The Gödel Geometry

The Gödel geometry is described by the line element \begin{equation} ds^2 = -\left( dt + e^x dz \right)^2 + dx^2 + dy^2 + \frac12 e^{2x} dz^2 . \end{equation} This geometry satisfies $\vec{G} - \frac12 d\vec{r} = \sigma^t \hat{e}_t$, corresponding to dust with energy density $\rho=1$, and with cosmological constant $\Lambda=-\frac12$. Define new coordinates via: \begin{align} e^x &= \cosh(2r) + \cos\phi \,\sinh(2r) ,\\ z &= e^{-x} \sin\phi \,\sinh(2r) ,\\ t &= 2\sqrt2 \arctan\left(e^{-2r}\tan\frac\phi2\right) - \sqrt{2}\,\phi + 2\tau , \end{align} which brings the line element to the form \begin{equation} ds^2 = -\left(d\tau+\sqrt2\sinh^2(r)\,d\phi\right)^2 + dr^2 + dy^2 + \sinh^2(r)\cosh^2(r)\,d\phi^2 . \end{equation} The coefficient of $d\phi^2$ in this line element is \begin{equation} \sinh^2(r)\left(\cosh^2(r)-2\sinh^2(r)\right) = \sinh^2(r)\left(1-\sinh^2(r)\right), \end{equation} which changes sign where $\sinh(r)=1$, that is, where $r=\ln(1+\sqrt2)$. When $r$ has this value, the $\phi$ direction is null; when $r$ is larger than this value, the $\phi$ direction is timelike. But $\phi$ is periodic, so there are closed timelike (and null) curves purely in the $\phi$ direction.

©2017 by Tevian Dray