Black Holes As Actual Infinities

[Agent Smith]

1. There was little "feigned" concern about CERN creating mini-black holes. It was a possibility, though a small one. Such black holes could be dangerous if they do not dissolve quickly enough.

2. Your comment about Laplace and black holes in the 1800s is full of holes. In 1800 stars still burned on coal and no one knew that c was the speed limit. Yes, I have heard of black holes mentioned back then, but they were not the objects that we now call black holes.

3. Two components of the Schwarzschild metric depend on [imath]\dfrac{GM}{R}[/imath]. When R goes to 0, the metric becomes undefined, ie. length and time scales become unmeasurable.

4. We have to switch to the Quantum scale long before we get to the Planck scale. Anything smaller than, say, [imath]10^{-8}[/imath] m starts showing suggestions of Quantum effects.

5. Non-sensical results: time and distance scales no longer existing. Everything is now composed of the most elementary particles, which we may not even understand. The Physics we understand about our Universe ends at the event horizon.

-Dan

(my previous post)

Gracias for keeping it simple.

1. Black holes were a possibility and still they built CERN? No safety regulations apply when stakes are that high, eh? Do you think there's a law in Europe that could find CERN liable for violating safety regulations?

2. I got that from Wikipedia. Didn't know that there was more to the story than the few lines I read. All I can say is that the germ of the idea was there; the difference is in the details, oui?

3. So division by \(\displaystyle 0\). Gotcha!

4. Here's what trips me up. It seems as though "what happens" depends on the scale of space and time(?). If I shrink the sun to the size of an atom then it becomes a quantum object, despite its mass still being the same (suppose), like as if the story of a movie depends on the size of the theater.

5. How about "adding" a fudge factor to the equations so they don't devolve into nonsense. Einstein did something similar with [imath]\lambda[/imath].

 

[Agent Smith]

[imath]\frac{\text{Lena}}x \to \frac{\text{Lena}}{x + 1}[/imath]

 

[Emma343]

A little philosophy to get this party going ...
Aristotle (Greek philosopher), way back in the BC era, claimed that only potential infinities exist and that there are no actual infinities. This issue isn't resolved to date, because even though it seems so tempting to say that the set of naturals is infinite, we assert the existence of at least 1 infinity (the naturals) as an axiom and not a theorem.

Then someone I met mentioned black holes, those that are collapsed stars or primordial ones that were formed during The Big Bang itself. My questions:

1. Is a black hole an actual infinity?
2. If it is how may we enrich our mathematics?

This is a fascinating mix of philosophy, mathematics, and physics. Let's break it down:


1. Is a black hole an actual infinity?
In physics, particularly in general relativity, the singularity at the center of a black hole is described as a point where density and spacetime curvature become infinite. However, most physicists agree that this “infinity” is not an actual infinity in the mathematical sense—it signals a breakdown in our current models rather than the existence of a truly infinite physical quantity. In this sense, a black hole is not an actual infinity, but rather a region where our understanding (and equations) cease to be valid.


From Aristotle’s view, and also in many areas of mathematics and physics today, infinities are often treated as conceptual tools rather than real, measurable entities. We use them in equations (like limits or unbounded integrals), but whether they exist in the physical universe is still up for debate.


2. If it is, how may we enrich our mathematics?
Even if black holes aren’t actual infinities, they do push the boundaries of mathematics. For example:

• Singularity theory, differential geometry, and topology have all been developed further due to the study of black holes.
• The search for a quantum theory of gravity, which would reconcile general relativity and quantum mechanics, is driving mathematical innovation. This might one day “resolve” the singularity and give us a finite, consistent picture.