Gravitational singularity

A conical singularity occurs when there is a point where the limit of some diffeomorphism invariant quantity does not exist or is infinite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere the coordinate system is used.

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

A simple illustration of a non-spinning black hole and its singularity

Solutions to the equations of general relativity or another theory of gravity (such as supergravity) often result in encountering points where the metric blows up to infinity. However, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point. In order to test whether there is a singularity at a certain point, one must check whether at this point diffeomorphism invariant quantities (i.e. scalars) 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 Schwarzschild solution that describes a non-rotating, uncharged black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon. However, spacetime at the event horizon is regular. The regularity becomes evident when changing to another coordinate system (such as the Kruskal coordinates), where the metric is perfectly smooth. On the other hand, in the center of the black hole, where the metric becomes infinite as well, the solutions suggest a singularity exists. The existence of the singularity can be verified by noting that the Kretschmann scalar, being the square of 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 Kerr black hole, the singularity occurs on a ring (a circular line), known as a "ring singularity". Such a singularity may also theoretically become a wormhole.[18]

More generally, a spacetime is considered singular if it is geodesically incomplete, meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the event horizon of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the Big Bang cosmological model of the universe contains a causal singularity at the start of time (t=0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite spacetime curvature.

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.