# Will solve the singularity Cancer

## Kerr solution

The Kerr solution is a vacuum solution of Einstein's field equations and describes the metric of rotating, uncharged black holes. More generally speaking, it describes the curved outer space of an electrically neutral, rotating mass. To a good approximation, rotating masses can be described with the Kerr solution on the basis of the general theory of relativity.

### A feat of arithmetic

The New Zealander *Roy Patrick Kerr* found this solution in 1963 in Cartesian coordinates, which must be seen as a mathematical masterpiece. Because on the one hand it is generally difficult to find an exact solution to the complicated Einstein field equations and on the other hand the Kerr solution in the Cartesian coordinates has a rather complicated structure. Coordinates that are better adapted to the symmetry of Kerr geometry also make their mathematical formulation easier. The Kerr geometry is usually noted in pseudo-spherical Boyer-Lindquist coordinates. The line element then reads:

Alternatively, it can also be written in matrix form, i.e. the metric tensor of the Kerr metric in Boyer-Lindquist form. The functions contained in the line element are listed in detail under the entry Boyer-Lindquist coordinates.

### Rotation makes things difficult

In principle, it can be said that rotating black holes are significantly more complex spacetime than their non-rotating counterpart, the Schwarzschild solution. Because space-time rotates. This is reflected tensorially in the fact that the metric tensor is no longer diagonal: there is a cross term in Boyer-Lindquist coordinates that couples azimuthal and temporal components. In the Kerr-Schild coordinates, which are also pseudo-spherical, there are even three cross terms! This is an additional complexity, but in numerical simulations of rotating holes the use of Kerr-Schild coordinates can be more useful, because boundary conditions (in hydrodynamics and magnetohydrodynamics) can then be implemented better.

### two properties: mass and angular momentum

So Kerr holes have the physical properties **Dimensions***M.* and **Angular momentum***J*. The**Robinson's theorem** dictates that spacetime with certain conditions must necessarily be identical to the Kerr solution. The rotation can be done by the so-called **Kerr parameters** (Rotation or spin parameters) *a* are characterized. In physical terms, this number is a specific angular momentum *a* = *J*/*Mc*. In geometrized units (*G* = *c* = 1) varies *a* between the values -*M.* (maximum retrograde rotation) and +*M.* (maximum prograde rotation). Sometimes theorists also bet out of convenience *M.* = 1 (sometimes also on my website), so that *a* varies between -1 and 1. The maximum values +1 and -1 are to be used with caution, because the singularity of the Kerr solution is then visible to the outside! The relativist *Kip Thorne* found out in 1974 that the rotation has a critical maximum value, namely *a* = 0.998*M.*. Applies exactly *a* = 0, the Kerr solution changes into the static Schwarzschild solution.

### curved spacetime

Black holes are spacetime that have very strong curvatures, especially when you get close to the curvature singularity. Outwardly, however, the curvature decreases rapidly and even disappears in the borderline case of an infinitely high distance to the hole. Both the Schwarzschild metric and the Kerr metric have this property, and it is called **asymptotic flatness**. That is, for very large distances from the black hole, the Schwarzschild or Kerr metric changes over to the Minkowski metric.

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