In 1949, Kurt Godel published a paper discussing a four-dimensional GR spacetime with closed timelike curves (CTCs) [Godel, 1949]. He started by writing

down the line element

where a is a positive constant with the dimensions of a length and t, x, y, and я are taken dimensionless. He explored the consequences physically and geometrically.

Physics of Godel’s spacetime

The physics of this spacetime is explored using Einstein’s field equations for GR (q.v. Appendix):

where the R^_{v} are the components of the Ricci tensor, R is the Ricci scalar field, g^_{v} are the components of the metric tensor that can be read off from the line element (20.11), Л is the cosmological constant, T^_{v} are the components of the stress-energy-momentum tensor, and к = 8n G/c^{4}, where G is the Newtonian constant of gravitation.

The Ricci tensor and the Ricci scalar can be readily calculated knowing the line element. The important physical properties of this spacetime are these:

1. The Ricci scalar R is found to be R = -a^{-2}, which means that this spacetime is homogeneous: there are no special points where there is a singularity.

2. Assuming this spacetime is that of the universe, we can model the stress- energy-momentum tensor as that due to a dustlike distribution of matter. This means that galaxies are regarded as individual grains of dust in a cloud of dust. Now one of the characteristics of a dustlike system is that there is no pressure in the system coming from the particles themselves: the grains of dust are thinly spread out and essentially do not interact with each other directly. In such a case, the stress-energy-momentum tensor for a dust cloud is assumed to have the form T^^{v} = pu^u^{v}, where p is the intrinsic mass density and u are the components of the flow four-vector field associated with the dust cloud, satisfying the constraint u^u^ = c^{2}.

3. Assuming the Ricci tensor can be written in the form

where a and в are to be determined, a simple calculation gives two solutions: (i) a_{1} = 0, в_{1} = 1/a^{2}c^{2} and (ii) a_{2} = 0, в_{2} = -1/a^{2}c^{2}. The first solution then gives p_{1} = -1/a^{2}c^{2}к, which is negative and therefore rejected.^{[1]}

Hence solution (ii) applies and we find

That the cosmological constant is negative here is of interest. When Einstein introduced such a term in his field equations, it was intended to provide a repulsive effect to counteract the inherent attractive nature of gravity, so that his model universe would be static. At that time, Einstein was unaware of the Lemaltre-Hubble expansion of the universe. In the weak field, Newtonian space-time limit, the gravitational field intensity for a mass M located at the origin of coordinates is given by

so we see that a positive Л represents a weak, long-range repulsion. Godel’s negative Л therefore acts in the opposite way, acting as a form of pressure serving to stabilize his spacetime. This pressure has to be regarded as an inherent property of empty space, if the stress-energy-momentum tensor is purely dust-like.

4. To understand the role of Godel’s negative cosmological constant, we need to look at the geodesics, or free particle motion, in his spacetime. Having established that his spacetime was acceptable in terms of energy density, Godel went on to discuss the compass of inertia, the behaviour of free particle worldlines in his spacetime. He showed that the off-diagonal terms in the metric tensor acted to create a ‘swirling’ effect, such that freely moving objects do not lock onto the distant stars but their trajectories start to veer away. The effect is unlike that found with some other spacetimes with intrinsic rotation, such as van Stockum’s rotating infinite cylinder metric [van Stockum, 1937], because Godel’s spacetime is homogeneous. Godel’s negative cosmological constant acts as a form of centripetal force, stabilizing his model universe.

[1] Godel’s spacetime has positive energy density everywhere. Negative energy density is a commonproblem with worm hole solutions in GR.