Singularity theorem

In the general theory of relativity in physics, the singularity theorem is a theorem from the group of mathematical theorems that deduce the existence of singularities in space-time from a few global assumptions . The conditions are on the one hand energy conditions for the mass and energy distribution in space and on the other hand causality conditions for the topology of spacetime.

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

Historical classification

Shortly after the publication of Einstein's field equations in 1915, the first exact solution was presented in 1916 with the Schwarzschild solution. This has strong symmetry assumptions as a distinctive feature and there is the possibility of a singularity of curvature in the center. This occurs because the unstoppable spherical collapse within the event horizon brings the area of ​​validity of the exterior solution closer and closer to the center of symmetry. This mass agglomeration proceeds until theoretically all mass is concentrated in one point and the curvature of space diverges at this point. Such a breakdown 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 it could be shown that singularities are a consequence of the attractive nature of gravitation.

Energy conditions

Main article: Energy conditions

In the general theory of relativity, the mass and energy distribution is described with an energy-momentum tensor . In the context of this theory, energy conditions are inequalities for contractions of this tensor. The different singularity theorems differ in the strength of the applied energy condition. A strong condition results in causal singularities that are easy to prove, but there may be forms of matter in the universe that contradict this and only obey weaker energy conditions. ${\displaystyle T_{ab}}$

The weakest energy conditions (light-like) are very likely fulfilled by all matter, but only light-like singularities follow from this.

Causality conditions

Causality encompasses all possibilities with which events in space-time can influence each other, and assigns events (space-time coordinates) to relations. These relations result from the tangential vectors on the curves that connect the events. If a point lies in the temporal future of another, there is at least one connecting curve between them with exclusively temporal tangential vectors. Other possibilities are that points are only related light-like or that they are not causally connected (there are only space-like connecting curves). Points are causally connected if there are light-like or time-like connection curves. Causality conditions restrict the relationships in which the entire events of a space-time can be.

Spacetime is chronological

Spacetime is called chronological if there are no closed time-like curves in it. This means that there is no point in its own timelike past or future.

Spacetime is causal

Spacetime is called causal if there are no closed time-like or light-like curves in it. This means that there is no point in its own causal past or future.

Spacetime is strictly causal

Spacetime is said to be strictly or strongly causal when no causal curve intersects a convex neighborhood in two unconnected sets. To put it clearly, a violation of this condition would mean that one could get as close as desired to this event from an event via a causal curve.

Hawking and Penrose singularity theorem

Space-time diagram, with an event horizon, a trapped area (condition 4.1. In the theorem) and in a plane x ¹ = constant the horism E ⁺ and the future I ⁺ limited by it are drawn. The compactness of the horism becomes clear and that both the outward and inward zero geodesics converge (arrows in q for the light cone portion in the plane shown)

The spacetime of the dimension is causally geodetically incomplete if the following four points are met: ${\displaystyle n}$

1. The strong energy condition applies along all causal curves (follows from the strong energy condition).
2. The generic condition is met. That is, each of these curves (with Tangentialvektorfeld ) contains a point with non-zero effective curvature: . ${\displaystyle u^{a}}$ ${\displaystyle u^{c}u^{d}u_{[a}R_{b]cd[e}u_{f]}\neq 0}$
3. Spacetime is chronological.
4. Spacetime includes at least one of the following:
   1. a closed space-like submanifold of the dimension , the mean curvature vector field of which is past-oriented and time-like. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle (n-2)}$
   2. a compact achronic (spatial or light-like) submanifold without border or ${\displaystyle {\mathcal {T}}}$
   3. a point in such a way that along every zero geodesic that is completely continued into the past (into the future) starting from the point with the tangential vector field , the trace of the covariant derivative tensor of the zero geodesic array becomes negative. ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle u^{a}}$ ${\displaystyle \theta =u_{\,\,;a}^{a}}$ ${\displaystyle u_{\,\,;b}^{a}}$ ${\displaystyle x}$

The aim of this theorem is therefore to prove the causal geodetic incompleteness of the manifold. Put simply, world lines of observers or light particles simply end at a point that no longer belongs to space. The proof is based on the fact that conditions 1 and 2 together with the assumed causal geodetic completeness cause a contradiction to conditions 3 and 4. If 1–4 apply, the causal geodetic completeness must be violated.

The first two conditions imply a focus on causal geodesics. This follows directly from the Raychaudhuri equation and the existence and uniqueness theorems about ordinary differential equations. The focusing has the effect that up to a certain finite distance from the origin of the geodetic congruence at least one point must appear conjugated to this origin. In a causal geodetically complete spacetime, all geodesics can be extended to infinite parameters. Since the focusing effect acts on all causal geodesic families, all causal geodesics then contain at least one pair of conjugate points in the maximally continued parameter interval.

On the other hand, from conditions 3 and 4, in a region whose causal development is only given by the values ​​on , one can construct a causal curve that has no conjugate points. Due to a very strong causal condition that applies in this area, the global hyperbolicity, a causal geodesic exists as a boundary curve to this constructed curve. With the receipt of this causal geodesic, which has no conjugate points in the entire parameter range, the conditions for contradiction are led and the proof is achieved. ${\displaystyle {\mathcal {T}}}$

literature

• Stephen Hawking, George F. Ellis: The Large Scale Structure of Space-Time . Cambridge University Press, Cambridge 1973, ISBN 0-521-09906-4 (English).
• Marcus Kriele: Spacetime. Foundations of general relativity and differential geometry (=  Lecture notes in physics. NS M: Monographs. Volume 59 ). Springer, Berlin et al. 1999, ISBN 3-540-66377-0 (English).

Individual evidence

1. ^ Stephen Hawking, The occurrence of singularities in cosmology. T. 1-3. In: Proceedings of the Royal Society. A. Vol. 294, 1966, ISSN  0962-8444 , pp. 511-521, Vol. 295, 1966, pp. 490-493, Vol. 300, 1967, pp. 187-201.
2. ↑ Stephen Hawking, Roger Penrose: The singularities of gravitational collapse and cosmology. In: Proceedings of the Royal Society. A. Vol. 314, 1970, ISSN  0962-8444 , pp. 529-548.
3. ↑ Marcus Kriele: Spacetime, foundations of general relativity and differential geometry. Lecture notes in physics. Vol. 59. Springer, Heidelberg 1999. ISBN 3-540-66377-0 , p. 383