In Schutz says When we have weak gravitaional fields then the line element *ds* is(adsbygoogle = window.adsbygoogle || []).push({});

$$

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

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

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