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I've read a few questions about the Kerr metric, and the ring singularity that the math discusses behind the inner Cauchy horizon. While we should be careful in taking anything past the Cauchy horizon too seriously, I have a question for the region where the $r$-coordinate becomes negative:

Is any gravity exerted repulsive within the negative-$r$ region as a whole, or, is it simply the ring singularity in the negative-$r$ region that exerts a repulsive gravitational effect?

My question concerns what the Kerr metric mathematically describes in the negative-$r$ region, not with what actually happens for rotating black holes.

Hokon
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The curve of the $\rm d^2r/d\tau^2$ is symmetric in positive and negative $\rm r$ (at least in the polar slice, but farther away also at every angle), so if the $\rm d^2r/d\tau^2$ is negative at $\rm r>x$ then it is also negative at $\rm r<-x$.

A negative $\rm d^2r/d\tau^2$ at positive $\rm r$ means acceleration toward the black hole, while at negative $\rm r$ it means acceleration away from the black hole, so at negative $\rm r$ the black hole is mostly repulsive while at positive $\rm r$ it is mostly attractive.

Inside the Cauchy horizon the gravity can be repulsive even at positive $\rm r$ though, and in the negative region where no horizons exist it can also be attractive at comparable distances, so close above or below the ring you would always be accelerated in the positive $\rm r$ direction. Farther away it is the other way around.

For a detailed explanation see Madore, for the equations of motion in various coordinates see here.

Yukterez
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