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Can anyone please explain Hawking-Penrose Singularity Theorems and geodesic incompleteness?

In easy to understand plain English please.

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One of the biggest surprises that General Relativity (GR) has given us is that under certain circumstances the theory predicts its own limitations. There are two physical situations where we expect for General Relativity to break down. The first is the gravitational collapse of certain massive stars when their nuclear fuel is spent. The second one is the far past of the universe when the density and temperature were extreme. In both cases, we expect that the geometry of spacetime will show some pathological behaviour.

The first step towards a mathematical characterisation under which circumstances GR breaks down was achieved in the seminal work of Penrose and Hawking in their singularity theorems). The general structure of the theorems establishes that if on a spacetime (${\cal{M}}, g_{ab}$):

  • the matter content satisfies an energy condition
  • gravity is strong enough in some region
  • and a global causal condition is met

then (${\cal{M}}, g_{ab}$) must be geodesically incomplete.

The energy conditions are general inequalities that relate the energy momentum tensor of matter, $T_{ab}$, with a certain class of vector fields. For example: the weak energy condition establishes that $T_{ab}u^{a}u^{b}\ge0$ for any timelike vector $u_{a}$ (by continuity this will then also be true for any null vector $v^{a}$); the dominant energy condition requires that in any orthonormal basis the energy density dominates all the other terms, $T^{00}\ge |T^{\alpha\beta}|$; and the strong energy condition states that $T_{ab}u^{a}u^{b}\ge u^{a}u_{a}g^{cb}T_{cb}$ for any timelike vector $u_{a}$.

The second requirement of the theorem can be stated sometimes by requiring the existence of a closed trapped surface, $\cal{T}$. By this is meant a $C^{2}$ closed spacelike 2-surface such that the two families of null geodesics orthogonal to ${\cal{T}}$ are converging. This is the formal description of the intuitive idea that the gravitational field becoming so strong in some region that light rays (and so all the other forms of matter) are trapped inside a succession of 2-surfaces of smaller and smaller area.

The global causal conditions come in different forms. The idea of chronological spacetime is that there are no closed timelike curves. In the other hand, a strongly causal satisfies that for every point $p\in{\cal{M}}$ there is a neighbourhood $\cal{V}$ of $p$ which no non-spacelike curve intersects more than once. Finally, one can require that there is surface $S$ which is any subset of spacetime which is intersected by every non-spacelike, inextensible curve, i.e. any causal curve, exactly once. This surface then is called a Cauchy Surface.

The notion of geodesic incompleteness can be better understood by defining what we mean by geodesic completeness. A geodesic complete spacetime is one where any geodesic admits an extension to arbitrarily large parameter values. Then, a spacetime that is not geodesically complete, must be geodesically incomplete. Geodesic incompleteness describes intuitively that there is an obstruction to free falling observers to continue traveling through spacetime. In some sense, they have reached the edge of spacetime in a finite amount of time; they have encountered a singularity.

As a note, these theorems do not show curvature blow-up which is the usual method to show that Black holes or the Big bang have a 'true' gravitational singularity and not just a lose in differentiability (as in shock waves or the tip of a cone).

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