Summary

Equations

The observed angular velocity of the frame drag in the system of the coordinate bookkeeper far away from the mass is^[1]

${\displaystyle {\rm {\omega ={\frac {2\ a\ r}{\Xi }}}}}$

in units of ${\displaystyle {\rm {G=M=c=1}}}$ and with

${\displaystyle {\rm {\Sigma =r^{2}+a^{2}\ \cos ^{2}\theta ,\ \Delta =r^{2}-2\ r+a^{2},\ \Xi =\left(a^{2}+r^{2}\right)^{2}-a^{2}\ \sin ^{2}\theta \ \Delta }}}$

The radius of gyration is given by^[2]

${\displaystyle {\rm {{\bar {R}}={\sqrt {\frac {\Xi }{\Sigma }}}}}}$

and the cartesian coordinates in the bookkeeper's frame of reference are

${\displaystyle {\rm {r_{xy}={\sqrt {(a^{2}+r^{2})\sin ^{2}\theta }}}},\ {\rm {r_{xyz}}}={\rm {\sqrt {r_{xy}^{2}+r^{2}\cos ^{2}\theta }}}}$

The gravitational time dilation component is

${\displaystyle {\rm {\varsigma ={\sqrt {\frac {\Xi }{\Sigma \ \Delta }}}}}}$

so the observed transverse velocity of a frame dragged zero angular momentum probe is

${\displaystyle {\rm {u_{\perp }=\Omega _{obs}=\omega \ r_{xy}}}}$

and the local velocity

${\displaystyle {\rm {v_{\perp }=\Omega _{loc}=\omega \ {\bar {R}}\ \varsigma }}}$

That is exactly the speed of light at the outer edge of the outer ergosphere at

${\displaystyle {\rm {r_{\text{ergo}}=1+{\sqrt {1-a^{2}\cos ^{2}\theta }}}}}$

which corresponds to a Boyer-Lindquist radius of ${\displaystyle 2}$ and a cartesian radius of ${\displaystyle {\sqrt {\rm {2^{2}+a^{2}}}}}$ at the equator.

References

1. ↑ Andrei & Valeri Frolov: Rigidly rotating ZAMO surfaces in the Kerr spacetime, page 1, eq. 5
2. ↑ Scott A. Hughes: Nearly horizon skimming orbits of Kerr black holes, pages 5, 6 etc

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

File usage on Wikipedia

• de.wikipedia.org/wiki/Kerr-Metrik
• de.wikipedia.org/wiki/Lense-Thirring-Effekt
• en.wikipedia.org/wiki/Lense-Thirring_precession