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According to Einstein's theory of general relativity (cf. [[Relativity theory|Relativity theory]]), [[Space-time|space-time]] can be modeled by a $4$-dimensional Lorentzian manifold (cf. [[Pseudo-Riemannian space|Pseudo-Riemannian space]]). Photons correspond to null geodesics and freely falling particles to time-like geodesics (cf. also [[Geodesic line|Geodesic line]]). Since an affine parameter of a time-like geodesic can be regarded as a normal clock carried by the freely falling particle, one says that a Lorentzian manifold contains a (future) singularity if there exists a time-like (or null) geodesic $\gamma$ which is future inextensible and whose affine parameter remains finite. The interpretation of a singularity as a physical singularity in the sense that space-time or gravity diverges in some sense may be misleading. As a simple example, consider the $2$-dimensional Lorentzian manifold $( M , g ) = ( \mathbf{R} ^ { 2 } \backslash \{ 0 \} , 2 / ( u ^ { 2 } + v ^ { 2 } ) d u d v )$. The curve $\nu ( t ) : = ( 1 / ( 1 - t ) , 0 )$ is an incomplete geodesic and the mapping $\psi : ( u , v ) \rightarrow ( 2 u , 2 v )$ is an isometry (cf. also [[Isometric mapping|Isometric mapping]]). Defining $x \sim y$ as

\begin{equation*} x \sim y \end{equation*}

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120010/n1200109.png"/></td> </tr></table>

one obtains the compact Lorentzian Clifford–Pohl torus $( \pi ( M ) , \pi_{*} g )$. The curve $\pi ( \nu )$ is an incomplete null geodesic in this compact torus and, therefore, in any physical sense a non-singular Lorentzian manifold. There are theorems which state that under "physically reasonable" conditions there are singularities in the Lorentzian manifolds representing space-times [[#References|[a2]]], [[#References|[a1]]] (but see also [[#References|[a4]]] and [[#References|[a3]]]). There is not much known about the properties of the singularities predicted by these theorems. In particular, it is not known whether these singularities are necessarily of a different nature than those in the Clifford–Pohl torus. More generally, it is not known whether they can be naked in the following sense.

A singularity is called naked if it is visible from some regular point in space-time $M$, i.e., if there is an $x \in M$ such that the geodesic $\gamma$ is contained in the past of $x$. It is presently (1998) believed by the majority of physicists that naked future singularities do not occur in physically realistic and stable space-times (cf. [[Penrose cosmic censorship|Penrose cosmic censorship]]). Note that in the physics literature there are various conflicting definitions of singularities and naked singularities. The version chosen here is motivated by the singularity theorems.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press (1973) {{MR|0424186}} {{ZBL|0265.53054}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> S.W. Hawking, R. Penrose, "The singularities of gravitational collapse and cosmology" ''Proc. R. Soc. London Ser. A'' , '''314''' (1970) pp. 529–548 {{MR|0264959}} {{ZBL|0954.83012}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> M. Kriele, "A generalization of the singularity theorem of Hawking and Penrose to spacetimes with causality violations" ''Proc. R. Soc. London A'' , '''431''' (1990) pp. 451–464 {{MR|1086352}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> R.P.A.C. Newman, "Black holes without singularities" ''Gen. Rel. Grav.'' , '''18''' : 11 (1989) pp. 981–995 {{MR|1016054}} {{ZBL|0677.53072}} </td></tr></table>