Undergrad Metric for Lambdavacuum EFE - Radial Coordinates

[redtree]

I am having trouble finding the equation for the metric for the Lambdavacuum solution to the EFE in radial coordinates. Any suggestions?

 

[davidge]

The metric doesn't change with the addition of the Lambda term. So, it's the same as the metric of a 2-sphere. Can you get that?

EDIT: I didn't see you were looking for a vacuum solution. The metric should then be the Minkowski Metric.

 

[redtree]

The vacuum field solution of empty space with no cosmological constant produces the Minkowski metric:

\begin{equation}

d\vec{s}^2=d\vec{x}^2-dt^2

\end{equation}In radial coordinates, the vacuum field solution around a spherical mass produces the Schwarzschild metric:

\begin{equation}

\begin{split}

d\vec{s}^2&=\left(\frac{1}{1-\frac{\vec{r}_s}{\vec{r}}} \right) d\vec{r}^2 + \vec{r}^2 \left(d\theta^2 + \sin^2\theta d\varphi^2 \right) -\left(1 - \frac{\vec{r}_s}{\vec{r}} \right) dt^2

\end{split}

\end{equation}Which as $\vec{r}\to \infty$ becomes the Minkowski metric, where $\vec{x}=[\vec{r}, \theta, \varphi]$.I am still not sure how to write metric of the Lambdavacuum solution. What is $d\vec{s}^2$ as a function of $\Lambda$ and $g_{\mu \nu}$?.

 

[vanhees71]

For the Schwarzschild solution including $\Lambda$ see the following problem and solution sheets from our General Relativity lecture in Frankfurt:

http://th.physik.uni-frankfurt.de/~hees/art-ws16/sheet06.pdf
http://th.physik.uni-frankfurt.de/~hees/art-ws16/lsg06.pdf

 

[Jorrie]

I am having trouble finding the equation for the metric for the Lambdavacuum solution to the EFE in radial coordinates. Any suggestions?

Also look at Carroll's lecture notes from eq. 8.7 onward and you will see the metric and how it relates to Einstein's field equations, including the cosmological constant.

 

[PeterDonis]

I am still not sure how to write metric of the Lambdavacuum solution

It's the de Sitter metric; see here:

https://en.wikipedia.org/wiki/De_Sitter_space

If by "radial coordinates" you mean coordinates with a radial coordinate $r$ defined the way it is in Schwarzschild coordinates (such that the area of a 2-sphere at radial coordinate $r$ is $4 \pi r^2$), those are the "static coordinates" described at that link.

 

[redtree]

That's great. Thanks!

Given:
\begin{equation}
\begin{split}
d\vec{s}^2&=\left(\frac{1}{1-\frac{\vec{r}_s}{\vec{r}}-\frac{\Lambda \vec{r}^2}{3}} \right) d\vec{r}^2 + \vec{r}^2 \left(d\theta^2 + \sin^2\theta d\varphi^2 \right) -\left(1 - \frac{\vec{r}_s}{\vec{r}}-\frac{\Lambda \vec{r}^2}{3} \right) dt^2
\end{split}
\end{equation}

Am I correct in the following, where $\tilde{\infty}$ denotes complex infinity?:
\begin{equation}
\begin{split}
\lim_{\vec{r}\to \infty} d\vec{s}&=\tilde{\infty}
\end{split}
\end{equation}

With a singularity at the transition between real and imaginary values for $\vec{s}$.

 

[vanhees71]

It's the de Sitter metric; see here:

https://en.wikipedia.org/wiki/De_Sitter_space

If by "radial coordinates" you mean coordinates with a radial coordinate $r$ defined the way it is in Schwarzschild coordinates (such that the area of a 2-sphere at radial coordinate $r$ is $4 \pi r^2$), those are the "static coordinates" described at that link.

Sure, this is what I quoted for $m=0$ (i.e., $r_{\text{S}}=0$).

 

[redtree]

Given:
\begin{equation}
\begin{split}
d\vec{s}^2&=\left(\frac{1}{1-\frac{\vec{r}_s}{\vec{r}}-\frac{\Lambda \vec{r}^2}{3}} \right) d\vec{r}^2 + \vec{r}^2 \left(d\theta^2 + \sin^2\theta d\varphi^2 \right) -\left(1 - \frac{\vec{r}_s}{\vec{r}}-\frac{\Lambda \vec{r}^2}{3} \right) dt^2
\end{split}
\end{equation}

Am I right in understanding $\Lambda$ is negative such that $\frac{\Lambda \vec{r}^2}{3}$ is positive?