The Mathematical Problem In Kerr Metric

Singularities, which occur in gravitational collapse, are not visible from outside but are hidden behind an event horizon. This means that one can still predict the future outside the event horizon. From the aspects of observers in the asymptotically flat region, the event horizon is the surface of black hole, which is the ultimate destiny of many massive stars undergoing violent gravitational collapse.This article can be divided in the main into two parts. In the first part, the point of view is stressed that null hypersurfaces and null directions are, in fact, much more convenient and may perhaps also be regarded as more fundamental, since it appears that the light-cone structure of space-times of the black-holes solutions is exactly of that kind that makes the approach through the null hypersurfaces and null directions more effective grasping the inherent symmetries of these space-times and revealing their analytical richness. A mathematical analysis of null hypersurfaces is present in order to offer more correct understanding of the nature of gravitational wave and the asymptotically flat space-times, to which the significant physical quantity(such as mass, momentum, and angular momentum) of subsystems of the universe can be attribute. Since, at this stage, our attention is primarily paid to asymptotically flat region, we deal with the construction of optical function and its associated light cones. Through the estimates of Ricci coefficients, we get to the conclusion that some types of constrain need to impose on the light cones of which the generator tend to null infinity. After achieving the aim, we can discuss the possibility of attach a standard sphere at infinity. For the sake of completeness, we shall first review the physical motivation of null hypersurface approach, derive these propagation equations, and describe the geometrical representations in mathematical terms. The second part of this article is concerned with the unique two-parameter family of solution-Kerr metric which describes the space-time around black-holes. The black holes of nature, the elements in which are our concepts of space and time, are the most perfect macroscopic objects there are in the universe. Since the general theory of relativity provides only a single unique family of solutions for their descriptions, they are the simplest objects as well. Many amazing features which were found to characterize this outstandingly remarkable solution of Einstein's equations have been always leading to some elegant mathematical theorem-No-hair theorems. Thanks to these theorems, a universally acknowledged strong belief that if an event horizon develops in an asymptotically flat space-time, then the solution exterior to this horizon approaches a Kerr solution asymptotically with time. Following this historical tradition, I give a new simple proof of black hole uniqueness theorem concerning an interesting class of static asymptotically flat solutions. Its features contains regular event horizons, and some physical conditions that a broad class of non-rotating equilibrium black hole metrics might plausibly be expected to satisfied. The conclusion is that the class was exhausted by the positive mass Schwarzschild family of metrics.