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View Full Version : Can black holes ever collide if they are infinitely small?

skintigh
2009-Mar-13, 08:46 PM
I had a thought the other day, and either I'm missing something or I deserve a Nobel Prize. I'm really hoping it's the latter, but I will hold off on including my mailing address for Stephen Hawking to send his medal to.

My thought:

Supposedly black holes are infinitely small.

Supposedly black holes have collided before.

If two objects are infinitely small and at a relative stand-still and are then attracted by gravity in a completely empty universe, they will be drawn exactly together and will collide. But as soon as you, for example, add a single extra atom to that universe somewhere that is not exactly even between them, the infinitely small black holes will each be dragged off course by just the tiniest and different amounts, they will miss the collision by just the tiniest amount and will orbit each other for eternity and never collide. Same goes for having one black hole start off with a velocity that is not exactly mirrored by the other.

Thus either black holes have never collided or they are not infinitely small.

Is my logic flawed? Is there some sort of "friction" I'm not taking into account that will make orbiting black holes collide? Am I way behind the times and scientists don't actually believe black holes are infinitely small anymore? Do we know if two have ever collided? Should I give up my Nobel dreams and stick to my day job?

loglo
2009-Mar-13, 08:56 PM
Thus either black holes have never collided or they are not infinitely small.

Its the latter. Event horizons have finite size, it is the singularity which is undefined in the centre which tends towards infinitely small. Once one black hole is within the others event horizon it can't escape so the collision intersection is not infinitely small after all.

Argos
2009-Mar-13, 08:57 PM
They may be infinitely small but their gravitational fields are not. It´s possible that they can be captured in their respective g fields, as in the case of supermassive BH´s at the center of two colliding galaxies.

nauthiz
2009-Mar-13, 09:15 PM
Isn't it also the case that the theory only describes the singularity compressing down to a mathematical point when the black hole is not spinning at all?

And, building on that, is it physically possible for a black hole with zero angular momentum to form? I know that we can describe them on paper, but it seems to me like any way you slice it, accreting matter or a very high-energy collision would always produce an end product that is spinning very, very fast.

grant hutchison
2009-Mar-14, 12:43 AM
If two objects are infinitely small and at a relative stand-still and are then attracted by gravity in a completely empty universe, they will be drawn exactly together and will collide. But as soon as you, for example, add a single extra atom to that universe somewhere that is not exactly even between them, the infinitely small black holes will each be dragged off course by just the tiniest and different amounts, they will miss the collision by just the tiniest amount and will orbit each other for eternity and never collide.The flaw is in the "orbit each other for eternity", which doesn't happen under general relativity.
There's no such thing as a close stable orbit around a black hole: the usual angular momentum "barrier" in the effective potential disappears, and things just fall through the event horizon and into the singularity if they get close enough. There is also the matter of gravitational waves, which will remove energy from a couple of black holes in close orbit, making them spiral towards each other and merge.

Grant Hutchison

Delvo
2009-Mar-14, 01:26 PM
...undefined... tends towards infinitely small...Notice the difference between this and "zero". Zero is a number, and undefinition is not. It's the short mathematical way to say "You can't do that, there's no such value, that equation can't answer this question." "Singularity" doesn't mean it actually goes to zero; it means it gets so small that the usual way we'd try to determine its size gets distorted into gibberish.

Jeff Root
2009-Mar-14, 03:15 PM
I think that "singularity" in this case means two things. One, it means
that the volume of the matter at the center of a black hole is decreasing
toward zero as it falls forever into a forever-deepening gravity well. Two,
the volume is constrained to be finite by the limitations of uncertainty.
Although the volume is very, very small, it can never be effectively smaller
than that allowed by the product of position times energy. However, that
limitation doesn't have any practical significance, since the matter is out
of view of the rest of the Universe anyhow due to its being inside the
event horizon it creates.

-- Jeff, in Minneapolis

loglo
2009-Mar-14, 08:36 PM
Notice the difference between this and "zero". Zero is a number, and undefinition is not. It's the short mathematical way to say "You can't do that, there's no such value, that equation can't answer this question." "Singularity" doesn't mean it actually goes to zero; it means it gets so small that the usual way we'd try to determine its size gets distorted into gibberish.

Yeah. I think the point/object at the centre of a black hole needs to be given a proper name to remove the singularity confusion. We need Wheeler back to come up with something pithy!

mugaliens
2009-Mar-14, 11:07 PM
"Planck mass is the mass for which the Schwarzschild radius is equal to the Compton length divided by pi. The radius of such a black hole would be roughly the Planck length." - Source (http://en.wikipedia.org/wiki/Planck_length#Physical_significance)

The rate at which BH's radiate away any percentage of their mass via Hawking radiation is inversely proportional to their size. When they get down to about a meter in size, they will radiate their entire mass away in energy in about 1 second.

I suspect the smallest possible black hole would have a dimension of around the Planck length, and a life of duration around that of Planck time (nearly instantaneously short).

Put all of this information together, and you wind up with the fact that two small BHs wouldn't be around long enough to do anything, much less collide with one another.

Jeff Root
2009-Mar-15, 01:49 AM
Mugs,

I think your figure of "a meter" is way, way too big. Maybe six or even
12 orders of magnitude too big. If the entire mass of the Earth were
squeezed into a black hole, the event horizon would have a diameter of

-- Jeff, in Minneapolis

mugaliens
2009-Mar-15, 01:15 PM
Mugs,

I think your figure of "a meter" is way, way too big. Maybe six or even
12 orders of magnitude too big. If the entire mass of the Earth were
squeezed into a black hole, the event horizon would have a diameter of

-- Jeff, in Minneapolis

You're absolutely right - I was quoting from memory, which was obviously way off - 22 orders of magnitude off (which is pretty far off...!)

From Wikipedia's Black Hole Evaporation (http://en.wikipedia.org/wiki/Hawking_radiation#Black_hole_evaporation):

...a 1-second-lived black hole has a mass of 2.28 × 105 kg, equivalent to an energy of 2.05 × 1022 J that could be released by 5 × 106 megatons of TNT. The initial power is 6.84 × 1021 W.

Given the above, a BH weighing about half a million pounds would evaporate in 1 second with a force of 100,000 Tsar Bomba events.

It's radius? Given by the Schwarzschild metric (http://en.wikipedia.org/wiki/Schwarzschild_black_hole#The_Schwarzschild_metric) , rs=2GM/c2, it's a mere 3.39x10-22m.

Pretty small indeed!

Jeff Root
2009-Mar-15, 02:18 PM
I was quoting from memory, which was obviously way off -
22 orders of magnitude off (which is pretty far off...!)
Wow!!! And weird. I was something like 22 orders of magnitude
off on some estimate I posted online several years ago. Probably
in sci.astro. Or it might have been in an e-mail. But 22 OoM
sounds awfully familiar. I think I wondered at the time how many
people have ever been *that* far wrong.

-- Jeff, in Minneapolis

skintigh
2009-Mar-16, 04:39 AM
loglo, argus:
I don't disagree that the event horizons will merge and it may even look like there is just one black hole, but if the cores are infinitely small they will never touch.

And I know infinitely small and zero are not the same thing, but a black hole of of either size would never be able to collide with another.

Personally, I don't believe anything in the physical universe can be infinitely-anything. Infinity is a human concept.

Anyway, I think the correct answer is that they are NOT infinitely small, they can only shrink to the Plank length. It would still be tough for them to collide, but it sounds like I just don't understand all this about angular momentum and gravity waves, and maybe some gamma radiation for good measure.

No Nobel for me :(

mugaliens
2009-Mar-16, 02:29 PM
Wow!!! And weird. I was something like 22 orders of magnitude
off on some estimate I posted online several years ago. Probably
in sci.astro. Or it might have been in an e-mail. But 22 OoM
sounds awfully familiar. I think I wondered at the time how many
people have ever been *that* far wrong.

-- Jeff, in Minneapolis

Well, in my case, it wasn't so much as being hugely wrong as it is dealing with dimensions about which I do not have a personal frame of reference. That and getting my wires crossed - I recalled a discussion I had elsewhere, wherein someone asked the mass of a 1 m diameter black hole. For some reason, I was recalling that 1 m dimension along with the 1 s time life of the other black hole, but not it's relatively miniscule mass of just half a million pounds.

gzhpcu
2009-Mar-16, 02:45 PM
Personally, I don't believe anything in the physical universe can be infinitely-anything. Infinity is a human concept.
Right. When we come up with a mathematical model of physics that delivers an infinite value for one of its equations in a special circumstance, then the model breaks down for this special case, and is no longer valid.

grant hutchison
2009-Mar-16, 04:04 PM
I don't disagree that the event horizons will merge and it may even look like there is just one black hole, but if the cores are infinitely small they will never touch.There's no reason two dimensionless points can't end up occupying exactly the same coordinates.
And general relativity predicts that if the two singularities haven't merged, their combined event horizon will radiate gravitational waves, removing energy from the system until perfect rotational symmetry is achieved; which occurs when the singularities have zero separation. Whether you consider that to be two dimensionless singularities right next to each other, or one merged singularity, seems like more of a semantic rather than a practical issue.

Grant Hutchison

chornedsnorkack
2009-Mar-16, 04:16 PM
The flaw is in the "orbit each other for eternity", which doesn't happen under general relativity.
There's no such thing as a close stable orbit around a black hole: the usual angular momentum "barrier" in the effective potential disappears, and things just fall through the event horizon and into the singularity if they get close enough. There is also the matter of gravitational waves, which will remove energy from a couple of black holes in close orbit, making them spiral towards each other and merge.

But nothing can get out of an event horizon. Not light, and not gravitational waves either. Therefore, a black hole that has fallen into the event horizon of another must stop shedding angular momentum by emission of gravitational waves.

If two Schwarzschild (zero angular momentum) point masses undergo off-center collision, how shall the point singularities start to spaghettify into the circle singularity of a Kerr black hole?

nauthiz
2009-Mar-16, 04:22 PM
But nothing can get out of an event horizon.

In general, matter can't (though I think Hawking radiation, if correct, is an exception to that rule). Gravitational waves aren't quite the same thing, though, since they're a gravitational phenomenon. A black hole's gravitational field has no problems crossing its event horizon.

grant hutchison
2009-Mar-16, 04:24 PM
But nothing can get out of an event horizon. Not light, and not gravitational waves either. Therefore, a black hole that has fallen into the event horizon of another must stop shedding angular momentum by emission of gravitational waves.See above: the asymmetrical event horizon generated by the separated singularities emits gravitational waves. The process is called "ringing down": you should be able to Google more detail using "black hole merger" and "ring-down".

Grant Hutchison

2009-Mar-19, 06:49 PM
This has been a fascinating discussion on the original question. It skipped one idea though. One of the traits of gravity is that it bends direction. Inside the event horizon, all directions lead to the singularity. Therefore, any movement within it is to that center. There can be no off-center motion. The two tiny black holes must collide.

This of course ignores the fact that that they cannot exist long enough to hit each other. Get into that discussion and you run up against an additional issue. Whos time are you talking about? Your time while watching the two black holes collide is different from the time of the singularities. In fact, at the event horizon an inbound traveler sees the outside observer experience all the time there will ever be in the universe. Inside the event horizon, but still distant from the singularity, that inbound traveler must see the outside observer experience more than all the time there is. Since I do not know what more than infinity means, I won't argue the case.

Then of course if the black hole evaporates, the inbound traveler would return to the universe before he fell in. I think I need a drink.

gzhpcu
2009-Mar-20, 03:11 PM
There's no reason two dimensionless points can't end up occupying exactly the same coordinates.
And general relativity predicts that if the two singularities haven't merged, their combined event horizon will radiate gravitational waves, removing energy from the system until perfect rotational symmetry is achieved; which occurs when the singularities have zero separation. Whether you consider that to be two dimensionless singularities right next to each other, or one merged singularity, seems like more of a semantic rather than a practical issue.

It is a mathematical issue, since most physicists do not believe singularities can exist. It is just the consequence of the breakdown of our mathematical model for gravity.
General relativity breaks down when there is some discontinuity in spacetime. Two important examples are

* The center of black holes and,
* The Big Bang.
It is unknown how to describe these spacelike singularities.

gzhpcu
2009-Mar-20, 03:37 PM
Here is an article on how weird it is to be in a black hole...
http://curious.astro.cornell.edu/question.php?number=652

In a region like the one here on Earth, you can only remember events that meet 2 criteria: (1) it has to have been in the past, and (2) it has to have happened at a distance no more than what light could have travelled since it happened. The second rule is just the familiar light speed limit. The first is called causality... Here comes the strange part. General relativity (that same theory supported by so many experiments and needed to make the GPS system work) predicts that, simply by compressing any piece of matter down enough to make a black hole, you create a region where this just isn't true. Inside the event horizon, time and space change places. Therefore the new restrictions go like this: in order for you to remember something, (1) it has to have happened farther from the center of the black hole than where you are now, and (2) if T is the time that it would take light to travel to you from the location of the event, then it happened either no more than T hours ago or T hours into the future...
First of all, note that restriction #1 prevents you from moving away from the center of the black hole, and therefore from going back across the event horizon. Also note that it says "farther", not "at least as far". This means that not only can't you move away from the center, you can't even stand still. Also we see that everyone inside the event horizon is a psychic. This happens because light can travel to you from events in the future, so you can quite literally see them. You can't see anything closer to the center than you are because light can't travel away from the center. If you look away from the center, though, you see two images of everything--one from T hours in the past and one from T hours in the future. For nearby objects, these two images will look just the same, since T will be very small due to the large speed of light. For faraway objects, though, they could be completely different..

Centaur
2009-Mar-20, 05:46 PM
Supposedly black holes are infinitely small.

The term infinitely refers to a huge size without bounds. The term infinitesimally (one divided by infinity) refers to a size of essentially zero.

mugaliens
2009-Mar-21, 11:02 PM
* The center of black holes and,
* The Big Bang.
It is unknown how to describe these spacelike singularities.

f(x)=1/x is a singularity. It's simply the point at which a mathematical object is undefined.

The term is misleading to the laymen, who often synonomize it with a point. Rather, a signularity is a discontinuity. The event horizon is a signularity, but not the same as the BH's gravitational singularity.

Again, the term singularity is not synonomous with a single point.

01101001
2009-Mar-21, 11:56 PM
The term is misleading to the laymen, who often synonomize it with a point..

In general, a singularity is a point at which an equation, surface, etc., blows up or becomes degenerate. Singularities are often also called singular points.

mugaliens
2009-Mar-22, 01:03 PM

Please have the fine folks at Mathworld use more than one variable in their equations... ;)

Q=1/x+1/y+1/z is undefined anywhere on any of three mutually orthoganal planes.

Q=1/(x2-2y+z) is undefined anywhere on a complex surface that permeats a three-dimensional area of space.

Many functions are undefined throughout multidimentional continuums.

gzhpcu
2009-Mar-22, 04:11 PM
f(x)=1/x is a singularity. It's simply the point at which a mathematical object is undefined.

The term is misleading to the laymen, who often synonomize it with a point. Rather, a signularity is a discontinuity. The event horizon is a signularity, but not the same as the BH's gravitational singularity.

Again, the term singularity is not synonomous with a single point.
I did not mean geometrically, I meant our physics breaks down and can not offer an accurate description, since an infinite value delivered by an equation means a division by zero.

mugaliens
2009-Mar-23, 01:09 AM
...since an infinite value delivered by an equation means a division by zero.

Well, that's the thing about it being "undefined." Infinity over any finite number is as undefined as any finite number over zero, yet the two remain different mathematical concepts.

Noclevername
2009-Mar-25, 01:21 AM
I thought black holes can't be infinitely small, though. Beyond a certain point their internal compression pressure outbalances the gravity holding them together, and BWOOMPH! there it went, all over the subatomic landscape. Am I nuts for thinking this? (please confine your speculations about my sanity to that relevant to the topic.)

Jeff Root
2009-Mar-25, 03:32 AM
Noclevername,

The presumed infinitesimal size applies to *all* black holes-- but it only
applies to the actual matter which is the cause of the black hole. The
singularity at the center of a supermassive BH, a solar-mass BH, a BH
with the mass of Earth, and a BH the mass of an elephant would all be
infinitesimal, but the sizes of their event horizons would vary enormously.

Event horizons can't collide. But they *can* merge.

-- Jeff, in Minneapolis