Expert: Philip A. Stahl - 1/22/2009

Question
Hello,

I was reading the latest Scientific American about naked singularities, and as far as I can understand it, the difference between a black hole and a naked singularity is that, since the latter doesn't have an event horizon, the pull of gravity does not, as it does when you pass beyond a black hole's event horizon, become infinite. So I'm wondering, can such a thing as "infinite gravity" even exist in nature, or is the whole concept of infinity really just a mathematical construct? How, when and why was the concept of infinity introduced into the math of astrophysics, and have we seen anything to indicate that "infinity" is a concept that makes sense in the real world? It seems to me that infinity is something that goes beyond the end, or beyond the whole way, so I don't see how it makes sense, for instance, to call something "infinitely close" to something else, or saying approaching something "to infinity". It is all woolly math, or can this terminology actually be applied to the real world? Whatever the case, I don't see why a mathematical equation that results in "infinite gravity" should necessarily be mirrored by the forces around an actual black hole. If there are any black holes. It's starting to look as if the whole idea of the event horizon is based on untenable assumptions, isn't it?

Sorry if I was rambling a bit - just answer to the best of your ability. I know it's all very theoretical, so I don't expect very definite answers.

- Tue

Answer
Hello,

First, I am not about to get into any debate over the meaning of “infinite” or “infinity”.  I think one can do no better, if seriously interested in the concept, than to consult Rudy Rucker’s excellent book: ‘Infinity: The Science and Philosophy of the Infinite’ (Bantam, 1982). Pay special attention to Chapter One, ‘Infinity’ – which not only delves into the history of the concept, but also gives many examples of how and why it can be significant. (for example, you will garner an excellent insight into what it means to be "infinitely close" to something else, or saying something exhibits an approach "to infinity")

Second, as for the SciAm article itself, pay special heed to p. 38:

“…..applying the theory to stellar collapse is still a formidable task. Einstein’s equations of gravity are enormously complex, and solving them requires physicists to make simplifying assumptions”.

This the author of the piece (Joshi) has surely done with his ancillary referenced arxiv paper (‘Quantum Evaporation of a Naked Singularity’)


Call me skeptical here but I tend more to side with Roger Penrose (‘The Road to Reality’, p. 768) that “timelike” naked singularities (those in which signals can enter or leave) are forbidden, by his mathematical conjecture known as the “cosmic censorship theorem”.

Joshi himself, does a fair job of presenting his side in his paper, but I am leery of the use of quantum gravity loops as applied to the cores of collapsed stars which can feasibly end up as “naked singularities”. (Basically, such “loops” are employed as a way to find solutions near singularities, by avoiding the “infinity”. One uses different loop diagrams, and then “regulates” or normalizes those which appear to display divergences).

In a way the procedure is analogous to what is done in certain cases in plasma physics, in which the Laplace transform function expressing the velocity u, angular frequency (w) and wave number vector (k) associated with a particular plasma dispersion function, points to a pole or “infinity” at certain regions.

Such is the case below, for the upper right half-plane for plotting velocities u on the imaginary (vertical) and real (horizontal) axis.

For some Laplace transform function E1(w) one has:

E1(w) ~  INT  du {(df/du) /  [u – w/k])}

And at certain value of u (= w/k) what do we find? Well, w/k – w/k = 0 so

E1(w) -> oo

Of course, this is verboten! It is an infinity! A singularity! As you can see they don’t merely arise with naked singularities!

To avoid this (E1(w) -> oo) in obtaining what we call the inverse transform, one then performs (as shown below with arrows) an “analytical continuation” process which escapes the singularity and arrives at a rational and reasonable solution.




Plotting the graph on the axes:

 u(i)
 !
 !
 !          pole x
 !
 !
 !
 !----->-!----------!--->----->u(r)
              (x)   Res
Landau contour


This “Landau contour” – after the contour integral, wends its way around the singular pole (infinity) and going along the horizontal axis, then downwards (picking up what we call a “residue” (2 pi(i)) and then back up and further along to the right of real axis u(r).

Detailed explanation is beyond the scope of this response, since we are dealing with the calculus of residues here. Not everyone’s cup of tea! But useful to prove my point even if you don’t grok the math. Thus, there is "method to the madness": to show that “singularities” crop up in many physics contexts, and there are always mathematical methods to deal with them. In quantum mechanics they also pop up and we call the method of dealing with them, “renormalization”. In quantum loop theory it’s often known as “regularization”.

My dispute with Joshi has to do with his application of these quantum gravity loops without knowing much more about their (topological) properties than what he has provided in his arxiv paper. I’d like to personally see the morphisms behind what he has presented. And then see if they hold up when his equation(s) are analyzed in detailed computer models for different topologies!

As for black holes themselves there is no issue that they exist. They do! Once Karl Schwarzschild discovered the particular (Schwazschild) solution to Einstein’s general relativity equations, the template was set. (You may find the full mathematical expression on page 74 of ‘The Future of Theoretical Physics and Cosmology’ (ed. by Gibbons, Shellard and Rankin.) If you can get the book, take heed as Kip Thorne points out, that what the solution is saying is NOT that there is an "infinite gravity" but rather that the warp of the metric is so incredibly large the region defined by 'r' is "cut off" from the rest of the universe.

I also heartily recommend getting hold of Penrose’s book and reading the section on ‘The Weyl Curvature Hypothesis’ from p. 765.

The Schwarzschild solution template merely had to await the first physical confirmation that an entity of sufficiently small physical dimensions and correlated x-ray signal could meet it, and this was found with the discovery of Cygnus X-1.

I know this stuff may be hazy and appear obscure, but you didn’t exactly ask what could be called a “basic” question, and it isn’t always possible to break convoluted questions (especially to do with “infinity”) into simple set- piece answers.  However, if you need further clarification about any of the points, feel free to follow-up.