What exactly is Göedel reffering to when he says "world lines of matter" in his 1949 article about his solution to Einstein Field Equations?

Question

Im writing my college thesis about Gödel's article "A New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation".

As far as I had undestand, there are important geometric object such as x^0-lines (spatial stationary particle world lines), closed timelike curves (or Gödel's helix as I like to call it) and three-spaces, but I cant figure out what is Kurt exactly reffering to when he writes "world lines of matter". As he claims in this article, he uses this object to talk about some of properties of his solution, properties 2 and 6 to be precise, but when he writes propertie 8 and its proof is critical to understand, both formally and heuristically, the very nature of this object.

Pardon me if I am being too emphatical, maybe my english is not as good to write in a more relaxed and ellegant manner.

Thanks for your help.

Comments

World line of matter refers to the trajectory of physical objects in spacetime. The Gödel's theory of the rotating universe theorizes that the WLM follow paths that could potentially go back in time. Another interesting consequence of this theory is here

Answer 1

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 $S_0$".