After a change of coordinate system on flat space from $x\rightarrow y$, we have the metric tensor:
$$g_{\mu \nu} = \frac{\partial y^{\alpha}}{\partial x^{\mu}} \frac{\partial y^{\beta}}{\partial x^{\nu}}\eta_{\alpha \beta}.$$
Now, after expanding $$y^{\alpha}= x^{\alpha}+\epsilon \xi^{\alpha},$$ I need the determinant $g$ in terms of the new variable $\xi$. Is there a standard method to do this?
Best Answer
Taking the determinant on both sides, you get:
$$g = -\left|\frac{\partial y(x)^\alpha}{\partial x^\beta}\right|^2$$
where $g = \text{det} (g_{\mu \nu})$ and $\text{det} (\eta_{\mu \nu}) = -1$. On the RHS is the Jacobian (squared) of the coordinate transformation. Can you take it from here?