Gödel metric

The Gödel metric, also known as the Gödel solution or Gödel universe, is an exact solution 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 (dust solution), and the second associated with a negative cosmological constant (see Lambdavacuum solution). 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. This solution was found by Kurt Gödel in 1949. 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 (see below), but this article uses the chart that Gödel originally used. In this chart, the metric (or, equivalently, the line element) 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. 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'. The computation of the Fermi–Walker derivatives with respect to shows that the spatial frames are spinning about with the angular velocity .