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

Question

I am 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 understood, there are important geometric objects such as $x^0$-lines (spatially stationary particle world lines), closed timelike curves (or Gödel's helix as I like to call it) and three-spaces, but I cannot figure out what is Kurt Gödel exactly referring to when he writes "world lines of matter". As he claims in this article, he uses this object to talk about some properties of his solution, properties 2 and 6 to be precise, but when he writes property 8 and its proof it is critical to understand, both formally and heuristically, the very nature of this object.

Pardon me if I am too emphatical, maybe my English is not good enough to write in a more relaxed and elegant manner.

Answer 1

The matter world line just means the path followed by a particle with non-zero mass as it moves freely through spacetime i.e. it is the line that is the solution to the geodesic equation.

He mentions it because it can be used to create a foliation. Any "well behaved" spacetime can be represented as a globally hyperbolic manifold that allows a foliation by Cauchy surfaces. Less cryptically this means at every point along an observer's world line we can take the tangent vector to the world line as the time like direction and split the spacetime into a time axis and a 3D spatial submanifold normal to the time direction.

However Gödel's spacetime is not "well behaved" because it contains closed time like curves i.e. world lines that can intersect with themselves to form a loop. This means the spacetime is not globally hyperbolic and cannot be globally foliated.

Answer 2

Within the context of that paper “world line of matter” is the same thing as “$x_0$-line” (as explicitly stated at the bottom of first column on page 448). But keep in mind, that the first term refers to a matter content (dust) of that solution and is thus coordinate independent, whereas the second term is coordinate dependent and coincides with the first for the specific coordinate system chosen by Gödel.

Gödel universe (GU) is a solution of Einstein field equations with a negative cosmological constant and dust matter. The last part means that the stress–energy tensor has the form: $$T^{\mu\nu}=\rho u^\mu u^\nu,$$where $u^\mu$ is a global timelike vector field, $u^\mu u_\mu = 1$ (also $\rho$ is constant since GU is homogeneous and stationary). World lines of matter are just integral curves of that vector field $u$ and form a unique timelike geodesic congruence.

To emphasize:

• not every timelike geodesic in GU is a world line of matter, but only those geodesics along which the actual matter content of GU evolves.

• there are widely used coordinate systems for GU where $x_0$-lines (or possibly $t$-lines) do not correspond to matter world lines, so physical (or mathematical) statements should avoid coordinate dependent terms.