Sunday, August 14, 2011

Gödel's universe

Kurt Gödel found a solution to Einstein's equations for General Relativity where time is meaningless, because every point in spacetime can access every other point in spacetime through Closed Timelike Curves.

In other words, he found a solution to Einstein's equation where you can trivially time travel to any point in history from any point in space, a clear violation of Einstein (and most every other physicist's) views on the nature of time.

It can be defined as follows:

where ω is a nonzero real constant, which turns out to be the angular velocity, as measured by a nonspinning observer riding any one of the arbitrary points.

Gödel never explained how he found his solution, but there are many possible derivations. Let's see one here:

Start with a simple frame in a cylindrical type chart, featuring two undetermined functions of the radial coordinate:


Think of the timelike unit vector field e0 as a tangent to the lines of arbitrary points.

Following Gödel, we can interpret the arbitrary points as galaxies, so that the Gödel interpretation becomes a cosmological model of a rotating universe. Because this model exhibits no Hubble expansion, it is not a realistic model of the universe in which we live, but can be taken as illustrating an alternative universe which would in principle be allowed by general relativity (if one admits the legitimacy of a nonzero cosmological constant).

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