Explaining Weak & Null Singularities in Black Holes

[Dmitry67]

I understand the concept of a spacetime (future) singularity in a BH
I understand what is a ring singularity in Kerr'sblack hole

Could anyone explain (for dummies) what is meant by "weak" and "null" singularity?

 

[xantox]

Null means that the singularity is not spacelike nor timelike, but light-like. Weak means that the tidal deformations do not diverge at the singularity. It may happen at the inner horizon of a perturbed charged black hole, as studied in Poisson and Israel, "Internal structure of black holes", Phys. Rev. D41 (1990), 1796-1809.

 

[Dmitry67]

Thank you
The what is 'singular' there?

 

[George Jones]

Thank you
The what is 'singular' there?

Spacetime.

It probably is easier to give somewhat general examples "singular spacetime" than to give a generic definition of a spacetime singularity.

For example, a spacetime is singular if there is a timelike curve having bounded acceleration (i.e, a worldline an observer could follow) that ends after a finite amount of proper time, and that is inextendable. Singular spacetimes have "edges".

What does inextendable mean? This depends on how "differentiable" the differentiable manifold used to model spacetime is.

Spacetimes usually are taken to be suitably smooth differentiable manifolds. If differentiability of the metric is relaxed to continuity of the metric, then spacetimes can be extended through some singularities. Since second derivatives of the metric are used to construct curvature, we could have a continuous metric (like the absolute value function) that, when differentiated twice, gives a distribution that involves a Dirac delta function. Then, tidal forces don't build up in a continuous way.

 

[Dmitry67]

Thank you
Still, it is not clear to me. I read in some sources that bluesheet is not fatal and observer can survive falling thru cauchy horizon. So for an observer metrics is not singular? As I understand, geodesics there just lead inside the second horizon, and they don't 'end' after finite time?

 

[George Jones]

Thank you
Still, it is not clear to me. I read in some sources that bluesheet is not fatal and observer can survive falling thru cauchy horizon.

If a curvature singularity blows up like a Dirac delta function, then integration produces only a finite contribution to the tidal deformation of an object, which, if the object is robust enough, it can withstand.

So for an observer metrics is not singular? As I understand, geodesics there just lead inside the second horizon, and they don't 'end' after finite time?

This depends on the differentiability condition imposed on the spacetime manifold.

 

[xantox]

Thank you
Still, it is not clear to me. I read in some sources that bluesheet is not fatal and observer can survive falling thru cauchy horizon. So for an observer metrics is not singular? As I understand, geodesics there just lead inside the second horizon, and they don't 'end' after finite time?

Poisson and Israel have shown that perturbations of a charged black hole due to ingoing radiation lead to a nonscalar singularity, and if also outgoing radiation is present then a phenomenon dubbed mass inflation arises and the singularity becomes scalar (the Weyl curvature scalar diverges), though the metric is still regular and tidal effects integrated on the infalling body worldline are finite (for this reason the singularity is called weak).

Thus, it seems possible that spacetime can be classically continued beyond the Cauchy horizon, even if general relativity cannot predict it. A fully quantum theory of gravity is required to exactly model the Cauchy horizon and its vicinity.

 

[stevebd1]

If a curvature singularity blows up like a Dirac delta function, then integration produces only a finite contribution to the tidal deformation of an object, which, if the object is robust enough, it can withstand.

Is the weak singularity at the Cauchy horizon pretty much an instantaneous infinite blip (like an upside-down capital T) on the scalar curvature or is there expected to be some gradient of change? If there is a gradient of change, would this have to be confined within the event horizon or might some small degree of mass-inflation be detected outside the BH?