A doubt respect Einstein-Rosen metric

  • Thread starter Breo
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Main Question or Discussion Point

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?
 
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Answers and Replies

  • #2
PeterDonis
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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.
 
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