- #1

yuiop

- 3,962

- 20

With reference to this wikipedia article http://en.wikipedia.org/wiki/Lemaitre_metric

it states

[tex]r = \left(\frac{3}{2}(p-\tau)\right)^{2/3}\ r_g^{1/3} [/tex]

and

[tex] r_g = \frac{3}{2}(p-\tau) [/tex]

Those two statements put together imply

[tex]r = \left(\frac{3}{2}(p-\tau)\right)^{2/3} \left(\frac{3}{2}(p-\tau)\right)^{1/3} [/tex]

[tex]r = r_g[/tex]

This in turn implies the Schwarzschild radial variable r is equal to the constant Schwarzschild radius [itex]r_s = 2gm/c^2[/itex] and this leads to the Lemaitre metric being equivalent to

[tex] ds^2 = d\tau^2 - dp^2 [/tex]

and when [itex]r_g/r = 1[/itex] is inserted into the Lemaitre coordinate definitions:

[tex]\begin{cases}

d\tau = dt + \sqrt{\frac{r_{g}}{r}}\frac{1}{(1-\frac{r_{g}}{r})}dr~,\\

d\rho = dt + \sqrt{\frac{r}{r_{g}}}\frac{1}{(1-\frac{r_{g}}{r})}dr~.

\end{cases}[/tex]

the result is:

[tex]\begin{cases}

d\tau = dt \pm\ \frac{dr}{0}~,\\

d\rho = dt \pm\ \frac{dr}{0}~.

\end{cases}[/tex]

Obviously I am missing something important here. Can anyone clarify?

it states

[tex]r = \left(\frac{3}{2}(p-\tau)\right)^{2/3}\ r_g^{1/3} [/tex]

and

[tex] r_g = \frac{3}{2}(p-\tau) [/tex]

Those two statements put together imply

[tex]r = \left(\frac{3}{2}(p-\tau)\right)^{2/3} \left(\frac{3}{2}(p-\tau)\right)^{1/3} [/tex]

[tex]r = r_g[/tex]

This in turn implies the Schwarzschild radial variable r is equal to the constant Schwarzschild radius [itex]r_s = 2gm/c^2[/itex] and this leads to the Lemaitre metric being equivalent to

[tex] ds^2 = d\tau^2 - dp^2 [/tex]

and when [itex]r_g/r = 1[/itex] is inserted into the Lemaitre coordinate definitions:

[tex]\begin{cases}

d\tau = dt + \sqrt{\frac{r_{g}}{r}}\frac{1}{(1-\frac{r_{g}}{r})}dr~,\\

d\rho = dt + \sqrt{\frac{r}{r_{g}}}\frac{1}{(1-\frac{r_{g}}{r})}dr~.

\end{cases}[/tex]

the result is:

[tex]\begin{cases}

d\tau = dt \pm\ \frac{dr}{0}~,\\

d\rho = dt \pm\ \frac{dr}{0}~.

\end{cases}[/tex]

Obviously I am missing something important here. Can anyone clarify?

Last edited: