Definitions

# Penrose-Hawking singularity theorems

The Penrose-Hawking singularity theorems are a set of results in general relativity which attempt to answer the question of whether gravity is necessarily singular. These theorems answer this question affirmatively for matter satisfying "reasonable" energy conditions. This means that a generic spacetime solution with reasonable matter in general relativity will contain singularities at which the theory "breaks down."

## Interpretation and Significance

In a physical theory of gravitation, a singularity is, roughly speaking, a point in spacetime where various physical quantities (such as the curvature or energy density) become infinite, and therefore physical laws "break down." Singularities can be found in various important spacetimes, such as the Schwarzschild metric for a black hole and the Big Bang in the Friedmann-Robertson-Walker metric thought to describe our universe. They present a problem, for since it is not clear how the equations of physics apply at a singularity, one cannot predict what might come "out" of a singularity in our past, or what happens to an observer that falls "in" to a singularity in the future.

Since the presence of singularities seems objectionable, one might hope that they do not form except under contrived circumstances. For example, in the collapse of a star to form a black hole, if the star is spinning and thus possesses some angular momentum, might not the centrifugal force partly counteract the force of gravity and keep a singularity from forming? The singularity theorems prove that this cannot happen, and that a singularity will form. In the collapsing star example, since all matter and energy is a source of gravitational attraction in general relativity, the additional angular momentum only pulls the star together more strongly as it contracts: it eventually settles down to a Kerr black hole. See also No-hair theorem.

An interesting "philosophical" feature of general relativity is revealed by the singularity theorems. Because general relativity predicts the inevitable occurrence of singularities, the theory in a sense predicts its own breakdown at a finite time to the future.

## Elements of the Theorems

In mathematics, there is a deep connection between the curvature of a manifold and its topology, which was exploited most notably by Gromov. One of Gromov's theorems states that a manifold which has a positive curvature everywhere must be compact. The condition of positive curvature is most conveniently stated as follows--- for every geodesic there is a nearby initially parallel geodesic which will bend toward it when extended, and the two will intersect at some finite length.

When two nearby parallel geodesics intersect, the extension of either one is no longer the shortest path between the endpoints. The reason is that two parallel geodesic paths necessarily collide after an extension of equal length, and if you follow one path until the intersection then the other, you are connecting the endpoints by a non-geodesic path of equal length. This means that for a geodesic to be a shortest length path, it must never intersect neighboring parallel geodesics.

If you start with a small sphere and send out parallel geodesics from the boundary, assuming that the manifold has a Ricci curvature bounded below by a positive constant, none of the geodesics are shortest paths after a while, since they all collide with a neighbor. This means that after a certain amount of extension, we have reached all the new points that we are ever going to reach. If all the points in a connected manifold are at a finite geodesic distance from a small sphere, the manifold must be compact.

Penrose argued analogously that in relativity, the null geodesics which are the paths of light rays generate the boundary of the proper future of a region. Once the null geodesics intersect, they are no longer on the boundary of the future, they are in the interior of the future. In relativity, the Ricci curvature is determined by the energy tensor, and its projection on light rays is always positive, in the sense that the volume of a congruence of parallel null geodesics once it starts decreasing, will reach to zero in a finite time. Once the volume is zero, there is a collapse in some direction, so every geodesic intersects some neighbor.

Penrose concluded that whenever there is a sphere where all the outgoing (and ingoing) light rays are initially converging, the boundary of the future of that region will end after a finite extension, because all the null geodesics will converge. This is significant, because the outgoing light rays for any sphere inside the horizon of a black hole solution are all converging, so the boundary of the future of this region is either compact or comes from nowhere. The future of the interior either ends after a finite extension, or has a boundary which is eventually generated by new light rays which can't be traced back to the original sphere.

# Nature of a Singularity

The singularity theorems use the notion of geodesic incompleteness as a stand-in for the presence of infinite curvatures. Geodesic incompleteness is the notion that there are geodesics, paths of observers through spacetime, that can only be extended for a finite time as measured by an observer traveling along one. Presumably, at the end of the geodesic the observer has fallen into a singularity or encountered some other pathology at which the laws of general relativity break down.

### Assumptions of the Theorems

Typically a singularity theorem has three ingredients:

1. An energy condition on the matter,
2. A condition on the global structure of spacetime,
3. Gravity is strong enough (somewhere) to trap a region.

There are various possibilities for each ingredient, and each leads to different singularity theorems.

### Tools Employed

A key tool used in the formulation and proof of the singularity theorems is the Raychaudhuri equation, which describes the divergence $theta$ of a congruence (family) of geodesics. The divergence of a congruence is defined as the derivative of the log of the determinant of the congruence volume the Raychaudhuri equation is

$dot\left\{theta\right\} = - sigma_\left\{ab\right\}sigma^\left\{ab\right\} - frac\left\{1\right\}\left\{3\right\}theta^2 - \left\{E\left[vec\left\{X\right\}\right]^a\right\}_a$

where $sigma_\left\{ab\right\}$ is the shear tensor of the congruence (see the congruence page for details). The key point is that $\left\{E\left[vec\left\{X\right\}\right]^a\right\}_a$ will be non-negative provided that the Einstein field equations hold and

When these hold, the divergence becomes infinite at some finite value of the affine parameter. Thus all geodesics leaving a point will eventually reconverge after a finite time, provided the appropriate energy condition holds, a result also known as the focusing theorem.

This is relevant for singularities thanks to the following argument

1. Suppose we have a spacetime which is Globally hyperbolic, and two points $p$ and $q$ that can be connected by a timelike or null curve. Then there exists a geodesic of maximal length connecting $p$ and $q$. Call this geodesic $gamma$.
2. The geodesic $gamma$ can be varied to a longer curve if another geodesic from $p$ intersects $gamma$ at another point, called a conjugate point.
3. From the focusing theorem, we know that all geodesics from $p$ have conjugate points at finite values of the affine parameter. In particular, this is true for the geodesic of maximal length. But this is a contradiction – one can therefore conclude that the spacetime is geodesically incomplete.

In general relativity, there are several versions of the Penrose-Hawking singularity theorem. Most versions state, roughly, that if there is a trapped null surface and the energy density is nonnegative, then there exist geodesics of finite length which can't be extended.

These theorems, strictly speaking, prove that there is at least one non-spacelike geodesic that is only finitely extendible into the past but there are cases in which the conditions of these theorems obtain in such a way that all past-directed spacetime paths terminate at a singularity. into the past.

## References

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