Penrose–Hawking singularity theorems

As singularities theorem is known in general relativity theory in physics a theorem from the group of mathematical propositions, deduced from the few global assumptions about a space-time singularities in the presence of her. The conditions are as energy constraints on the mass and energy distribution in space and on the other hand causality conditions on the topology of spacetime.

Such theorems were first proved in the late 1960s by Stephen Hawking and Roger Penrose.

Historical classification

Shortly after the publication of Einstein's field equations in 1915 was in 1916 presented a first exact solution to the Schwarzschild solution. This has as a distinctive feature of strong symmetry assumptions, and as a result there is the possibility for a curvature singularity at the center. This occurs because the unstoppable spherical collapse inside the event horizon the scope of the exterior solution ever closer to the center of symmetry. This mass agglomeration proceeds until all the theoretical mass is concentrated at a point and diverges the curvature of space at this point. Such a collapse of the theory in such simple models can easily be classified as an artifact of the symmetry assumptions. With the formulation and the proof of the first singularity theorems by Hawking and Penrose, however, indicated that singularities are a consequence of the attractive nature of gravity.

Energy conditions

Main article: Energy conditions

In the general theory of relativity, mass and energy distribution is described with an energy -momentum tensor. Energy conditions are inequalities for contraction of this tensor in the framework of this theory. The various singularity theorems differ in the strength of the applied energy condition. A strong condition results in easy to be proved causal singularities, but there may be forms of matter in the universe which contradict this and obey only weaker energy conditions.

The weakest energy conditions ( light -like ) are very likely to be met by all matters, however, the situation will only follow light -like singularities.

Causality conditions

Causality includes all possibilities which can influence events in space-time and assigns events ( space-time coordinates ) Relations to. These relations are derived from the tangent vectors to the curves that connect the events. If a point in the temporal future of another, so there is at least one compound curve between them with only timelike tangent vectors. Other possibilities are that only light-like points are related or that they are not causally related ( ie it only spacelike curves are connecting ). Causally related are points when there are light -like or time- like compound curves. Causality conditions limit in which relations the entire events of a spacetime can relate to each other.

The space-time is chronologically

The space-time is referred to as chronological, if there are no closed time-like curves in it. This means that no point lies in its own time-like past or future.

The spacetime is causally

The space-time will be referred to as a causal if there is no time-like closed or light-like curves in it. This means that there is no point is in its own causal past or future.

The space-time is strictly causal

The space-time is referred to as a strictly causal or strong, if no causal curve intersects a convex area in two unrelated quantities. Figuratively speaking, would be a violation of this condition that one could come close to return to this event any of an event over a causal curve.

Singularitätentheorem by Hawking and Penrose

The space-time of dimension n is kausalgeodätisch incomplete if the following four conditions have been met:

The objective of this theorem, it is thus to prove the kausalgeodätische incompleteness of the manifold. In simplified terms, ie world lines of observers or particles of light simply end at a point that is no longer part of the room. The proof is based on that condition 1 and 2 bring together with assumed kausalgeodätischer be complete, a contradiction to the conditions 3 and 4. So if 1-4 are true, the kausalgeodätische completeness must be violated.

The first two conditions imply a focus of causal geodesics. This follows directly from the Raychaudhuri equation and the existence and uniqueness theorems on ordinary differential equations. Focusing means that up to a certain finite distance from the origin of the geodetic congruence conjugated at least one point should occur at this origin. In a kausalgeodätisch full spacetime can be continued all geodesics in infinite parameters. Since the focusing effect to all causal Geodätenscharen acts, then include all causal geodesics in a maximum continuous parameter interval at least one pair of conjugate points.

On the other hand can be made ​​of the conditions 3 and 4, in a region whose causal development is given only by the values ​​to construct a causal graph, which does not conjugate points. Due to a very strong causal condition which applies in this area, the global hyperbolicity, there exists a causal geodesic curve as a limit to this constructed curve. By obtaining these causal geodesic which is the entire parameter range without conjugate points, the conditions are led to a contradiction and reached the proof.