformation $x^alpha = x^{alpha}(xi)$ the metric becomes

$$
g_{munu} = frac{partial xi^{alpha}}{partial x^mu}frac{partial xi^{beta}}{partial x^nu} eta_{munu} tag{1}
$$

So for example, if you take $xi^1 = x$ and $xi^2 = y$ the cartesian coordinates, then the local matrix is the identity $eta_{munu} = delta_{munu}$ and Eq. (1) becomes

$$
g_{munu} = J^{alpha}_{;mu}J^{beta}_{;nu}delta_{alphabeta} = (J^T J)_{munu} tag{2}
$$

where the entries of the matrix $J$ are defined as

$$
J^{alpha}_{;beta} = frac{partial xi^alpha}{partial x^beta} tag{3}
$$

As an example take $x^1 = r$ and $x^2 = theta$ (your case), so that $xi^1 = rcostheta = x^1 cos x^2$ and $xi^2 = x^1sin x^2$, it is easy to calculate

$$
frac{partial xi^1}{partial x^1} = frac{partial }{partial x^1}(x^1 cos x^2) = cos x^2 ~~~(cdots)
$$

so when you evaluate that in (1) you get

$$
g_{11} = 1, ~ g_{12} = g_{21} = 0 ~mbox{and}~ g_{22} = (x^1)^2
$$

It is important to keep track of the order of things here. In this representation we are labeling the rows with the super-index $alpha$ and the columns with the sub-index $beta$.

Now, the contravariant version of Eq. (2) is

$$
g^{munu} = eta^{alpha beta}frac{partial x^mu}{partial xi^alpha} frac{partial x^nu}{partial xi^beta} tag{4}
$$

and again, under the same assumptions as before, this transforms to

$$
g^{munu} = delta^{alphabeta}mathcal{J}^mu_{;alpha}mathcal{J}^nu_{;beta} = (mathcal{J}mathcal{J}^T)^{munu} tag{5}
$$

where

$$
mathcal{J}^{alpha}_{;beta} = frac{partial x^alpha}{partial xi^beta} tag{6}
$$

Taking the same changes of coordinates you will get $x^1 = [(xi^1)^2 + (xi^2)^2]^{1/2}$ and $x^2 = arctan(xi^2/xi^1)$ so that

$$
frac{partial x^1}{partial xi^1} = frac{xi^1}{[(xi^1)^2 + (xi^2)^2]^{1/2}} ~(cdots)
$$

which results in

$$
g^{11} = 1, ~ g^{12} = g^{21} = 0, ~mbox{and}~ g^{22} = frac{1}{(xi^1)^2 + (xi^2)^2} = frac{1}{(x^1)^2}
$$

as expected