Gödel metric
Its definition is somewhat artificial, since the value of the cosmological constant must be carefully chosen to correspond to the density of the dust grains, but this spacetime is an important pedagogical example.
Like any other Lorentzian spacetime, the Gödel solution represents the metric tensor in terms of a local coordinate chart.
"Non-spinning" means that the observer does not feel centrifugal forces, but in this coordinate system, it would rotate about an axis parallel to the y-axis.
To investigate the properties of the Gödel solution, the frame field can be assumed (dual to the co-frame read from the metric as given above), This framework defines a family of inertial observers that are 'comoving with the dust grains'.
It follows that the 'non spinning inertial frame' comoving with the dust particles is The components of the Einstein tensor (with respect to either frame above) are Here, the first term is characteristic of a Lambdavacuum solution and the second term is characteristic of a pressureless perfect fluid or dust solution.
The cosmological constant is carefully chosen to partially cancel the matter density of the dust.
The Gödel spacetime is a rare example of a regular (singularity-free) solution of the Einstein field equations.
In any Lorentzian spacetime, the fourth rank Riemann tensor is a multilinear operator on the four-dimensional space of tangent vectors (at some event), but a linear operator on the six-dimensional space of bivectors at that event.
This can be rewritten as the symmetry group containing three-dimensional subgroups with examples of Bianchi types I, III, and VIII.
The Gödel solution is the Cartesian product of a factor R with a three-dimensional Lorentzian manifold (signature −++).
The Weyl tensor of the Gödel solution has Petrov type D. This means that for an appropriately chosen observer, the tidal forces are very close to those that would be felt from a point mass in Newtonian gravity.
, which components evaluated in our frame) has the form That is, they measure isotropic tidal tension orthogonal to the distinguished direction
If the past light cone of a given observer is studied, it can be found that null geodesics moving orthogonally to
spiral inwards toward the observer, so that if one looks radially, one sees the other dust grains in progressively time-lagged positions.
) appears static in the chart, the Fermi–Walker derivatives show that it is spinning with respect to gyroscopes.
It turns out that in addition, optical images are expanded and sheared in the direction of rotation.
According to Hawking and Ellis, another remarkable feature of this spacetime is the fact that, if the inessential y coordinate is suppressed, light emitted from an event on the world line of a given dust particle spirals outwards, forms a circular cusp, then spirals inward and reconverges at a subsequent event on the world line of the original dust particle.
This causal anomaly seems to have been regarded as the whole point of the model by Gödel himself, who was apparently striving to prove that Einstein's equations of spacetime are not consistent with what we intuitively understand time to be (i. e. that it passes and the past no longer exists, the position philosophers call presentism, whereas Gödel seems to have been arguing for something more like the philosophy of eternalism).
In this section, we introduce another coordinate chart for the Gödel solution, in which some of the features mentioned above are easier to see.
Start with a simple frame in a cylindrical type chart, featuring two undetermined functions of the radial coordinate: Here, we think of the timelike unit vector field
This is equivalent to requiring that it match a perfect fluid; i.e., we require that the components of the Einstein tensor, computed with respect to our frame, take the form This gives the conditions Plugging these into the Einstein tensor, we see that in fact we now have
The simplest nontrivial spacetime we can construct in this way evidently would have this coefficient be some nonzero but constant function of the radial coordinate.
in order to focus our attention on this three-manifold, let us examine how the appearance of the light cones changes as we travel out from the axis of symmetry
This is not a geodesic congruence; rather, each observer in this family must maintain a constant acceleration in order to hold his course.
According to Hawking and Ellis (see monograph cited below), all light rays emitted from an event on the symmetry axis reconverge at a later event on the axis, with the null geodesics forming a circular cusp (which is a null curve, but not a null geodesic): This implies that in the Gödel lambda dust solution, the absolute future of each event has a character very different from what we might naively expect.
Besides rotating, this model exhibits no Hubble expansion, so it is not a realistic model of the universe in which we live, but can be taken as illustrating an alternative universe, which would in principle be allowed by general relativity (if one admits the legitimacy of a negative cosmological constant).
Less well known solutions of Gödel's exhibit both rotation and Hubble expansion and have other qualities of his first model, but traveling into the past is not possible.
[4] The quality of these observations improved continually up until Gödel's death, and he would always ask "Is the universe rotating yet?"
Some have interpreted the Gödel universe as a counterexample to Einstein's hopes that general relativity should exhibit some kind of Mach's principle,[4] citing the fact that the matter is rotating (world lines twisting about each other) in a manner sufficient to pick out a preferred direction, although with no distinguished axis of rotation.
Others[citation needed] take Mach principle to mean some physical law tying the definition of non-spinning inertial frames at each event to the global distribution and motion of matter everywhere in the universe, and say that because the non-spinning inertial frames are precisely tied to the rotation of the dust in just the way such a Mach principle would suggest, this model does accord with Mach's ideas.
