Thread: "Light cones tipping over" View Single Post
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 Quote by cesiumfrog Could you elaborate a little on exactly where those closed timelike curves are in the Kerr solution?
Let me elaborate a bit on what Chris said.

O'Neill, in his book The Geometry of Kerr Black Holes, proves:

there is a closed timelike curve through any event inside the inner (Cauchy) horizon, i.e., through any event for which r < r-.

Carroll gives the following simple example. Consider a curve for which $\phi$ varies, and for which $t$, $r$, $\theta$ are held constant. Because of periodicity with respect to $\phi$, any such curve is closed.

Now, the timelike part.

Take $r < 0$ with $|r|$ small, and $\theta = \pi/2$. Note $r$ is a coordinate, not a radial distance, and negative $r$ is part of (extended) Kerr. Because $0 = dt = dr = d \theta$, the line element along the curve is

$$ds^2 = \left( r^2 + a^2 + \frac{2Mr a^2}{r^2} \right) d\phi^2$$

For $r$ negative and small. the last term, whcih is negative, dominates, and thus $ds^2$ is the line element for a timilike curve.