Hello everyone, I already know that the solution to this question is obvious but I can't find it.(adsbygoogle = window.adsbygoogle || []).push({});

Consider an expanding universe following the FRW metric [itex] ds^2=-dt^2-a^2(t)dx^2[/itex] (1 space dimension for simplicity). We know that the physical spatial distance [itex]x_p[/itex] is related to the comoving spatial distance [itex]x_c[/itex] by [itex]x_p=ax_c[/itex]. First of all, is this correct?

Now we also know that the metric in the physical coordinates is related to the metric in the comoving coordinates by:

$$

g_{\mu\nu}=\frac{\partial x_c^\alpha}{\partial x_p^\mu}\frac{\partial x_c^\beta}{\partial x_p^\nu}\eta_{\alpha\beta},

$$

where [itex]\eta_{\alpha\beta}[/itex] is the flat metric. Now, computing explicitely the transformation matrix I got:

$$

\frac{\partial x_c^\alpha}{\partial x_p^\mu}=\left(\begin{array}{cc}

1 & 0 \\ 0 & 1/a \end{array}\right).

$$

Is this correct?

But if this is true then it seems to me that applying it to the equation for [itex]g_{\mu\nu}[/itex] one gets:

$$

g_{\mu\nu}=\left(\begin{array}{cc} -1 & 0 \\ 0 & 1/a^2 \end{array}\right).

$$

How is this possible? What am I doing wrong?

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# Dumb question on FRW metric

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