2
$\begingroup$

A 2D space has coordinates $x^1$ and $x^2$ with line element

$$\mathrm{d} l^{2}=\left(\mathrm{d} x^{1}\right)^{2}+\left(x^{1}\right)^{2}\left(\mathrm{~d} x^{2}\right)^{2}.$$

I'm looking to find the metric tensor and its dual metric. I understand the basics of summing over $g_{ij}$ for $n=2$ and using the Kronecker delta but the second non $dx^1$ term is tripping me up. I don't know how to use it in summing as its not a derivative.

$\endgroup$
5
$\begingroup$

Your question is a bit unclear but as I understand it you want to derive the metric tensor of this line element and its inverse. You have to remember that $dl^2 = g_{i j}dx^i dx^j$. So your metric is $\text{diag}(1,(x^1)^2)$. And so its inverse is $\text{diag}(1,(x^1)^{-2})$.

Edit for clarity :

\begin{equation} dl^2=g_{i j}dx^i dx^j=g_{11}(dx^1)^2+2g_{12}dx^1 dx^2+g_{22}(dx^2)^2 \end{equation}

So one has $g_{11}=1$, $g_{12}=0$ and $g_{22}=(x^1)^2$. Since the metric is diagonal finding its inverse is straightforward : $g^{11}=1$, $g^{12}=g^{21}=0$ and $g^{22}=(x^1)^{-2}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.