Universe of Gödel

The universe of Gödel is a solution with the equations of the General relativity published by the Mathématicien Kurt Gödel in 1949. This solution has several remarkable properties, though physically unrealistic persons, and was at the origin of the search for a greater number of exact solutions to the equations of Einstein.

Metric of the universe of Gödel

This solution describes a Lorentzian four-dimensional space (just like our Espace-temps) filled up nonrelativistic matter of null Pression and a cosmological Constante. The Métrique (or the element length) of this space is written

${\ rm D} d^2 = - {\ rm D} t^2 + {\ rm D} x^2 - \ frac {1} {2} \ exp (2 \ sqrt {2} \ Omega X) \; {\ rm D} y^2 + {\ rm D} z^2 - 2 \ exp (\ sqrt {2} \ Omega X) \; {\ rm D} T \; {\ rm D} there,$

where $\ omega$ is a constant position representing the Vorticité fluid which is at rest compared to the coordinated $x$ , $y$ , $z$ . The Density of energy $\ rho$ of the fluid and the cosmological constant $\ Lambda$ are connected to the vorticity by

$4 \ pi \ rho = - \ Lambda = \ omega^2$

in a system of units such as the Speed of light and the Constante of gravitation are worth 1.

Properties

The universe of Gödel represents a homogeneous Espace, i.e. all its points are equivalent.

Its principal characteristic is that it comprises curves of time kind closed. By the change of coordinates

$\ exp (\ sqrt {2} \ Omega X) = \ cosh 2r + \ cos \ phi \ sinh 2r,$

$\ Omega there \ exp (\ sqrt {2} \ Omega X) = \ sin \ phi \ sinh 2r,$

$\ tan \ left (\ frac {1} {2} (\ varphi + \ Omega T - \ sqrt {2} T) \ right) = \ exp (- 2 r) \ tan \ frac {1} {2} \ varphi,$

the element length is rewritten

${\ rm D} d^2 = 2 \ omega^ {- 2} \ left (- {\ rm D} t'^2 + {\ rm D} r^2 - (\ sinh^ 4 r - \ sinh^ 2 r) \; {\ rm D} \ varphi^2 + 2 \ sqrt {2} \ sinh^ 2 r \; {\ rm D} \ varphi \; {\ rm D} you \ right) + {\ rm D} z^2.$

The fluid is always at rest compared to the coordinates $r$ , $\ varphi$ , $z$ and around the origin is symmetrical compared to the axis $r = 0$ spaces it. Space being homogeneous, this property is found for all the other points. In $r = 0$ , the future Cone of light is directed upwards, just like in a system of polar Coordonnées ordinary in the Espace of Minkowski, and does not include the lines of coordinates of $r$ and $\ varphi$ . As one considers points for larger values of $r$ , the cones of light are inclined little by little until including the line of coordinate of $\ varphi$ starting from $r = \ ln (1 + \ sqrt {2})$ . The lines of coordinates of $\ varphi$ are thus for the great values of $r$ of the closed curves of time kind. For this reason, the space of Gödel is not regarded as a physically acceptable solution of the equations of Einstein.

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