Kerr metric

The Kerr metric is a vacuum solution of Einstein's field equations for uncharged, rotating black holes. This solution is named after Roy Kerr, who calculated first. Fully spelled out is the line element

In general, the line element is shortened in Boyer- Lindquist coordinates given as follows:

With the sizes

Here is the vacuum speed of light, the proper time, the mass of the field producing body and the Schwarzschild radius. The parameter is also called Kerrparameter. He is by definition proportional to the angular momentum of the rotating mass. Is the Kerrparameter positive, the body of mass performs a prograde rotation. In the case of a negative Kerrparameters the rotation is retrograde. Disappears the Kerrparameter, it follows as a limiting case of the Schwarzschild metric. Shortly after the discovery of the Schwarzschild or Kerr metric and the corresponding generalizations for the case of electrically charged black holes have been found. The following table gives an overview of the metrics used, where Q denotes the electric charge.

Gradient

As a review of the Kerr metric only significant computation is possible, here are the contravariant components of the metric tensor on the square of the four - gradient operator are shown:

Applications

• The Kerr metric describes the infinitesimal relation between the proper time of a test body ( a clock, which is rigidly connected to the test body ) in the gravitational field of a rotating, uncharged black hole and the parameters of the spacetime along the world line. The integration of these infinitesimal proper times along a given path can be calculated, the changes in the proper time of the test body along this path. Because the proper time - as known from the theory of relativity - this is the state of motion of the test body depends on this calculation, however, is usually not trivial. The influence of the mass of the test body to the entire space-time structure is neglected.

• Acting on the test body along its path is no external force, the shape of the orbit can be calculated (based on clearly defined initial conditions ) in principle by integrating the geodesic equation.

Special Areas

Kerr metric is singular on multiple surfaces. The denominator of the purely radial component of the Kerr metric zero and thus the metric is singular example. This condition is exactly satisfied when

Both values ​​coincide with the Schwarzschild radius and therefore, those two surfaces are also referred to as internal and external event horizon. Although two event horizons is the radial coordinate has a constant value, shows the bending behavior of the event horizons that this rather have the properties of a rotation ellipsoid. Since eludes the inner event horizon of direct observation due to the outer horizon, the inner horizon has no physical meaning.

Two other singular surfaces arise gtt due to a sign change of the time-like component. The condition gtt = 0 leads also to a quadratic equation with the solutions

These two areas can because of the term under the square root cos2θ as flattened spheres or spheroids are presented. The outer surface touches the external event horizon of the two poles of which are defined by the axis of rotation. The two poles correspond to an angle θ from 0 or π.

The space between the two outer surfaces is called singular ergosphere. Normally, each particle experiences a positive proper time along its world line. Within the sphere of this Summary is however only possible when the particle co-rotates with a certain angular velocity of at least the inner mass m. There can therefore be no particles that rotate in the opposite direction as the inner mass.

As with the Schwarzschild metric, the singularity of the event horizon of the Kerr metric is also just a coordinate singularity. By a choice of the coordinates of the space-time Kerr metric may also be described to the inside of the event horizon and steadily without singularities in the metric.