# Einstein field eqns, with cosmological const, Newtonian limit

1. Mar 31, 2009

### Mmmm

1. The problem statement, all variables and given/known data
This question is a slightly customised version of Q18(a) P212 of Schutz

Prove that
$$G_{\alpha \beta} + \Lambda g_{\alpha \beta} = 8 \pi T_{\alpha \beta}$$
in the newtonian limit reduces to
$$\nabla^2 \phi = 4\pi \rho + \Lambda$$

(I found this result in another text, using it the remaining parts of the question in the book work nicely)

2. Relevant equations

for weak gravity
$$g^{\alpha \beta} = \eta_{\alpha \beta} + h_{\alpha \beta}$$

where
$$\eta_{\sigma \alpha} h^\sigma _\beta = h_{\alpha \beta}$$

using lorentz guage for stationary T
$$G_{\alpha \beta} = -\frac{1}{2}\nabla^2 \overline{h} _{\alpha \beta}$$

where
$$\overline{h}_{\alpha \beta} = h_{\alpha \beta}-\frac{1}{2} \eta_{\alpha \beta}{h^\lambda} _\lambda$$

Newtonian limits
$$\left T_{00}\right > \left T_{0i}\right > \left T_{ij}\right$$

$$\left \overline{h}_{00}\right > \left \overline{h}_{0i}\right > \left \overline{h}_{ij}\right$$

$$T_{00} \approx \rho$$
$$\overline{h}_{00} \approx -4\phi$$
$${h}_{00} \approx -2\phi$$

3. The attempt at a solution

$$G_{\alpha \beta} + \Lambda g_{\alpha \beta} = 8 \pi T_{\alpha \beta}$$

using the above:

$$\Rightarrow -\frac{1}{2}\nabla^2 \overline{h}_{\alpha \beta} + \Lambda (\eta_{\alpha \beta} + h_{\alpha \beta}) = 8 \pi T_{\alpha \beta}$$

non trivial eqn whaen $\alpha = \beta =0$

$$\Rightarrow -\frac{1}{2}\nabla^2 \overline{h}_{00} + \Lambda (\eta_{00} + h_{00}) = 8 \pi T_{00}$$

Newtonian limit
$$\Rightarrow -\frac{1}{2}\nabla^2 (-4\phi) + \Lambda (-1 + -2\phi) = 8 \pi \rho$$

$$\Rightarrow \nabla^2 (\phi) = 4 \pi \rho + \frac{1}{2}\Lambda +\Lambda \phi$$

it should be
$$\nabla^2 \phi = 4\pi \rho + \Lambda$$

as you can see something has gone a bit wrong somewhere.
if that $h_{00}$ were -1 it would work....

Last edited: Mar 31, 2009