I've recently been looked a little more into the ergoregion and the static limit of a rotating black hole and would appreciate any feedback regarding the question below in respect of the transition of time-like to space-like intervals at the static limit.(adsbygoogle = window.adsbygoogle || []).push({});

The Killing vector field* for a black hole ranges from 0 to -1 outside the event horizon meaning spacetime with time-like (c^2 t^2 > r^2) intervals to infinite. Divergence of the Killing field is at the event horizon for a static (Schwarzschild) black hole (where it become positive) and at the ergosphere for a Kerr rotating black hole; co-ordinate intervals become light-like (c^2 t^2 = r^2) at the point of divergence and then space-like (c^2 t^2 < r^2) beyond (in the case of the rotating black hole, within the ergoregion) implying that spacetime is being dragged faster than c and there is no static observer relative to the universe once past the ergosphere. This also coincides with the divergence of the time dilation equation which is also at the ergosphere edge for a rotating black hole (hence the term the outer surface of infinite redshift).

But when it comes to calculating the Lense-Thirring effect, the angular velocity of frame-dragging at the ergosphere edge (irrespective of sol mass) ranges from 0.1c at a/M=0.2 to 0.333c at a/M=1 which implies the spacetime is not rotating faster than c and should remain time-like well into the ergosphere.

Is there another factor that comes into play that rotates spacetime fast enough that spacetime intervals becomes space-like within the ergoregion?

Steve

*Source- 'Compact Objects in Astrophysics' by Max Camenzind

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# Space-like intervals and the ergoregion

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