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I've read that all paths inside the EH lead to singularity. ALL paths. Even the ones pointing away from it, right? Because there's NO pointing away from singularity, since ALL paths point to it.

So how could it be at the exact center? Or am I misunderstanding "center" here?

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Inside the event horizon all timelike paths lead to the singularity. A timelike path is one along which you never travel at a speed greater than c.

A static black hole is spherically symmetric; any asymmetries are radiated away as gravitational waves as the black hole forms. Therefore the singularity must be at the centre or it would break the spherical symmetry.

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@WillieWong: yes, OK, thanks :-) –  John Rennie Mar 15 '13 at 14:50
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The singularity lies in the future of all worldlines inside the event horizon. In Schwarzschild coordinates, it lies at $r=0$. Once you've crossed the horizon the radial coordinate $r$ becomes timelike while $t$ becomes spacelike. So trying to avoid the decreasing $r$ is as pointless as trying to stop yourself from growing older.

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In a Kerr metric (rotating black hole) the singularity is a ring or circle, not a point.
The center of the ring is at the center of the black hole, but at least in standard Boyer-Lindquist coordinates, the ring singuarity itself is not at the center of the black hole.

It gets worse. Some authors have suggested that matter can traverse inside the ring to an extended metric in 'another universe".

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