In chapter §84 of Classical Theory of Fields, they introduce the so called spatial metric to measure the distance $dl$ between two infinitesimally close points in space. They send a beam of light from one point, it goes to the second point and reflects back. They write the interval between the two events as:
$ds^2 = g_{ab}dx^adx^b + 2g_{0a}dx^adx^0 + g_{00}(dx^0)^2$
Here the indices $a$ and $b$ range from $1$ to $3$. Since we have a light beam $ds^2=0$. They solve the equation to get:
$dx^0 = \frac{1}{2g_{00}} \left( -2g_{0a}dx^a \pm \sqrt{(2g_{0a}dx^a)^2 - 4g_{00}g_{ab}dx^adx^b} \right)$
The negative root is the time from sending the beam to its reflection from the second point. And the positive root is the time from reflection to going back to the initial point. The time of travel from one point to another is then, half of the difference between the two roots:
$dt=\frac{1}{g_{00}} \left( \sqrt{(g_{0a}g_{0b} - g_{00}g_{ab})dx^adx^b} \right)$
The proper time in a reference frame standing still at the point where the light beam was sent from is $d\tau = \frac{\sqrt{g_{00}}}{c}dt$
Thus the distance $dl=cd\tau$ is
$dl^2 = \left( \frac{g_{0a}g_{0b}}{g_{00}} - g_{ab} \right)dx^adx^b = \gamma_{ab} dx^a dx^b$
The coefficients $\gamma_{ab}$ they call the spatial metric.
But then, they proceed to say it makes no sense to integrate the $dl$'s between two points, because the metric generally depends on $x^0$. But don’t you integrate along constant $x^0$? Thus making the dependence of the metric on $x^0$ irrelevant.