WOLFRAM|DEMONSTRATIONS PROJECT

Gödel's Universe

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r
0
-0.017
ϕ
0
2.756
Lighter
:Darker[ColorData[97][2],0.7] should be a valid color directive, an image, or a list of them.
Lighter
:Darker[ColorData[97][1],0.58] should be a valid color directive, an image, or a list of them.
Lighter
:Darker[ColorData[97][3],0.75] should be a valid color directive, an image, or a list of them.
General
:Further output of Lighter::arg will be suppressed during this calculation.
MapThread
:Object ColorData[97,ColorList]〚{2,1,3,4,5,6,7,8,9,10}〛 at position {2, 2​} in MapThread[{Specularity[White,#1],#2,Lighting#3}&,{{3,3,3,6,6,6,6,3,6,6},1,{{{Ambient,Lighter[Darker[ColorData[1][1],0.7],0.05]},{Directional,Darker[ColorData[97][2],0.7],ImageScaled[{0,2,2}]},{1},{Directional,Darker[ColorData[97][2],0.7],ImageScaled[{2,0,2}]}},9}}] has only 0 of required 1 dimensions.
In 1949, Kurt Gödel found a solution of Einstein's field equations and gave it to Einstein on his birthday [1]. This solution describes a homogeneous, anisotropic, and rotationally symmetric spacetime in which matter takes the form of a pressure-free perfect fluid with negative cosmological constant, where closed timelike world lines exist. Theoretically, traveling along such a world line allows an observer to travel into their own past. For an astounding and detailed visualization of how an observer would perceive such a travel, see [2, 3].
The manipulate graphic reproduces the famous illustration contained in the book by Hawking and Ellis [4], showing photons emitted by an observer on the axis of symmetry at the bottom; the photons spiral out, expand, and reconverge at the top. Light-like geodesics are shown as well as light cones that tip as you increase
r
0
,
producing the closed timelike curves. Further interpretations can be found in [5].