Regularization of black hole singularities

Question

Hi I have a question: when dealing with the gravitational Lorentz factor from schwarzchild solution to EFE, used in defining gravitaional time dilation and one encounters singularities at $r=0$ or $r=Rs$, would it be possible to apply a regulator term $£$ such that we'll be having $Rs/r+£$ instead of the usual $Rs/r$, I'm asking because regulators are used when dealing with similar divergences In QFT and Feynman diagrams and I'd like to know if this is also possible with Black hole singularities. Although I am thinking minimal Length like Planck's length or some Form of minimal radius may be applicable in this case of blackhole's event horizon and quantum gravity, but it's just a thought.

Answer 1

Disclaimer: I'm not a physicist and don't claim to be. But I am intimately familiar with divergent series regularization.

Even if you were to do this the utility would be questionable at best in your Schwarzchild example.

Suppose you enter a blackhole, the force you experience increases and increases as you approach the singularity. Finally you reach the singularity and your hyper spaghettified stream of atoms suddenly discovers that the force is only $-\frac{1}{12}$ Newtons repelling you. But only at the point $r = 0$. As soon as you are repelled slightly its again a force of $10^{100000}$ pulling you toward the singularity.

It kind of doesn't matter WHAT regularization you do at the point. The fact the singularity is there in LIMIT is the issue. Even if at the point you can do some regularization magic.

Leaving my humorous handwaving aside, Let's consider a concrete mathematical example.

Consider the function $y = \frac{1}{1-x}$. At $x=1$ this has a singularity of sorts. This function can be described as $$ y = \frac{1}{\ln(x)} + \frac{1}{2} - \frac{1}{12} \ln(x)^2 ... $$

While the regularized value of $y(1)$ is $\frac{1}{2}$. That $\frac{1}{2}$ has almost NO meaning in terms of how $y$ behaves at $1$. If $y(x)$ described the pressure you experienced as a function of $x$ then approaching $x=1$ you would feel the infinity WELL before you notice $\frac{1}{2}$. Now you might say "hey there is an infinite amount here of the nature $\frac{1}{\ln(x)}$ so if I subtract that away only $\frac{1}{2}$ is left" but that is besides the point.

Consider another function $y = \ln(x)$. The regularized value of $\ln(0) = \gamma$ but again if $\ln(x)$ described some kind of physical parameter as a function of $x$ you wouldn't physically care that the regularization at $0$ is $\gamma$. What matters is it goes to negative infinity FIRST.

FYI: this can be derived by noticing that $\ln(1-x) = x + \frac{x^2}{2} + \frac{x^3}{3} + ... $ therefore $\ln(0) = \ln(1-1) = 1 + \frac{1}{2} + \frac{1}{3} + ... $ and the regularized value of THIS is $\gamma$. See here: in particular the answer by reuns.

So where might regularization be relevant?

If you have a boundary, that you want to extend past, then regularization becomes your friend. This happens often in analytic continuation. I believe in QFT this the thing which is coming up.

No one is trying to look at isolated singularities, assigning numbers to them, and then pretending limits don't exist. They are trying to extend functions beyond their natural domain and to do so in a way that can be compatible with any underlying smooth/complex structure.

To recover your scharzchild example. If there is a suitable embedding of the schwarzchild solution into some higher dimensional space followed by analytic continuation and a coordinate transformation such that the singularity at $r=0$ becomes a natural boundary of dimension $\ge 1$ then you might start looking at regularization to EXTEND beyond it.