What Are the Radial and Tangential Accelerations in the Kerr Metric?

[Jorrie]

Pervect has https://www.physicsforums.com/showpost.php?p=1046874&postcount=17" for radial and tangential gravitational accelerations of a moving particle in Schwarzschild coordinates.

$$\frac{d^2 r}{d t^2} = \frac {3 m{{\it v_r}}^{2}}{ \left( r-2\,m \right) r} + \left( r-2\,m \right) \left( {{\it v_\phi}}^{2}-{\frac {m}{{r}^{3}}} \right)$$

$$\frac{d^2\phi}{d t^2} = -\frac {2 {\it v_r}\,{\it v_\phi}\, \left( r -3\,m \right) }{ \left( r-2\,m \right) r}$$

where $v_r =dr/dt$, $v_\phi = d\phi/dt$, $m$ the mass of the primary and $r,\phi$ the Schwarzschild coordinate parameters.

Does anyone know of an equivalent set of equations for the Kerr metric, at least for movement along the equator of a rotating black hole?

 

[stevebd1]

I know that this is a seriously old thread but it remains unanswered. The following paper looks at gravitational acceleration for moving objects in Kerr metric-

http://arxiv.org/abs/gr-qc/0407004

'Geometric transport along circular orbits in stationary axisymmetric spacetimes'

Donato Bini, Christian Cherubini, Gianluca Cruciani, Robert T. Jantzen

(Submitted on 1 Jul 2004)

'Parallel transport along circular orbits in orthogonally transitive stationary axisymmetric spacetimes is described explicitly relative to Lie transport in terms of the electric and magnetic parts of the induced connection. The influence of both the gravitoelectromagnetic fields associated with the zero angular momentum observers and of the Frenet-Serret parameters of these orbits as a function of their angular velocity is seen on the behavior of parallel transport through its representation as a parameter-dependent Lorentz transformation between these two inner-product preserving transports which is generated by the induced connection. This extends the analysis of parallel transport in the equatorial plane of the Kerr spacetime to the entire spacetime outside the black hole horizon, and helps give an intuitive picture of how competing "central attraction forces" and centripetal accelerations contribute with gravitomagnetic effects to explain the behavior of the 4-acceleration of circular orbits in that spacetime.'

 

[Jorrie]

Thanks Steve, it looks promising!