general relativity - About the 1D singularity of black hole

I saw some responses here saying that the singularity into the black hole is one dimension object so my question is : is it possible that the singularity is simply a merger of the 4 dimensions of the spacetime? Is it possible physically and mathematically that many dimensions can merge into one ?

Answer

I'll attempt to address your question in the context of classical general relativity since the dimensionality of the relevant manifold is more involved in some of the more recent holographic pictures, or may not even be well defined at all in the fundamental theory (whatever that is) until some sort of low energy limit is taken.

When people try to conceptualize a singularity, they often consider the case of a point charge in classical electromagnetism, in which the electric field is given by $${\bf{E}}({\mathbf{r}}) = \frac{e}{4\pi\epsilon_0}\frac{({\mathbf{r}}-{\mathbf{r_0}})}{|{\mathbf{r}}-{\mathbf{r_0}}|^3}$$ Here the electric field diverges as $\mathbf{r}\rightarrow \mathbf{r_0}$. In this case , it's perfectly adequate to define something like "a singularity is a location where one or more components of the electric field diverges".

The natural approach is to attempt to define a GR singularity as a location where some measure of the spacetime curvature diverges. The first example that comes to mind might be $r=0$ in a Schwarzschild metric. Unfortunately, $r=0$ isn't a location where the manifold has a smooth Lorentz signature metric, so it is not part of spacetime, i.e. the problem is with the highlighted words. This is more than just a mathematical technicality: if the metric isn't well behaved, spacetime just doesn't have the properties we expect it (classically) to have.

One approach that has been tried to partially resurrect the idea of a singular location is to provide a prescription for attaching extra points to the spacetime manifold. These are thought of as representing some sort of "boundary" to spacetime, which represents the singular points. This works for straightforward cases like the Schwarzschild solution, but doesn't work in general.

Another approach, and perhaps the most successful one, to the definition of singularity in GR has been to examine the behaviour of curves in the spacetime. A physical pointlike object moving through spacetime traces out a curve. As you wind the object's proper time forward, it traces out a path through spacetime. This path is a map from the real numbers, representing this proper time (or some other parameter) into the manifold. Now normally, this parameter can be extended into the infinite future (we're considering classical objects, not particles in QFT which can disappear in the sense that they turn into something else at a given time). If there are curves in the spacetime for which the parameter cannot be extended into the infinite future, this is a sign that the spacetime is singular.

You can produce a singularity like this by just removing points from spacetime. For example, if you remove some points from Minkowski space, then curves which would otherwise have run into those points now can't have their parameters extended indefinitely, so the spacetime is flagged as singular. However, in this case where we removed some points, the manifold can be extended by putting the points back, and the singularity goes away! To circumvent this difficulty, you only apply the "inextendible curves" criterion to maximally extended spacetimes - i.e. ones which use analytic extension of the metric to "put back" any points that could possibly be put back.

Singular spacetimes can exhibit some extremely bizarre behaviour. For example one observer freely falling through a compact region of spacetime, can experience unbounded curvature forces. This observer's worldline has a limit point p. Another observer can travel through p with no such problems (Hawking and Ellis prop 8.5.2)!

Returning to the specific question, whereas loss of a spatial dimension in a manifold certainly would be singular behaviour in a differential-geometric sense, it doesn't capture all the peculiar behaviours which are part of singular spacetimes in GR. Perhaps even without looking at these exotic singular behaviours, it is clear that the "simple" case of the Schwarzschild singularity doesn't involve a straightforward dimensional reduction. If you take a look at the conformal diagram of the maximally extended Schwarzschild solution you can see that the spacetime has topology $\mathbb{R}^2XS^2$. The future singularity is represented as part of an attached boundary, and timelike curves in that region approaching from any spatial direction will encounter it. There are no directions for which the curves can "slip through" as there would be if the singularity consisted of a set of points where the dimensionality of spacetime was reduced.