Editing Gravitational singularity (section)

===Curvature===
[[File:Black hole details.svg|thumb|upright=0.7|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 (general relativity)|metric]] blows up to infinity. However, many of these points are completely [[Smooth function|regular]], and the infinities are merely a result of [[Coordinate singularity|using an inappropriate coordinate system at this point]]. To test whether there is a singularity at a certain point, one must check whether at this point [[Diffeomorphism invariance|diffeomorphism invariant]] quantities (i.e. [[scalar (physics)|scalar]]s) 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 metric|Schwarzschild]] solution that describes a non-rotating, [[Electric charge|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 [[Smooth function|regular]]. The regularity becomes evident when changing to another coordinate system (such as the [[Kruskal coordinates]]), where the metric is perfectly [[Smooth function|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. <math>R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}</math>, 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]].<ref>If a rotating singularity is given a uniform electrical charge, a repellent force results, causing a [[ring singularity]] to form. The effect may be a stable [[wormhole]], a non-point-like puncture in spacetime that may be connected to a second ring singularity on the other end. Although such wormholes are often suggested as routes for faster-than-light travel, such suggestions ignore the problem of escaping the black hole at the other end, or even of surviving the immense [[tidal force]]s in the tightly curved interior of the wormhole.</ref>

More generally, a spacetime is considered singular if it is [[Geodesic (general relativity)#Geodesic incompleteness and singularities|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]] [[physical cosmology|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.