Doubt: Why Quadratic in Matrix but Power 4 in Einstein-Rosen Metric?

[Breo]

I have a doubt since I see the next equation and the corresponding matrix:

$$ ds^2 = \Bigg( \frac{1-\frac{r_s}{4\rho}}{1+\frac{r_s}{4\rho}}\Bigg)^2 dt^2 - \Big(1+\frac{r_s}{4\rho}\Big)^4 (d\rho^2 + p^2 d\Omega_2^2) $$$$ g_{\mu\nu} =
\left( \begin{array}{ccc}
\Bigg( \frac{1-\frac{r_s}{4\rho}}{1+\frac{r_s}{4\rho}}\Bigg)^2 & 0 & 0 & 0 \\
0 & -\Big(1+\frac{r_s}{4\rho}\Big)^2 & 0 & 0 \\
0 & 0 & -\rho^2 & 0 \\
0 & 0 & 0 & -sin^2 \theta \end{array} \right) $$

My doubt comes because I see a quadratic term in the matrix: $$ g_{11} = -\Big(1+\frac{r_s}{4\rho}\Big)^2 $$ however, a power 4 term in the ds² equation. Why?

 

[PeterDonis]

the corresponding matrix

Where are you getting the "corresponding matrix" from? It doesn't look right; since the matrix is diagonal, its diagonal elements should match the terms in the line element, so $g_{11}$ should have a fourth power. Also, $g_{22}$ and $g_{33}$ are not right; they should have the fourth power factor multiplying them as well, and the factors of $\rho^2$ are incorrect. (I assume that the $p^2$ in the line element is a typo and should be $\rho^2$ .) Please give a specific reference (book or article and chapter/page/section/etc.) for the metric you posted.