Weak field Newtonian limit

[lailola]

Hello,

I have to calculate the gravitational field strength g given by the De Sitter-Schwarzschild spacetime.

If G=c=1 I get:

$R_{00}\simeq -kT_{00}+\frac{1}{2}kT\eta_{00}+\Lambda \eta_{00}\simeq$

$-kT_{00}+\frac{1}{2}kT_{00}\eta^{00}\eta_{00} +\Lambda \eta_{00}\simeq -\frac{1}{2}kT_{00}+ \Lambda =-\frac{1}{2}k\rho+ \Lambda$

On the other side:

$R_{00}\simeq -\Gamma_{00,j}^j \simeq -\frac{1}{2}\bigtriangledown^2g_{00}\simeq -\bigtriangledown^2 \phi$

Equaling:

$\bigtriangledown^2 \phi=\frac{1}{2}k\rho-\Lambda=4piG\rho-\Lambda$

The first problem I have is that I don't know how to get this result in normal units, and the second problem is that, from here, I have to find the gravitational field strength g, and I don't know how to do it.

thanks for any help!

 

[haushofer]

What precisely are "normal units"? I would say the units as you use here are just fine. Second, the gravitational field strength is given by

$$g^i \equiv \partial^i \phi$$

whereas you have found an expression for $\partial_i \partial^i \phi$. Looking at the general method to solve the Poisson equation for lambda=0 should help you, e.g. here:

http://cass.ucsd.edu/~ppadoan/new_website/physics105b/Lecture3.pdf

You have an extra constant, which modifies this solution. See also

http://arxiv.org/abs/gr-qc/0004037,

eqn.2.6.

 

[lailola]

With 'normal' units I mean G=c=1

 

[lailola]

What precisely are "normal units"? I would say the units as you use here are just fine. Second, the gravitational field strength is given by

$$g^i \equiv \partial^i \phi$$

whereas you have found an expression for $\partial_i \partial^i \phi$. Looking at the general method to solve the Poisson equation for lambda=0 should help you, e.g. here:

http://cass.ucsd.edu/~ppadoan/new_website/physics105b/Lecture3.pdf

You have an extra constant, which modifies this solution. See also

http://arxiv.org/abs/gr-qc/0004037,

eqn.2.6.

Thank you haushofer, those links have helped me a lot.

 

[haushofer]

With 'normal' units I mean G=c=1

Well, there is no c in your equations, so that's easy: just put G=1 :P

Glad I could help :)