# Space-like intervals and the ergoregion

1. May 12, 2008

### stevebd1

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.

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

2. May 13, 2008

### stevebd1

After further reading, I gather the frame-dragging figure of ω 'parameterizes the rotation of space as viewed from infinity'*, which implies that regardless of quantity of ω, space is being dragged around at c at the ergosphere edge. The question I have now is at what point do the results of the calculations become relativistic (i.e. observed information rather than actual/local information) and does ω still hold as an actual quantity of frame-dragging (as apposed to observed quantity) for objects with less extreme spin such as planets, stars and possibly neutron stars?

There is an equation that quantifies redshift in conjunction with the frame-dragging equations-

$$\alpha=\frac{r}{\Sigma}\sqrt{\Delta}$$

$$\Sigma=\sqrt{(r^{2}+a^{2})^{2}-a^{2}\Delta}$$

$$\Delta=r^2+a^2-2r_g r$$

$\alpha$- redshift, $r_g$-gravitational radius, $a$-spin parameter in m, $r$- radius in m

source- http://cr4.globalspec.com/blogentry/1670/Extreme-Frame-Dragging

For objects such as planets and main sequence stars, the redshift figure is 1 (i.e., the figures for frame-dragging are not relativistic). This changes for white dwarves (for a 1.14 sol wd with a spin parameter of 0.168, the redshift at the star’s surface is 0.9996) so it appears the quantity of ω for frame-dragging becomes relativistic the more compact an object becomes. This increases significantly for neutron stars and black holes-

3 sol neutron star with a/M=0.2, redshift=0.5143 at star’s surface

3 sol black hole with a/M=0.95, redshift=0.327 at ergosphere (perceived velocity of dragging at ergosphere edge is 0.327c when in actual fact it’s travelling at c)

The question I have now is, is it possible to calculate the actual velocity of frame-dragging at the neutron star’s surface or the edge of the ergosphere (providing proof that the angular velocity of f-d does indeed equal c) rather than relativistic figure produced using the frame-dragging calcs?

(One small issue with the redshift calc is that while at a/M=0, the Kerr equation produces the exact same results the Schwarzschild redshift equation produces for a static black hole, as a/m increases, divergence isn’t at the ergosphere as expected, it’s at the event horizon of the rotating black hole which seems conflictive with the Killing field and the term ‘surface of infinite redshift’ being applied to the static limit.

*'Experimental Evidence of Black Holes' by Andreas Müller

Last edited: May 13, 2008
3. May 13, 2008

### stevebd1

Below contains some corrections to the above, corrections in italics-

3 sol black hole with a/M=0.95, redshift=0.394 at ergosphere (perceived velocity of dragging at ergosphere edge is 0.327c when in actual fact it’s travelling at c)

The question I have now is, is it possible to calculate the actual velocity of frame-dragging at the neutron star’s surface or the edge of the ergosphere (providing proof that the angular velocity of f-d at the ergosphere edge does indeed equal c) rather than relativistic figure produced using the frame-dragging calcs?

4. May 21, 2008

### stevebd1

Can anyone confirm if the above is correct for compact objects (neutron stars, black holes) that the quantity of frame-dragging calculated is not the actual quantity of frame-dragging but the quantity as perceived from infinity (the equations used are in the link in the above post) and is there anyway of calculating the actual quantity of frame dragging in close proximity to neutron stars and black holes, proving that regardless of the quantity of spin, the angular velocity within the ergoregion exceeds c. Any feedback would be welcome.