Gravitational singularity

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A gravitational singularity or spatiotemporal, informally and from a physical point of view, can be defined as an area of space-time where no magnitude can be defined Physics related to gravitational fields, such as curvature, or others. Numerous examples of singularities appear in realistic situations within the framework of general relativity in solutions to Einstein's equations, including the description of black holes (such as the Schwarzschild metric) or the description of the origin of the universe. (Robertson-Walker metric).

From a mathematical point of view, adopting a definition of singularity can be complicated, because if we think of points where the metric tensor is not defined or not differentiable, we are talking about points that do not automatically belong to space-time. To define a singularity we must look for the traces that these excluded points leave in the fabric of spacetime. We can think of several types of strange behavior:

Temporary geodetic (or null) that after a time of its own (or affin parameter) cannot be prolonged (what is called incompleteness of causal geodesics).
Curve values that are made arbitrarily large near the excluded point (what is called singularity of curvature).

Types of singularities

Singularities can be, in their most general aspects:

Coordinates. They're the result of choosing a bad coordinate system. Some of these coordinate singularities indicate physical places that are special, for example, in Schwarzschild metric the uniqueness of coordinates in $r=2GM/c^2 ,!$ represents the horizon of events.
Physics. They are timeless singularities of full right. It differs from coordinates because in some of the contractions of the curvature tensor, this diverge ( $R_{{mu nu rho lambda }}R^{{mu nu rho lambda }},!$ , $R_{{mu nu }}R^{{mu nu }},!$ etc.)

Geometrically, physical singularities can be:

Open surfaces: This type of singularity can be found in black holes that have not kept the angular moment as is the case of a black Schwarzschild hole or a black Reissner-Nordstrøm hole.
Closed hypersurfaces: As the toroidal or ring-shaped singularity, which normally makes its appearance in black holes that have preserved its angular moment, as may be the case of a black Kerr hole or a black Kerr-Newman hole, here the matter, due to the twist, leaves a space to the middle forming a structure similar to that of a donut.

Depending on their nature, physical singularities can be:

Temporary, like the one found in a Schwarzschild hole in which a particle ceases to exist for a certain moment of time; depending on its speed, fast particles take longer to reach singularity while the slower ones disappear before. This type of singularity is inevitable, as sooner or later all particles must pass through the singular temporal hypersurface.
Singularitieslike the one in Reissner-Nordstrom holes, Kerr and Kerr-Newman. Being space hypersurfaces a particle can escape them and therefore it is avoidable singularities.

Depending on the visibility for asymptotically inertial observers far from the black hole region (Minkowski space-time) these can be:

Naked Singularities: There are cases in the black holes where due to high loads or at turning speeds, the area surrounding singularity disappears (in other words the horizon of events) leaving this visible in the universe we know. This case is supposed to be prohibited by the cosmic censorship hypothesis, which states that all singularity must be separated from space.
Singularities inside black holes. In other words, matter is compressed to an unimaginably small or singular region, whose density within it is infinite. That is to say that all that falls within the horizon of events is swallowed, devoured by a point that we could call "without return", and this is so that neither the light can escape this celestial phenomenon, even traveling to 300,000 km/s. And according to Einstein's theory of Relativity, as nothing can travel at a speed greater than that of light, nothing can escape.

Singularity Theorems

Singularity theorems, by Stephen Hawking and Roger Penrose, predict the occurrence of singularities under very general conditions about the shape and characteristics of space-time.

Expansion of the universe and the Big Bang

The first of the theorems, which is stated below, seems applicable to our universe; informally states that if we have a globally hyperbolic expanding space-time, then the universe came into existence from a singularity (Big Bang) a finite time ago:

Theorem 1. Sea (M,g) a globally hyperbolic time space that fulfills $scriptstyle R_{{ab}}xi ^{a}xi ^{b}geq 0$ for all temporary vectors $scriptstyle xi ^{a}$ (as would happen if Einstein's field equations are satisfied, the strong condition of energy for matter being fulfilled). Suppose there is a hypersurface of Space Cauchy (and class at least C2) for which the trace of the intrinsic curvature satisfies K. C 0, where C It's a certain constant. Then no time curve starting from σ and directed to the past can be longer than $scriptstyle 3/|C|$ . In particular, all temporal geodesics towards the past are incomplete.

The above theorem is therefore the mathematical statement that under the conditions observed in our universe, in which Hubble's law is valid, and admitting the validity of the theory of general relativity, the universe must have started at some point.

Black holes and singularities

The following theorem relates the occurrence of "trapped surfaces" with the presence of singularities. Since trapped surfaces occur in a Schwarzschild black hole, and presumably holes with similar geometries, the following theorem predicts the occurrence of singularities inside a very large class of black holes. A trapped surface is a compact two-dimensional Riemannian manifold that has the property that both its causal future and its causal past have a negative expansion at all points. It is not difficult to prove that any sphere, indeed any closed surface contained in a sphere, within the black hole region of a Schwarzschild space-time is a trapped surface, and therefore a singularity must appear in that region. The statement of this theorem, due to Roger Penrose (1965), is the following:

Theorem 2. Sea (M,g) a globally hyperbolic space-time in which $scriptstyle R_{{ab}}k^{a}k^{b}geq 0$ for all light-type vectors $scriptstyle k^{a}$ (such as it would happen if Einstein's field equations are satisfied by meeting the strong condition or the weak condition of energy, for the matter of such space-time). Suppose there is a Space Cauchy hypersurface (and class at least C2) and a trapped surface and it is θ₀ the maximum value of the expansion on it, if θ₀ ≤ 0; then there is at least one geodesic of light type, unextendable to the future, which will also be orthogonal to the trapped surface. In addition the parameter value afine to the point from which it is not extensible is less than $scriptstyle 2/|theta _{0}|$ .

The existence of an inextensible light-type geodesic implies that there will be a photon leaving said surface after a travel time proportional to 2/c|θ₀ | will run into a future temporal singularity. Although we do not know the real physical nature of singularities due to the lack of a quantum theory of gravity, the photon will either "disappear" or it will experience some phenomenon associated with said theory of quantum gravity whose nature we do not know. In addition to being able to neutralize a singularity in its smallest size with a force equivalent to or greater than said singularity and once with that force trying to reverse the direction of the gravitational force, a (nuclear) fusion is needed within the nucleus before being absorbed.

For which, the intrinsic curvature trace satisfies K < C < 0, where C is some constant. Then no time curve starting from Σ and directed towards the past can have a length greater than 3/|C|. In particular, all time geodesics into the past are incomplete.

Conservation of black hole area

Although without being strictly singularity theorems, there is a collection of results proven by Hawking (1971) that establish that, within the framework of the general theory of relativity:

A related black hole cannot disappear or be divided into two. Therefore if two black holes collided, after their interaction would necessarily be merged.
The total area of black holes in the universe is a growing monotonous function, more specifically the area of the two-hole collision event horizon is greater than the sum of original areas.
The temporal evolution of a surface trapped in a black hole region will forever be contained in that black hole.

The above theorems are important because they guarantee, that even in real situations where exact calculations are complicated or impossible, the topological properties of a space-time containing black holes guarantee certain facts, no matter how complicated the geometry is. Of course we know that in a quantum theory of gravity the first two results probably do not hold. Hawking himself suggested that the emission of Hawking radiation is a quantum-mechanical process through which a black hole could lose area or evaporate; Therefore, the previous results are only the predictions of the general theory of relativity.

Occurrence of singularities

Einstein's theory of general relativity's description of spacetime and matter cannot adequately describe singularities. In fact, the general theory of relativity only gives an adequate description of gravitation and spacetime on scales larger than the Planck length l_P:

l_{P}={sqrt {{frac {hbar G}{c^{3}}}}}approx 10^{{-33}}{mbox{cm}}

Where: $hbar$ is the constant of Planck reduced, $G,$ constant universal gravitation, $c,$ It's the speed of light.

From this quantum limit it should be expected that the theory of relativity will also stop being adequate when it predicts a spatial curvature of the order of l_P^-2 which happens very close to curvature singularities such as those inside various types of black holes.

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