From section 1, property (6), footnote 4 "world lines of matter" means "world lines of matter occurring as such in the solution". This at least hints that, for example, world lines of observers in spacecraft with propulsion, or wordlines of particles under applied forces are not considered "world lines of matter".

In section 3, immediately after the equation $w^j=\ldots$ it is stated that "in our case" $v_i=u_i$. In section 2, $u$ has been introduced as the unit vector "in the direction of the $x_0$-lines". This fits with the 1st paragraph of section 3, where the non-existence of a one-parametric system of three-spaces orthogonal on the $x_0$-lines is mentioned, and then related to the more general case of three-spaces everywhere orthogonal to a vector field $v$.

The 1st paragraph of section 3 relates to the 1st paragraph of section 1 "... owing to the fact that there exists a one-parametric system of three-spaces everywhere orthogonal on the world-lines of matter".

That is, the "world lines of matter" are the $x_0$-lines.

This is in keeping with property (8) as stated in section 1, where the three-spaces considered are such that they intersect each "world line of matter" in one point, if related to the proof of property (8), where $U$ is such a three-space that has to have a "point in common with each $x_0$-line situated on $S_0$".