Show that in a constant gravitational field $g_{\mu\nu} = g_{\mu\nu}(\vec{x})$, a transformation $x_0 \rightarrow x'_0 = x_0 + f(\vec{x})$ leads to the metric change in the form of
$g'_{00} = g_{00}$,
$g'_{0i} = g_{0i} - g_{00}f_{,i}$,
$g'_{ij} = g_{ij} + g_{00}f_{,i}f_{,j} - g_{0i}f_{,j} - g_{0j}f_{,i}$.
(1)
Here $f(\vec{x})$ is the time independent scalar function and $f_{,i} = \partial f/\partial x^i$.
2.
Show that during the above mentioned transformations 3<sup>d</sup> space metric
$\gamma_{ij} = -g_{ij} + \frac{g_{0i}g_{0j}}{g_{00}}$
(2)
will not change.
