# I Help with Newtonian Gravity as Limit case of General Relativ

1. Nov 28, 2016

### needved

In Schutz says When we have weak gravitaional fields then the line element *ds* is
$$ds^{2}=-(1+2\phi)dt^{2}+(1-2\phi)(dx^{2}+dy^{2}+dz^{2})$$
so the metric is
$${g_{\alpha\beta}} =\eta_{\alpha\beta}+h_{\alpha\beta}= \left( \begin{array}{cccc} -(1+2\phi) & 0 & 0 & 0\\ 0 & (1-2\phi) & 0 & 0\\ 0 & 0 & (1-2\phi) & 0\\ 0 & 0 & 0 & (1-2\phi)\end{array} \right)$$
where
$$\phi=\frac{M}{r}$$
so *h* is

$${h_{\alpha\beta}} = \left( \begin{array}{cccc} -2\phi & 0 & 0 & 0\\ 0 & -2\phi & 0 & 0\\ 0 & 0 & -2\phi & 0\\ 0 & 0 & 0 & -2\phi\end{array} \right)$$
the element

$$h_{00}= -2\phi$$

and the elements out of the diagonal are zero because the condition weak gravitational fields it implies

$$T_{i,j}=0$$

but i dont get it how in the book do

$$h_{xx}=h_{yy}=h_{zz}=-2\phi$$

i believe they use de definition of *trace reverse*

$$\bar h^{\alpha\beta}=h^{\alpha\beta}-\frac{1}{2}\eta^{\alpha\beta}h$$
and the *trace* definition

$$h = h^{\alpha}_{\alpha}$$
but how they do? what im missing?

Thanks

2. Nov 28, 2016

### Quantum_gravity

We are just pulling each diagonal term out of h. h1 = hx. Repeated subscript is an element along the diagonal.