# Find the coordinate transformation given the metric

## Homework Statement

Given the line element $ds^2$ in some space, find the transformation relating the coordinates $x,y$ and $\bar x, \bar y$.

## Homework Equations

$ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy$

$ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar x^2 + (1 + (b\bar y + c\bar x)^2) d\bar y^2 + 2(a\bar x + c\bar y)(b\bar y + c\bar x) d\bar xd\bar y$

## The Attempt at a Solution

My idea was to use the relations
$g_{\bar x\bar x} = g_{xx} (\frac{\partial {x}}{\partial {\bar x}})^2 + g_{yy} (\frac{\partial {y}}{\partial {\bar x}})^2 + 2g_{xy} (\frac{\partial {x}}{\partial {\bar x}}\frac{\partial {y}}{\partial {\bar x}})$, etc

Since the components $g_{ij}$'s of the metric are given, maybe I can isolate the x's and y's, but I'm not sure if this is correct and I can't seem to isolate and integrate. Can anyone give me any hint?

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Orodruin
Staff Emeritus
Homework Helper
Gold Member
You are missing the off diagonal terms in your relations between the metric components, but yes. That would be the general approach.

You are missing the off diagonal terms in your relations between the metric components, but yes. That would be the general approach.
I keep forgetting that off diagonal term, I've edited my post. So generally I need to isolate the variable then integrate?

Orodruin
Staff Emeritus