# Accelerations in Kerr Metric

• Jorrie
In summary, Pervect has provided a link to a paper that discusses gravitational acceleration in the Kerr metric for objects moving along the equator of a rotating black hole. The paper describes parallel transport along circular orbits in stationary axisymmetric spacetimes and explains how competing "central attraction forces" and centripetal accelerations contribute to the behavior of 4-acceleration in the Kerr spacetime.
Jorrie
Gold Member
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?

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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.'

Thanks Steve, it looks promising!

## 1. What is the Kerr metric?

The Kerr metric is a mathematical description of the geometry of spacetime around a rotating black hole. It is a solution to Einstein's field equations in general relativity, and it describes the curvature of spacetime and the gravitational effects of a rotating mass.

## 2. How is acceleration calculated in the Kerr metric?

Acceleration in the Kerr metric is calculated using the geodesic equation, which describes the paths of particles in curved spacetime. This equation takes into account the effects of both the gravitational field and the rotation of the black hole, and it can be solved to determine the acceleration of a particle at any given point in the Kerr spacetime.

## 3. Does the Kerr metric predict any unique effects on acceleration near a black hole?

Yes, the Kerr metric predicts the existence of an effect known as frame-dragging, where the rotation of the black hole causes nearby objects to experience a twisting or dragging force. This can lead to changes in the acceleration of particles in the vicinity of the black hole.

## 4. How does the Kerr metric differ from the Schwarzschild metric?

The Kerr metric and the Schwarzschild metric are both solutions to Einstein's field equations, but they describe different types of black holes. The Schwarzschild metric is for a non-rotating, spherically symmetric black hole, while the Kerr metric is for a rotating black hole. This difference leads to unique features in the geometry and acceleration calculations for each metric.

## 5. Can the Kerr metric be applied to other objects besides black holes?

Yes, the Kerr metric can also be used to describe the geometry and acceleration near other rotating objects, such as neutron stars. However, the effects may be weaker or stronger depending on the mass and rotation of the object compared to a black hole. The Kerr metric can also be used in theoretical models of rotating universes or other cosmological scenarios.

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