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An analytic extension embeds a spacetime isometrically as a proper open region in a larger analytic Lorentzian manifold with a nondegenerate metric tensor. A divergence of coordinate components does not obstruct extension if a new chart makes all metric coefficients analytic and the determinant nonzero. For a static spherical metric with an analytic lapse $f$ and a positive simple horizon radius, ingoing coordinates replace $-fd{t}^2+d{r}^2/f$ by $-fd{v}^2+2dvdr$. This removes the coordinate singularity at $f=0$; transverse metric coefficients must also remain regular. An extension across such a horizon is distinct from extending through a curvature singularity.

A black hole is a region from whose events no future-directed causal curve can reach the distant exterior. In an asymptotically flat spacetime with future null infinity ${\mathcal{I}}^{+}$, the standard definition isHere $J^{-}$ denotes the causal past and ${\mathcal{H}}^{+}$ is the future event horizon. This is a statement about the complete causal geometry, rather than a material surface or a local force threshold.

For a Schwarzschild black hole, ingoing Eddington-Finkelstein coordinates give outgoing radial null geodesics the slope $dr/dv=(1-2M/r)/2$ in units $G=c=1$. Inside $r=2M$ even this outgoing family moves toward smaller areal radius; the other future radial null family does too. A regular horizon can therefore prevent escape without being a curvature singularity.^[ref]

Rotating examples are described by a Kerr black hole. A dynamically forming event horizon need not coincide with a slice's apparent horizon; locating the former requires global escape information.

In an FLRW metric with zero separate cosmological constant, $\varrho +3p\ge 0$ makes the Friedmann acceleration equation imply $\ddot{a}\le 0$. If $a_0=a(t_0)>0$ and $v_0=\dot{a}(t_0)>0$, then the scale factor is a concave function andthroughout its regular past interval. Thus a regular positive scale factor cannot persist earlier than $t_0-a_0/v_0$. If the solution continues regularly whenever $a>0$, its first past zero occurs within $a_0/v_0=H(t_0{)}^{-1}$, where $H$ is the Hubble parameter.

The continuation condition matters: without restrictions on the equation of state, an earlier curvature singularity can occur at positive $a$. For example, in fixed units $a(t)=1+\sqrt{t}$, $t>0$, with $k=0$, defines energy density $\varrho =3(\dot{a}/a{)}^2/(8\pi G)$ and pressure $p=-\varrho /3-\ddot{a}/(4\pi Ga)$. These satisfy the Friedmann equations and $\varrho +3p>0$, yet $a\to 1$ at the singular past boundary. Conversely the Milne model reaches zero without divergent curvature. A finite duration, a zero of the scale factor, and a curvature singularity are distinct conclusions.

A coordinate singularity is a failure of a coordinate description which disappears in a regular chart. The horizon singularity of the Schwarzschild metric in $(t,r)$ coordinates is removable; its $r=0$ curvature singularity is not.

A physicist who uses numerical relativity to study strong gravitational fields, especially curvature singularities and gravitational collapse. His simulations test how the Einstein field equations behave close to a singularity, including whether the behavior found in symmetric models survives when symmetry restrictions are removed.^[ref]

With José M. Martín Senovilla, he reviewed how the Penrose singularity theorem connects gravitational collapse with geodesic incompleteness.^[ref]

At maximum expansion of a closed matter-dominated Friedmann solution, the Hubble parameter vanishes while the physical mass density stays positive and finite. Its critical density therefore vanishes and the cosmological density parameter diverges. This is a singularity of that ratio rather than a curvature singularity. In the cycloid variable $\theta$, ${\Omega }_M={\sec }^2(\theta /2)$, diverging at $\theta =\pi$ and approaching one at the Big Bang and Big Crunch ends.

For a captured nonradial null geodesic with $u=1/r$, the large-$u$ orbit equation is $(du/d\varphi {)}^2\sim 2\mu u^3$. Integration on the inward branch givesThe curvature singularity is reached at finite azimuth. This does not prevent a near-critical ray from winding repeatedly near the photon sphere before its final plunge. A radial ray has constant azimuth, so this asymptotic coordinate parametrization is inapplicable to it.

In five spacetime dimensions, consider the static spherical charged metricHere $M>0$ and $Q$ are geometric parameters, with dimensions of squared length; the physical ADM mass and electric charge depend on the gravitational and electromagnetic normalization. The spherical sections are three-spheres.

Putting $y=r^2$ reduces $f=0$ to $y^2-2My+Q^2=0$. For $0<|Q|<M$, the two positive roots giveAt $M=|Q|>0$ they coincide, giving an extremal black hole; for $M<|Q|$ no positive horizon root hides the central curvature singularity. The sign of electric charge cannot change these metric conditions because only $Q^2$ occurs. When $Q=0$, the positive horizon is $r_{+}=\sqrt{2M}$ and the other formal root is the singular centre, rather than a second regular horizon. This uncharged limit is the five-dimensional Schwarzschild-Tangherlini metric.

The lecture notes by David Tong derive Friedmann equations in section 4.6.2 and classify Reissner-Nordstrom spacetime in sections 6.2.1–6.2.5. Section 6.2 distinguishes the central curvature singularity from regular horizons, including the extremal limit.

For $0<|Q|<M$, a Reissner-Nordstrom spacetime has an outer event horizon and an inner Cauchy horizon. Between them $r$ is timelike; inside the inner horizon $r$ is spacelike and the $r=0$ curvature singularity is timelike. The exact analytic extension contains repeated exterior blocks beyond the Cauchy horizons. Small perturbations produce severe inner-horizon blueshift and mass inflation, so that exact continuation is not a robust prediction for collapse. At $|Q|=M$ the horizons merge into a degenerate Killing horizon; for $|Q|>M$ the spherical electrovacuum solution instead has a naked singularity.

A gravitational singularity theorem gives sufficient conditions for geodesic incompleteness: some freely falling worldline or light-ray geodesic cannot continue through an unlimited proper time or affine parameter within the stated spacetime. The conclusion concerns the failure of complete geodesic evolution; it does not by itself establish a curvature singularity.

A typical argument combines a focusing assumption such as the null energy condition, the focusing of a geodesic congruence, and assumptions on the global causal structure. The Penrose singularity theorem uses a trapped surface to show that collapse need not preserve exact spherical symmetry to force null incompleteness under its other hypotheses. This is useful because highly symmetric exact solutions alone cannot establish what happens during generic collapse.^[ref]

The Kretschmann scalar is the scalar tensor contraction of two copies of the Riemann curvature tensor. In Schwarzschild spacetime, with metric coefficient $1-2M/r$ and $G=c=1$, it is $48M^2/r^6$. Divergence of this scalar along a curve approaching a proposed regular point proves that there is no twice continuously differentiable, nondegenerate metric extension through that point: such an extension would have continuous finite curvature components and hence a finite scalar contraction. This gives a coordinate-independent test of a curvature singularity. Boundedness of this single scalar does not prove regularity.

For $V=1-2M/r^2+Q^2/r^4$, an orthonormal-frame curvature calculation gives $(V^{\prime \prime }{)}^2+6(V^{\prime }/r{)}^2+12((1-V)/r^2{)}^2$. Expansion gives the displayed invariant, diverging at zero radius. It is finite at every positive root of V. Thus horizon coordinate singularities are distinct from the central curvature singularity.

The Kruskal spacetime is the maximal analytic extension of the positive-mass vacuum Schwarzschild spacetime. It contains two exterior regions, a black hole, and a white hole. Kruskal–Szekeres coordinates remove both horizon coordinate singularities, while $r=0$ remains a curvature singularity.

A maximal analytic extension of an analytic spacetime is an analytic spacetime extension admitting no further proper analytic extension as a regular spacetime of the same dimension. It is different from a maximal Cauchy development, which can end at a smooth Cauchy horizon. Additional global choices or quotients can affect uniqueness. The simply connected extension of positive-mass Schwarzschild spacetime through both horizons is the Kruskal spacetime; its curvature singularities remain boundaries.

An FLRW metric with $p=-{\varrho }_0$ and constant negative energy density ${\varrho }_0$ has ${\alpha }^2=-8\pi G{\varrho }_0/3>0$ and obeys ${\dot{a}}^2+k=-{\alpha }^2{a}^2$. Positive solutions exist only for $k=-1$. They are $a={\alpha }^{-1}\sin [\alpha (t-t_B)]$ on $0<\alpha (t-t_B)<\pi$. This is an open slicing of Anti-de Sitter spacetime, whose Ricci scalar is $-12{\alpha }^2$ and Kretschmann scalar is $24{\alpha }^4$. The zeros mark boundaries of this slicing rather than curvature singularities.

For the three cases, distinguish a curvature singularity from a coordinate singularity. The Reissner-Nordstrom Kretschmann scalar isIt diverges at $r=0$ when either $M$ or $Q$ is nonzero. Any positive root of $F$ instead admits Eddington-Finkelstein coordinates: set $v=t+\int dr/F$ to obtainThe determinant of its $(v,r)$ block is $-1$, including at $F=0$. Hence positive roots of $F$ are removable coordinate singularities of the static chart. The apparent degeneracy at the poles of the angular chart is also a coordinate singularity.

For $0<Q^2<M^2$ there are two distinct positive roots:Both roots are coordinate singularities of the static chart. In the standard extension, $r_{+}$ is the outer event horizon and $r_{-}$ is an inner Cauchy horizon. The curvature singularity is $r=0$. Between the horizons $F<0$, so the radial direction is timelike; for $0<r<r_{-}$, $F>0$ again and the central curvature singularity is timelike. These are the three radial regions of the global structure of a charged spherical black hole.^[ref]

For completeness, if $Q=0$ and $M>0$, the Schwarzschild spacetime limit has just the positive horizon $r=2M$ and a spacelike curvature singularity at $r=0$; the formal root $r_{-}=0$ of $r^{2}F$ is not a regular inner horizon. If $M=Q=0$, the metric tensor is flat and the radial origin is regular in Cartesian coordinates.

If $M^2=Q^2$ with $M>0$, thenThe curvature singularity remains at $r=0$. At $r=M$ the static chart has a coordinate singularity, removed by the Eddington-Finkelstein coordinates above. The Kretschmann scalar there equals $8/M^4$, which is finite. The coincident horizons form a degenerate Killing horizon, and its future outer branch is the event horizon of the extremal black hole. A double zero of $F$ does not remove the horizon.

If $M^2<Q^2$, thenThere is no positive root and no horizon hiding $r=0$ in the extended Reissner-Nordstrom spacetime. Its Kretschmann scalar diverges there, so $r=0$ is a naked singularity. It is a timelike curvature singularity, as the normal to nearby $r=\text{constant}$ hypersurfaces has positive squared norm $g^{rr}=F$.