Gravitational singularity

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
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A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between general relativity and quantum mechanics; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity. Trying to find a complete and precise definition of singularities in the theory of general relativity, the current best theory of gravity, remains a difficult problem.^[1][2] A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite^[3] or, better, by a geodesic being incomplete.^[4]

Gravitational singularities are mainly considered in the context of general relativity, where density would become infinite at the center of a black hole without corrections from quantum mechanics, and within astrophysics and cosmology as the earliest state of the universe during the Big Bang. Physicists are undecided whether the prediction of singularities means that they actually exist (or existed at the start of the Big Bang), or that current knowledge is insufficient to describe what happens at such extreme densities.^[5]

General relativity predicts that any object collapsing beyond a certain point (for

In classical field theories, including special relativity but not general relativity, one can say that a solution has a singularity at a particular point in spacetime where certain physical properties become ill-defined, with spacetime serving as a background field to locate the singularity. A singularity in general relativity, on the other hand, is more complex because spacetime itself becomes ill-defined, and the singularity is no longer part of the regular spacetime manifold. In general relativity, a singularity cannot be defined by "where" or "when".[10]

Some theories, such as the theory of loop quantum gravity, suggest that singularities may not exist.^[11] This is also true for such classical unified field theories as the Einstein–Maxwell–Dirac equations. 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.

Motivated by such philosophy of loop quantum gravity, recently it has been shown^[12] that such conceptions can be realized through some elementary constructions based on the refinement of the first axiom of geometry, namely, the concept of a point ^[13] by considering Klein's prescription of accounting for the extension of a small spot that represents or demonstrates a point,^[14] which was a programmatic call that he called as a fusion of arithmetic and geometry.^[15] Klein's program, according to Born, was actually a mathematical route to consider `natural uncertainty in all observations' while describing `a physical situation' by means of `real numbers'.^[16]

Types

There is only one type of singularity, each with different physical features that have characteristics relevant to the theories from which they originally emerged, such as the different shapes of the singularities, conical and curved. They have also been hypothesized to occur without event horizons, structures that delineate one spacetime section from another in which events cannot affect past the horizon; these are called naked.

Conical

A conical singularity occurs when there is a point where the limit of some

An example of such a conical singularity is a cosmic string and a Schwarzschild black hole.^[17]

Curvature

Solutions to the equations of

) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by a change of coordinates.

An example is the

Riemann tensor

i.e. ${\displaystyle R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }}$, which is diffeomorphism invariant, is infinite.

While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity", in a rotating black hole, also known as a

Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship hypothesis. However, in 1991, physicists Stuart Shapiro and Saul Teukolsky performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make the simulation were removed. However, it is hypothesized that light entering a singularity would similarly have its geodesics terminated, thus making the naked singularity look like a black hole.^[19][20][21]

Disappearing event horizons exist in the Kerr metric, which is a spinning black hole in a vacuum, if the angular momentum (${\displaystyle J}$) is high enough. Transforming the Kerr metric to Boyer–Lindquist coordinates, it can be shown^[22] that the coordinate (which is not the radius) of the event horizon is, ${\displaystyle r_{\pm }=\mu \pm \left(\mu ^{2}-a^{2}\right)^{1/2}}$, where ${\displaystyle \mu =GM/c^{2}}$, and ${\displaystyle a=J/Mc}$. In this case, "event horizons disappear" means when the solutions are complex for ${\displaystyle r_{\pm }}$, or ${\displaystyle \mu ^{2}<a^{2}}$. However, this corresponds to a case where ${\displaystyle J}$ exceeds ${\displaystyle GM^{2}/c}$ (or in Planck units, ${\displaystyle J>M^{2}}$); i.e. the spin exceeds what is normally viewed as the upper limit of its physically possible values.

Similarly, disappearing event horizons can also be seen with the Reissner–Nordström geometry of a charged black hole if the charge (${\displaystyle Q}$) is high enough. In this metric, it can be shown^[23] that the singularities occur at ${\displaystyle r_{\pm }=\mu \pm \left(\mu ^{2}-q^{2}\right)^{1/2}}$, where ${\displaystyle \mu =GM/c^{2}}$, and ${\displaystyle q^{2}=GQ^{2}/\left(4\pi \epsilon _{0}c^{4}\right)}$. Of the three possible cases for the relative values of ${\displaystyle \mu }$ and ${\displaystyle q}$, the case where ${\displaystyle \mu ^{2}<q^{2}}$ causes both ${\displaystyle r_{\pm }}$ to be complex. This means the metric is regular for all positive values of ${\displaystyle r}$, or in other words, the singularity has no event horizon. However, this corresponds to a case where ${\displaystyle Q/{\sqrt {4\pi \epsilon _{0}}}}$ exceeds ${\displaystyle M{\sqrt {G}}}$ (or in Planck units, ${\displaystyle Q>M}$); i.e. the charge exceeds what is normally viewed as the upper limit of its physically possible values. Also, actual astrophysical black holes are not expected to possess any appreciable charge.

A black hole possessing the lowest ${\displaystyle M}$ value consistent with its ${\displaystyle J}$ and ${\displaystyle Q}$ values and the limits noted above; i.e., one just at the point of losing its event horizon, is termed extremal.

Entropy

Before

References