Gödel metric

The Gödel metric 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 nonzero cosmological constant (see lambdavacuum solution). It is also known as the Gödel solution or Gödel universe.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. The Gödel metric 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 nonzero cosmological constant (see lambdavacuum solution). It is also known as the Gödel solution or Gödel universe. This solution has many unusual properties—in particular, the existence of closed timelike curves that would allow time travel in a universe described by the solution. Its definition is somewhat artificial in that the value of the cosmological constant must be carefully chosen to match the density of the dust grains, but this spacetime is an important pedagogical example. The solution was found in 1949 by Kurt Gödel. Like any other Lorentzian spacetime, the Gödel solution presents the metric tensor in terms of some local coordinate chart. It may be easiest to understand the Gödel universe using the cylindrical coordinate system (presented below), but this article uses the chart that Gödel originally used. In this chart, the metric (or equivalently the line element) is where ω {displaystyle omega } is a nonzero real constant, which turns out to be the angular velocity of the surrounding dust grains around the y axis, as measured by a 'non-spinning' observer riding one of the dust grains. 'Non-spinning' means that it doesn't feel centrifugal forces, but in this coordinate frame it would actually be turning on an axis parallel to the y axis. As we shall see, the dust grains stay at constant values of x, y, and z. Their density in this coordinate chart increases with x, but their density in their own frames of reference is the same everywhere. To study the properties of the Gödel solution, we will adopt the frame field (dual to the coframe read off the metric as given above), This frame defines a family of inertial observers who are comoving with the dust grains. However, computing the Fermi–Walker derivatives with respect to e → 0 {displaystyle {vec {e}}_{0}} shows that the spatial frames are spinning about e → 2 {displaystyle {vec {e}}_{2}} with angular velocity − ω {displaystyle -omega } . It follows that the nonspinning 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. Notice that the cosmological constant is carefully chosen to partially cancel the matter density of the dust.