- #1
Einj
- 464
- 59
Hello everyone, I already know that the solution to this question is obvious but I can't find it.
Consider an expanding universe following the FRW metric ds^2=-dt^2-a^2(t)dx^2 (1 space dimension for simplicity). We know that the physical spatial distance x_p is related to the comoving spatial distance x_c by x_p=ax_c. 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 \eta_{\alpha\beta} 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 g_{\mu\nu} 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?
Consider an expanding universe following the FRW metric ds^2=-dt^2-a^2(t)dx^2 (1 space dimension for simplicity). We know that the physical spatial distance x_p is related to the comoving spatial distance x_c by x_p=ax_c. 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 \eta_{\alpha\beta} 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 g_{\mu\nu} 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?
