Gödel metric


The Gödel metric, also known as the Gödel solution or Gödel universe, is an exact solution, found in 1949 by Kurt Gödel, of the Einstein field equations in which the stress–energy tensor contains two terms: the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a negative cosmological constant.
This solution has many unusual properties—in particular, the existence of closed time-like curves that would allow time travel in a universe described by the solution. 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.

Definition

Like any other Lorentzian spacetime, the Gödel solution represents the metric tensor in terms of a local coordinate chart. It may be easiest to understand the Gödel universe using the cylindrical coordinate system, but this article uses the chart originally used by Gödel. In this chart, the metric is
where is a non-zero real constant that gives the angular velocity of the surrounding dust grains about the y-axis, measured by a "non-spinning" observer riding on one of the dust grains. "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. In this rotating frame, the dust grains remain at constant values of x, y, and z. Their density in this coordinate diagram increases with x, but their density in their own frames of reference is the same everywhere.

Properties

To investigate the properties of the Gödel solution, the frame field can be assumed,
This framework defines a family of inertial observers that are 'comoving with the dust grains'. The computation of the Fermi–Walker derivatives with respect to shows that the spatial frames are spinning about with the angular velocity. It follows that the 'non spinning inertial frame' comoving with the dust particles is

Einstein tensor

The components of the Einstein tensor 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.

Topology

The Gödel spacetime is a rare example of a regular solution of the Einstein field equations. Gödel's original chart is geodesically complete and free of singularities. Therefore, it is a global chart, and the spacetime is homeomorphic to R4, and therefore, simply connected.

Curvature invariants

In any Lorentzian spacetime, the fourth rank Riemann tensor is a multilinear operator on the four-dimensional space of tangent vectors, but a linear operator on the six-dimensional space of bivectors at that event. Accordingly, it has a characteristic polynomial, whose roots are the eigenvalues. In Gödelian spacetime, these eigenvalues are very simple:
This spacetime admits a five-dimensional Lie algebra of Killing vectors, which can be generated by 'time translation', two 'spatial translations', plus two further Killing vector fields:
and
The isometry group acts 'transitively', so spacetime is 'homogeneous'. However, it is not 'isotropic', as can be seen.
The given demonstrators show that the slices admit a transitive abelian three-dimensional transformation group, so that a quotient of the solution can be reinterpreted as a stationary cylindrically symmetric solution. The slices allow for an SL action, and the slices admit a Bianchi III. This can be rewritten as the symmetry group containing three-dimensional subgroups with examples of Bianchi types I, III, and VIII. Four of the five Killing vectors, as well as the curvature tensor do not depend on the coordinate y. The Gödel solution is the Cartesian product of a factor R with a three-dimensional Lorentzian manifold.
It can be shown that, except for the local isometry, the Gödel solution is the only perfect fluid solution of the Einstein field equation which admits a five-dimensional Lie algebra of the Killing vectors.

Petrov type and Bel decomposition

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.
To study the tidal forces in more detail, the Bel decomposition of the Riemann tensor can be computed into three pieces, the tidal or electrogravitic tensor, the magnetogravitic tensor, and the topogravitic tensor.
Observers comoving with the dust particles would observe that the tidal tensor has the form
That is, they measure isotropic tidal tension orthogonal to the distinguished direction.
The gravitomagnetic tensor vanishes identically
This is an artifact of the unusual symmetries of this spacetime, and implies that the putative "rotation" of the dust does not have the gravitomagnetic effects usually associated with the gravitational field produced by rotating matter.
The principal Lorentz invariants of the Riemann tensor are
The vanishing of the second invariant means that some observers measure no gravitomagnetism, which is consistent with what was just said. The fact that the first invariant is constant reflects the homogeneity of the Gödel spacetime.

Rigid rotation

The frame fields given above are both inertial,, but the vorticity vector of the timelike geodesic congruence defined by the timelike unit vectors is
This means that the world lines of nearby dust particles are twisting about one another. Furthermore, the shear tensor of the congruence vanishes, so the dust particles exhibit rigid rotation.

Optical effects

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. However, the solution is stationary, so it might seem that an observer riding on a dust grain will not see the other grains rotating about oneself. However, recall that while the first frame given above appears static in the chart, the Fermi–Walker derivatives show that it is spinning with respect to gyroscopes. The second frame appears to be spinning in the chart, but it is gyrostabilized, and a non-spinning inertial observer riding on a dust grain will indeed see the other dust grains rotating clockwise with angular velocity about his axis of symmetry. It turns out that in addition, optical images are expanded and sheared in the direction of rotation.
If a non-spinning inertial observer looks along his axis of symmetry, one sees one's coaxial non-spinning inertial peers apparently non-spinning with respect to oneself, as would be expected.

Shape of absolute future

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 means that observers looking orthogonally to the direction can see only finitely far out, and also see themselves at an earlier time.
The cusp is a non-geodesic closed null curve.

Closed timelike curves

In a flat spacetime, a circle drawn around a timelike axis, would itself be a spacelike loop. That is also true in Gödel spacetime for small enough circles, but if we increase their size, they will shift to further frames, that are twisted relative to the close ones. This twisting eventually turns the loop's direction into the time cone, so the larger loops become closed timelike curves. Since Gödel spacetime is homogeneous, that means there are CTCs through every event in it. 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.
Einstein was aware of Gödel's solution and commented in Albert Einstein: Philosopher-Scientist that if there are a series of causally-connected events in which "the series is closed in itself", then this suggests that there is no good physical way to define whether a given event in the series happened "earlier" or "later" than another event in the series:

In that case the distinction "earlier-later" is abandoned for world-points which lie far apart in a cosmological sense, and those paradoxes, regarding the direction of the causal connection, arise, of which Mr. Gödel has spoken.
Such cosmological solutions of the gravitation-equations have been found by Mr. Gödel. It will be interesting to weigh whether these are not to be excluded on physical grounds.

Globally nonhyperbolic

If the Gödel spacetime admitted any boundaryless temporal hyperslices, any such CTC would have to intersect it an odd number of times, contradicting the fact that the spacetime is simply connected. Therefore, this spacetime is not globally hyperbolic.

A cylindrical chart

In this section, we introduce another coordinate chart for the Gödel solution, in which some of the features mentioned above are easier to see.

Derivation

Gödel did not explain how he found his solution, but there are many possible derivations. We will sketch one here, and at the same time verify some of the claims made above.
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 as tangent to the world lines of the dust particles, and their world lines will in general exhibit nonzero vorticity but vanishing expansion and shear. Let us demand that the Einstein tensor match a dust term plus a vacuum energy term. 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. Specifically, with a bit of foresight, let us choose. This gives
Finally, let us demand that this frame satisfy
This gives, and our frame becomes