SPACE-TIME SINGULARITY

Gravitational Singularity.

Animated simulation of gravitational lensing caused by a Schwarzschild black hole passing in a line-of-sight planar to a background galaxy. Around and at the time of exact alignment extreme lensing of the light is observed.

A singularity in solutions of the Einstein field equations is one of two things:

1. a situation where matter is forced to be compressed to a point (a space-like singularity)
2. a situation where certain light rays come from a region with infinite curvature (a time-like singularity)

A gravitational singularity or space-time singularity is a location in space-time where the gravitational field of a celestial body becomes infinite in a way that does not depend on the coordinate system. The quantities used to measure gravitational field strength are the scalar invariant curvatures of space-time, which includes a measure of the density of matter. Since such quantities become infinite within the singularity, the laws of normal space-time could not exist.

According to modern general relativity, the initial state of the universe, at the beginning of the Big Bang, was a singularity.Both general relativity and quantum mechanics break down in describing the earliest moments of the Big Bang, but in general, quantum mechanics does not permit particles to inhabit a space smaller than their wavelengths.

A gravitational singularity as predicted by general relativity is at the center of a black hole: any star collapsing beyond a certain point (the Schwarzschild radius) would form a black hole, inside which a singularity (covered by an event horizon) would be formed.The Penrose–Hawking singularity theorems define a singularity to have geodesics that cannot be extended in a smooth manner.The termination of such a geodesic is considered to be the singularity

GENERAL RELATIVITY:

${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }}$

Space-like singularities are a feature of non-rotating uncharged black-holes, while time-like singularities are those that occur in charged or rotating black hole exact solutions. Both of them have the property of geodesic incompleteness, in which either some light-path or some particle-path cannot be extended beyond a certain proper-time or affine-parameter (affine-parameter being the null analog of proper-time).

The Penrose theorem guarantees that some sort of geodesic incompleteness occurs inside any black hole whenever matter satisfies reasonable energy conditions (It does not hold for matter described by a super-field, i.e., the Dirac field). The energy condition required for the black-hole singularity theorem is weak: it says that light rays are always focused together by gravity, never drawn apart, and this holds whenever the energy of matter is non-negative.

Hawking's singularity theorem is for the whole universe, and works backwards in time: in Hawking's original formulation, it guaranteed that the Big Bang has infinite density. Hawking later revised his position in A Brief History of Time (1988) where he stated that "there was in fact no singularity at the beginning of the universe" (p. 50). This revision followed from quantum mechanics, in which general relativity must break down at times less than the Planck time. Hence general relativity cannot be used to show a singularity.

Penrose's theorem is more restricted and only holds when matter obeys a stronger energy condition, called the dominant energy condition, in which the energy is larger than the pressure. All ordinary matter, with the exception of a vacuum expectation value of a scalar field, obeys this condition. During inflation, the universe violates the stronger dominant energy condition (but not the weak energy condition), and inflationary cosmologies avoid the initial big-bang singularity. However, inflationary cosmologies are still past-incomplete,and require physics other than inflation to describe the past boundary of the inflating region of spacetime.

It is still an open question whether time-like singularities ever occur in the interior of real charged or rotating black holes, or whether they are artifacts of high symmetry and turn into spacelike singularities when realistic perturbations are added.

Interpretation

Many theories in physics have mathematical singularities of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the Ultraviolet Catastrophe, re-normalization, and instability of a hydrogen atom predicted by the Larmor formula.

Some theories, such as the theory of loop quantum gravity suggest that singularities may not exist.The idea can be stated in the form that due to quantum gravity effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter, or alternatively that interpenetrating particle waves mask gravitational effects that would be felt at a distance.

How singularities may arise?

In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.

Other ways in which singularities occur is by degeneration of manifold structure. The presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired "corners" in a process of folding up, resembling the creasing of a table napkin.


BLACK HOLE SINGULARITIES

In the centre of a black hole is a gravitational singularity, a one-dimensional point which contains a huge mass in an infinitely small space, where density and gravity become infinite and space-time curves infinitely, and where the laws of physics as we know them cease to operate. As the eminent American physicist Kip Thorne describes it, it is "the point where all laws of physics break down".
Current theory suggests that, as an object falls into a black hole and approaches the singularity at the centre, it will become stretched out or “spaghettified” due to the increasing differential in gravitational attraction on different parts of it, before presumably losing dimensionality completely and disappearing irrevocably into the singularity. An observer watching from a safe distance outside, though, would have a different view of the event. According to relativity theory, they would see the object moving slower and slower as it approaches the black hole until it comes to a complete halt at the event horizon, never actually falling into the black hole.

(Click for a larger version) A gravitational singularity is hidden within a black hole (Source: Northern Arizona University: http://www4.nau.edu/meteorite/

The existence of a singularity is often taken as proof that the theory of general relativity has broken down, which is perhaps not unexpected as it occurs in conditions where quantum effects should become important. It is conceivable that some future combined theory of quantum gravity (such as current research into superstrings) may be able to describe black holes without the need for singularities, but such a theory is still many years away.
According to the "cosmic censorship" hypothesis, a black hole's singularity remains hidden behind its event horizon, in that it is always surrounded by an area which does not allow light to escape, and therefore cannot be directly observed. The only exception the hypothesis allows (known as a “naked” singularity) is the initial Big Bang itself.
It seems likely, then, that, by its very nature, we will never be able to fully describe or even understand the singularity at the centre of a black hole. Although an observer can send signals into a black hole, nothing inside the black hole can ever communicate with anything outside it, so its secrets would seem to be safe forever.