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A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is so intense that spacetime itself breaks 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]

The initial state of the universe, at the beginning of the Big Bang, is also predicted by modern theories to have been a singularity.[7] In this case, the universe did not collapse into a black hole, because currently-known calculations and density limits for gravitational collapse are usually based upon objects of relatively constant size, such as stars, and do not necessarily apply in the same way to rapidly expanding space such as the Big Bang. Neither general relativity nor quantum mechanics can currently describe the earliest moments of the Big Bang,[8] but in general, quantum mechanics does not permit particles to inhabit a space smaller than their wavelengths.[9]

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]

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. R μ ν ρ σ R μ ν ρ σ {\displaystyle R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }} , which is diffeomorphism invariant, is infinite.

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.

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