Penrose-Hawking singularity theorems

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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."

1 Interpretation and Significance
2 Elements of the Theorems
3 References

[edit] Interpretation and Significance

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.

[edit] Elements of the Theorems

[edit] Nature of a Singularity

Usually 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 (ie, 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 physics break down. This is because it is often technically simpler to discuss the properties of geodesics, rather than the curvatures of a spacetime manifold.

[edit] Assumptions of the Theorems

Typically a singuarity theorem has three ingredients ^[1]:

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

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

[edit] Tools Employed

A key tool used in the formulation and proof of the singularity theorems is the Raychaudhuri equation, which describes the divergence $θ$ of a congruence (family) of geodesics. For convenience, the Raychaudhuri equation is

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

where $σ a b$ is the shear tensor of the congruence (see the congruence page for details). The key point is that ${E[\vec{X}]^a}_a$ with be non-negative provided that the Einstein equations hold and^[1]

the weak energy condition holds and the geodesic congruence is null, or
the strong energy condition holds and the geodesic congruence is timelike.

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

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 $γ$ .
The geodesic $γ$ can be varied to a longer curve if another geodesic from $p$ intersects $γ$ at another point, called a conjuagate point.
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. [1]

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.

[edit] References

^ ^a ^b Stephen Hawking and Roger Penrose, The Nature of Space and Time, Princeton University Press, 1996.

Hawking, Stephen; and Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. ISBN 0-521-09906-4. The classic reference.