# Are black holes explained by complex analysis?

by Hercuflea
Tags: analysis, black, complex, explained, holes
 P: 347 Is the center of a black hole essentially a pole, or a "point at infinity"? I always thought about this in my complex analysis class because one variable complex functions are 4 dimensional, which could translate into space-time. Black holes have to have infinite density in their center, too, right? And the extremely strong gravitational field would basically make time go to zero?
P: 734
 Quote by Hercuflea Is the center of a black hole essentially a pole, or a "point at infinity"? I always thought about this in my complex analysis class because one variable complex functions are 4 dimensional, which could translate into space-time. Black holes have to have infinite density in their center, too, right? And the extremely strong gravitational field would basically make time go to zero?
I'm not sure what you mean by "pole" but most recent accepted theory is that the "center" of a black hole is not a zero-point singularity. The "center" must have a finite size of at least a Planck length: (1.61624 × 10-35 m).

Infinities simply do not work in math, and a zero-size point of any mass would require infinite mass, which we know is not the case since black hole masses can be measured and are definitely not infinite.

The "old" zero-point/infinite mass "singularity" definition has been dropped as far as I have read recently.
 P: 347 So does time practically stop at the center of a black hole because of the immense gravitational field?
P: 734
Are black holes explained by complex analysis?

 Quote by Hercuflea So does time practically stop at the center of a black hole because of the immense gravitational field?
To any "outside observer", time appears to stop at the Event Horizon (EH) which is an "effect" at: 2Gm/c^2 or at the poles (only) of an "Ergosphere" where the gravity curves space-time back into a "loop" returning to the interior of the EH boundry. (That is definitely a non-technical description)...

Inside the BH, time would seem to continue for any observer who might survive. Most PF posters are aware (I think) that an observer (human?) could survive inside the EH if the BH was large enough so that the gravitational pull at the EH was not strong enough to cause the "spaghetti effect". Near a small BH, the "spaghetti effect" would tear any matter apart, even individual atoms, before the EH was reached.
P: 534
 Quote by Labguy To any "outside observer", time appears to stop at the Event Horizon (EH) which is an "effect" at: 2Gm/c^2 or at the poles (only) of an "Ergosphere" where the gravity curves space-time back into a "loop" returning to the interior of the EH boundry. (That is definitely a non-technical description)... Inside the BH, time would seem to continue for any observer who might survive. Most PF posters are aware (I think) that an observer (human?) could survive inside the EH if the BH was large enough so that the gravitational pull at the EH was not strong enough to cause the "spaghetti effect". Near a small BH, the "spaghetti effect" would tear any matter apart, even individual atoms, before the EH was reached.
Yes, but the OP is asking about the center of the black hole, which he presumably meant the singularity, not the horizon.
P: 534
 Quote by Hercuflea So does time practically stop at the center of a black hole because of the immense gravitational field?
Mathematically you cannot extend any curve in spacetime pass the black hole singularity, so practically time "ends".
 P: 534 Having said that, of course we don't really know what happens at (and very close to) the singularity since general relativity should get corrections in that regime.
PF Gold
P: 6,514
 Quote by Labguy Infinities simply do not work in math
Huh? I don't get that. Infinities work perfectly well in math. It's in PHYSICS that they don't work.

 , and a zero-size point of any mass would require infinite mass,

I don't get that either. It does imply infinite DENSITY, but how do you get that it implies infinite mass?

 which we know is not the case since black hole masses can be measured and are definitely not infinite.
agreed
P: 734
 Quote by yenchin Yes, but the OP is asking about the center of the black hole, which he presumably meant the singularity, not the horizon.
Agreed, but what I meant to explain is that a "signularity" as defined 10-15 years ago doesn't exist. The "center" of a BH has a finite size with a very large density but not an infinite density.

The gravity of the large mass still exists and the measure of the mass determines the EH boundry, but the mass (or density) is not infinite.
P: 734
 Quote by phinds I don't get that either. It does imply infinite DENSITY, but how do you get that it implies infinite mass?
Take any sized finite volumn you can think of; maybe a sphere with a diameter of 1 Planck length, or a basketball. How can you fill that volumn to infinite density without using an infinite amount of mass? I can't think of a way to accomplish that.(?) That's how it implies infinite mass.

p=m/V.
P: 5,632
 ...Black holes have to have infinite density in their center, too, right? And the extremely strong gravitational field would basically make time go to zero?
No. Nobody knows either of those.

GR and QM breakdown in the vicinity of the singularity....In any case, as you approach such a 'point', quantum foam seems to blur space, time, mass etc....We just do not yet have a theory that works in those conditions.
PF Gold
P: 6,514
 Quote by Labguy Take any sized finite volumn you can think of; maybe a sphere with a diameter of 1 Planck length, or a basketball. How can you fill that volumn to infinite density without using an infinite amount of mass? I can't think of a way to accomplish that.(?) That's how it implies infinite mass. p=m/V.
Exactly. P=m/V, so finite mass and zero volume gives infinite density. Why do you think infinite mass is required?

This seems to tie in with your being wrong about infinities not working in math, which they do. Do you not get that 3/0 = infinity?
P: 734
 Quote by phinds Exactly. P=m/V, so finite mass and zero volume gives infinite density. Why do you think infinite mass is required?
But, my point is that there is nothing with zero volumn! If zero-volumn is considered, then whatever it may be called just doesn't exist. This is "General Astronomy" so go to some other more advanced forums, or many recent (last 5 years) papers published by "well-known" astrophysicists and see what they have to say about "zero-point" singularities.

 Quote by phinds This seems to tie in with your being wrong about infinities not working in math, which they do. Do you not get that 3/0 = infinity?
Yes, I get that one. How about ∞/∞? Is that = to 1? Is that a prime number? How about ∞-1/∞? Same number? Mathematicians can can play around with many "infinity tricks" but as someone once said, Infinities might work in math but not in physics.

Black Holes are physical entities with only 4 "detectible" properties, as opposed to Wheeler's original 3 "no-hair" properties. I have (long ago) posted at least 3-4 separate posts regarding the four properties, yet still today books, and PF posts, ignore that. And the gravity of a BH is proportional to its finite mass. If we have to use any infinities to describe any properties of a BH, then all that can be said is that "we don't know". (I saw that on TV)
P: 1,000
 Quote by Labguy Take any sized finite volumn you can think of; maybe a sphere with a diameter of 1 Planck length, or a basketball. How can you fill that volumn to infinite density without using an infinite amount of mass? I can't think of a way to accomplish that.(?) That's how it implies infinite mass.
That argument is erroneous.

The way you deduce infinite density is by assuming zero volume and a finite mass. If you accept the conclusion of infinite density but then reject the assumption of zero volume in favor of a different assumption of non-zero volume, you are committing an error.

You can't have your cake and eat it too.

A real mathematician would probably say that a finite mass divided by a zero volume is undefined rather than infinite. Or he would admit that he's not working in the field of real numbers but rather in some compactification thereof. A physicist is not bound by those rules. But he is bound to not call the volume zero in one breath and non-zero in the next.
P: 734
 Quote by jbriggs444 That argument is erroneous. The way you deduce infinite density is by assuming zero volume and a finite mass. If you accept the conclusion of infinite density but then reject the assumption of zero volume in favor of a different assumption of non-zero volume, you are committing an error.
You didn't get my meaning. I will never "accept the conclusion of infinite density" OR accept any assumption of zero volume.

 Quote by jbriggs444 A real mathematician would probably say that a finite mass divided by a zero volume is undefined rather than infinite. Or he would admit that he's not working in the field of real numbers but rather in some compactification thereof. A physicist is not bound by those rules. But he is bound to not call the volume zero in one breath and non-zero in the next.
I never called any "volume" zero; it is exactly that which I have been posting about.(Against?)

I think that we generally agree but have a "semantics" problem going on here.
P: 1,000
 Quote by Labguy You didn't get my meaning. I will never "accept the conclusion of infinite density" OR accept any assumption of zero volume. I never called any "volume" zero; it is exactly that which I have been posting about.(Against?)
Post 2 by you. "Zero-size point".

That you accepted it for purposes of a reductio ad absurdum is irrelevant. You are not allowed to have it both ways in the same argument.