# Difference between space-like and time-like singularities [duplicate]

See the Wikipedia article on Penrose-Hawking singularity theorems.

It says that

A singularity in solutions of the Einstein field equations is one of two things:

1. a situation where matter is forced to be compressed to a point (a space-like singularity)

2. a situation where certain light rays come from a region with infinite curvature (a time-like singularity)

If matter is forced to be compressed to a point, why is this called a space-like singularity?

If certain light rays come from a region with infinite curvature, why is this called a time-like singularity?

## marked as duplicate by tparker, John Rennie, Jon Custer, Kyle Kanos, user191954 Sep 6 '18 at 15:30

• Spacelike and timelike singularities mean that spacelike (resp. Timelike) curves are incomplete and stop at that point. – Slereah May 29 '17 at 21:51
• When you say that a curve is geodesically incomplete, do you mean that the curve starts at infinity but ends at a finite point? – nightmarish May 29 '17 at 21:55
• It means that, given any point on that curve, if the curve is normalized, it will only have a finite range in either the future or past direction – Slereah May 29 '17 at 21:57

The spacelike singularity is a spacelike region where curvature diverges. It is not a point, though it occurs when the radial distance in the metric approaches zero $r~\rightarrow~0$. This is a three dimensional region. The time like singularity is a one dimensional region, think of it a a point evolving in time, where curvature diverges. This has a duality with the spacelike singularity, being one dimensional and orthogonal to the spatial directions of a spacelike singularity.
Think of the Schwarzschild metric with signature $(+,~-,~-,~-)$ $$ds^2~=~\left(1~-~\frac{2GM}{rc^2}\right)dt^2~-~\left(1~-~\frac{2GM}{rc^2}\right)^{-1}dr^2~-~r^2(d\theta^2~+~sin^2\theta d^2\phi),$$ and see that the metric term $\left(1~-~\frac{2GM}{rc^2}\right)$ flips sign when $r~<~\frac{2GM}{c^2}$. This means the interior of the black hole has signature $(-,~+,~-,~-)$, and in a sense the radial direction serves at a time direction. The $t,~\theta,~\phi$ part of the metric defines the spatial surface.
The case for the timelike singularity occurs with charge or angular momentum. I will use the Reissnor-Nordstrom metric since it is simpler to understand than the Kerr metric. In fact it is a modification of the Schwarzschild metric $$ds^2~=~\left(1~-~\frac{2m}{r}~+~\frac{Q^2}{r^2}\right)dt^2~-~\left(1~-~\frac{2m}{r}~+~\frac{Q^2}{r^2}\right)^{-1}dr^2~-~r^2(d\theta^2~+~sin^2\theta d^2\phi),$$ Since the metric coefficient $\left(1~-~\frac{2m}{r}~+~\frac{Q^2}{r^2}\right)$ is quadratic in $r$ is vanishes at two radii. There is an inner and outer horizon determined by $r_{\pm}~=~m~\pm~\sqrt{m^2~-~Q^2}$. There is then an inner horizon where the metric flips signature again. This means the region is timelike and the region $r~=~0$ is a point in space evolving in time. This is seen in the Penrose diagram