Well, they’re black, and they’re like bottomless holes. What would you call them?
-Me, when a friend asked me why they’re named what they are
Ah, black holes. The ultimate shiver-inducer of the cosmos, out-jawing sharks, out-ooking spiders, out-scaring… um, something scary. But we’re fascinated by ‘em, have no doubt — even if we don’t understand a whole lot about them.
 |
But then, that’s why I’m here. Allow me to be your tour guide to infinity. Or the inverse of it, I suppose. Since it’s Halloween this seems appropriate… and my book Death from the Skies! just came out, and there’s lots of ways a black hole can destroy the Earth. Mwuhahahaha.
So below I present ten facts about black holes — the third in my series of Ten Things You Don’t Know (the first was on the Milky Way; the second about the Earth). Regular readers will know a few of these since I’ve talked about them before, but I’m hoping you don’t know all of these. And if you do, then feel free to leave a comment preening about your superior intellect. Mind you, this list is nowhere near complete: I could have picked probably 50 things that are weird about black holes. But I like these.
1) It’s not their mass, it’s their size that makes them so strong.
OK, first, a really quick primer on black holes. Bear with me!
The most common way for a black hole to form is in the core of a massive star. The core runs out of fuel, and collapses. This sets off a shockwave, blowing up outer layers of the star, causing a supernova. So the star’s heart collapses while the rest of it explodes outwards (this is the Cliff’s notes version; for more details on the process — which is way cool, so you should read it — check out my description of it).
As the core collapses, its gravity increases. At some point, if the core is massive enough (about 3 times the mass of the Sun), the gravity gets so strong that right at the surface of the collapsing core the escape velocity increases to the speed of light. That means that nothing can escape the gravity of this object, not even light. So it’s black. And since nothing can escape, well, read the quotation at the top of the page.
The region around the black hole itself where the escape velocity equals the speed of light is called the event horizon. Any event that happens inside it is forever invisible.
OK, so now you know what one is, and how they form. Now, I could explain why they have such strong gravity, but you know what? I’d rather let this guy do it. I hear he’s good.
So there you go. Sure, the mass is important, but sometimes it’s the little things that count.
2) They’re not infinitely small.
So OK, they’re small, but how small are they?
I was writing about black holes in my previous job, and we got in a fun discussion over just what we meant by black hole: did we mean the object itself that collapses down to a mathematical point, or the event horizon surrounding it? I said the event horizon, but my boss said it was the object. I decided she had a point (HAHAHAHAHA! A "point"! Man, I kill me), and made sure that when I wrote about the event horizon versus the black hole itself I was making myself clear.
Like I said above, to the collapsing core, its clock keeps ticking, so it sees itself collapsing all the way down to a point, even if the event horizon has some finite size.
What happens to the core? The actual mass that collapsed?
Out here, we’ll never know for sure. We can’t see in, and it sure enough isn’t gonna send any info out. But our math in these situations is pretty good, and we can at least apply them to the collapsing core, even when it’s smaller than the event horizon.
It will continue to collapse, and the gravity increases. Smaller, smaller… and when I was a kid I always read that it collapses all the way down to a geometric dot, an object with no dimensions at all. That really bugged me, as you can imagine… as well it should. Because it’s wrong.
At some point, the collapsing core will be smaller than an atom, smaller than a nucleus, smaller than an electron. It’ll eventually reach a size called the Planck Length, a unit so small that quantum mechanics rules it with an iron fist. A Planck Length is a kind of quantum size limit: if an object gets smaller than this, we literally cannot know much about it with any certainty. The actual physics is complicated, but pretty much when the collapsing core hits this size, even if we could somehow pierce the event horizon, we couldn’t measure its real size. In fact, the term "real size" doesn’t really mean anything at this kind of scale. If the Universe itself prevents you from measuring it, you might as well say the term has no meaning.
And how small is a Planck Length? Teeny tiny: about 10-35 meters. That’s one one-hundred quintillionth the size of a proton.
So if someone says a black hole has zero size, you can be all geeky and technical and say, not really, but meh. Close enough.
3) They’re spheres. And they’re definitely not funnel shaped.
The gravity you feel from an object depends on two things: the object’s mass, and your distance from that object. This means that anyone at a given distance from a massive object — say, a million kilometers — would feel the same force of gravity from it. That distance defines a sphere around an object: anyone on that sphere’s surface would feel the same gravity from the object at the center.
The size of an event horizon of a black hole depends on the gravity, so really the event horizon is a sphere surrounding the black hole. From the outside, if you could figure out how to see the event horizon in the first place, it would look like a pitch black sphere.
Some people think of black holes as being circles, or worse, funnel-shaped. The funnel thing is a misconception from people trying to explain gravity as a bending in space, and they simplify things by collapsing 3D space into 2D; they say the space is like a bed sheet, and objects with mass bend space the same way that a massive object (a bowling ball, say) will warp a bed sheet. But space is not 2D, it’s 3D (even 4D if you include time) and so this explanation can confuse people about the actual shape of a black hole event horizon.
I’ve had kids ask me what happens if you approach a black hole from underneath! They sometimes don’t get that black holes are spheres, and there is no underneath. I blame the funnel story. Sadly, it’s the best analogy I’ve seen, so we’re stuck with it. Use it with care.
4) Black holes spin!
It’s kind of an odd thought, but black holes can spin. Stars rotate, and when the core collapses the rotation speeds way, way up (the usual analogy is that of an ice skater who brings in his arms, increasing his rotation rate). As the core of the star gets smaller it rotates more rapidly. If it doesn’t quite have enough mass to become a black hole, the matter gets squeezed together to form a neutron star, a ball of neutrons a few kilometers across. We have detected hundreds of these objects, and they tend to spin very rapidly, sometimes hundreds of times a second!
The same is true for a black hole. Even as the matter shrinks down smaller than the event horizon and is lost to the outside Universe forever, the matter is still spinning. It’s not entirely clear what this means if you’re trying to calculate what happens to the matter once it’s inside the event horizon. Does centrifugal force keep it from collapsing all the way down to the Planck length? The math is fiendish, but do-able, and implies that matter falling in will hit matter inside the event horizon trying to fall further but unable to due to rotation, This causes a massive pile up and some pretty spectacular fireworks… that we’ll never see, because its on the other side of infinity. Bummer.
5) Near a black hole, things get weird
The spin of the black hole throws a monkey in the wrench of the event horizon. Black holes distort the fabric of space itself, and if they spin that distortion itself gets distorted. Space can get wrapped around a black hole — kind of like the fabric of a sheet getting caught up in a rotating drill bit.
This creates a region of space outside the event horizon called the ergosphere. It’s an oblate spheroid, a flattened ball shape, and if you’re outside the event horizon but inside the ergosphere, you’ll find you can’t sit still. Literally. Space is being dragged past you, and carries you along with it. You can easily move in the direction of the rotation of the black hole, but if you try to hover, you can’t. In fact, inside the ergosphere space is moving faster than light! Matter cannot move that fast, but it turns out, according to Einstein, space itself can. So if you want to hover over a black hole, you’d have to move faster than light in the direction opposite the spin. You can’t do that, so you have to move with the spin, fly away, or fall in. Those are your choices.
I suggest flying away. Fast. Because…
6) Approaching a black hole can kill you in fun ways. And by fun, I mean gruesome, horrifying, and really really ookie.
Sure, if you get too close, plop! You fall in. But even if you keep your distance you’re still in trouble…
 |
Gravity depends on distance. The farther you are from an object, the weaker its gravity. So if you have a long object near a massive one, the long object will feel a stronger gravitational force on the near end versus a weaker force on the far end! This change in gravity over distance is called the tidal force (which is a bit of a misnomer, it’s not really a force, it’s a differential force, and yes, it’s related to why we have ocean tides on Earth from the Moon).
The thing is, black holes can be small — a BH with a mass of about three times the Sun has an event horizon just a few kilometers across — and that means you can get close to them. And that in turn means that the tidal force you feel from one can get distressingly big.
|
| Praying to this guy won’t help. |
Let’s say you fall feet first into a stellar-mass BH. It turns out that as you approach, the difference in gravity between your head and your feet can get huge. HUGE. The force can be so strong that your feet get yanked away from your head with hundreds of millions of times the force of Earth’s gravity. You’d be stretched into a long, thin strand and then shredded.
Astronomers call this spaghettification. Ewwww.
So getting near a black hole is dangerous even if you don’t fall in. Evidently, there really is a tide in the affairs of men.
7) Black holes aren’t always dark
The thing is, black holes can kill from a long way off.
|
| Disk of DOOOOOM! Image credit: NASA/CXC |
Matter falling into a black hole would rarely if ever just fall straight in and disappear. If it has a little bit of sideways motion it’ll go around the black hole. As more matter falls in, all this junk can pile up around the hole. Because of the way rotating objects behave, this matter will create a disk of material whirling madly around the hole, and because the gravity of the hole changes so rapidly with distance, matter close in will be orbiting much faster than stuff farther out. This matter literally rubs together, generating heat through friction. This stuff can get really hot, like millions of degrees hot. Matter that hot glows with intense brightness… which means that near the black hole, this matter can be seriously luminous.
Worse, magnetic and other forces can focus two beams of energy that go plowing out of the poles of the disk. The beams start just outside the black hole, but can be seen for millions or even billions of light years distant.
They’re bright.
In fact, black holes that are eating matter in this way can glow so brightly that they become the brightest continuously-emitting objects in the Universe! We call these active black holes.
And as if black holes aren’t dangerous enough, the matter gets so hot right before it makes the final plunge that it can furiously emit X-rays, high-energy forms of light (and the beams can emit even higher energy light than that). So even if you park your spaceship well outside the event horizon of a black hole, if something else falls in and gets shredded, you get rewarded by being fried by the equivalent of a gazillion dental exams.
I may have mentioned this: black holes are dangerous. Best to stay away from them.
8) Black holes aren’t always dangerous.
 |
| I’m right there with you, dude. |
Having said that, let me ask you a question: if I were to take the Sun and replace it with Folgers crystals a black hole of the exact same mass, what would happen? Would the Earth fall in, be flung away, or just orbit like it always does?
Most people think the Earth would fall in, sucked inexorably down by the black hole’s powerful gravity. But remember, the gravity you feel from an object depends on the mass of the object and your distance from it. I said the black hole has the same mass as the Sun, remember? And the Earth’s distance hasn’t changed. So the gravity we’d feel from here, 150 million kilometers away, would be exactly the same! So the Earth would orbit the solar black hole just as nicely as it orbits the Sun now.
Of course, we’d freeze to death. You can’t have everything.
9) Black holes can get big.
Q: What happens if two stellar-mass black holes collide?
A: You get one bigger black hole.
You can extrapolate from there. Black holes can eat other objects, including other black holes, so they can grow. We think that early on in the Universe, when galaxies were just forming, matter collecting in the center of the nascent galaxy can collapse to form a very massive black hole. As more matter falls in, the hole greedily consumes it, and grows. Eventually you get a supermassive black hole, one with millions or even billions of times the mass of the Sun.
However, remember that as matter falls in it can get hot. It can be so hot that the pressure from light itself can blow off material that’s farther out, a bit like the solar wind but on a much grander scale. The strength of the wind depends on many things, including the mass of the black hole; the heftier the hole, the windier the, uh, wind. This wind prevents more matter from falling in, so it acts like a cutoff valve for the ever-increasingly girthy hole.
Not only that, but over time the gas and dust around the black hole (well, pretty far out, but still near the center of the galaxy) gets turned into stars. Gas can fall into a black hole more easily than stars (if gas clouds collide head-on their motion relative to the black hole can stop, allowing them to fall in; stars are too small and too far apart for this to happen). So eventually the black hole stops consuming matter because nothing more is falling into it. It stops growing, the galaxy becomes stable, and everyone is happy.
|
| Don’t panic! OK, maybe a little. |
In fact, when we look into the Universe today, we see that pretty much every large galaxy has a supermassive black hole in its heart. Even the Milky Way has a black hole at its core with a mass of four millions times that of the Sun. Before you start running around in circles and screaming, remember this: 1) it’s a long way off, 26,000 light years (260 quadrillion kilometers), 2) its mass is still very small compared to the 200 billion solar masses of our galaxy, and therefore 3) it can’t really harm us. Unless it starts actively feeding. Which it isn’t. But it might start sometime, if something falls into it. Though we don’t know of anything that can fall into it soon. But we might miss cold gas.
Hmmm.
Anyway, remember this as well: even though black holes can cause death and destruction on a major scale, they also help galaxies themselves form! So we owe our existence to them.
10) Black holes can be low density.
Of all the weirdnesses about black holes, this one is the weirdest to me.
As you might expect, the event horizon of a black hole gets bigger as the mass gets bigger. That’s because if you add mass, the gravity gets stronger, which means the event horizon will grow.
If you do the math carefully, you find that the event horizon grows linearly with the mass. In other words, if you double the black hole’s mass, the event horizon radius doubles as well.
That’s weird! Why?
The volume of a sphere depends on the cube of the radius (think way back to high school: volume = 4/3 x π x radius3). Double the radius, and the volume goes up by 2 x 2 x 2 = 8 times. Make the radius of a sphere 10 times bigger and the volume goes up by a factor of 10 x 10 x 10 = 1000.
So volume goes up really quickly as you increase the size of a sphere.
Now imagine you have two spheres of clay that are the same size. Lump them together. Is the resulting sphere twice as big?
No! You’ve doubled the mass, but the radius only increases a little bit. Because volume goes as radius cubed, to double the radius of your final clay ball, you’d need to lump together eight of them.
But that’s different than a black hole. Double the mass, double the size of the event horizon. That has an odd implication…
Density is how much mass is packed into a given volume. Keep the size the same and add mass, and the density goes up. Increase the volume, but keep the mass the same, and the density goes down. Got it?
So now let’s look at the average density of matter inside the event horizon of the black hole. If I take two identical black holes and collide them, the event horizon size doubles, and the mass doubles too. But volume has gone up by eight times! So the density actually decreases, and is 1/4 what I started with (twice the mass and eight times the volume gives you 1/4 the density). Keep doing that, and the density decreases.
A regular black hole — that is, one with three times the Sun’s mass — with have an event horizon radius of about 9 km. That means it has a huge density, about two quadrillion grams per cubic cm (2 x 1015). But double the mass, and the density drops by a factor of four. Put in 10 times the mass and the density drops by a factor of 100. A billion solar mass black hole (big, but we see them this big in galaxy centers) would drop that density by a factor of 1 x 1018. That would give it a density of roughly 1/1000 of a gram per cc… and that’s the density of air!
A billion solar mass black hole would have an event horizon 3 billion km in radius — roughly the distance of Neptune to the Sun.
See where I’m going here? If you were to rope off the solar system out past Neptune, enclose it in a giant sphere, and fill it with air, it would be a black hole!
That, to me, is by far the oddest thing about black holes. Sure, they warp space, distort time, play with our sense of what’s real and isn’t… but when they touch on the everyday and screw with that, well, that’s what gets me.
I first thought of this at a black hole conference at Stanford a few years back. I was walking with noted black hole expert Roger Blandford when it hit me. I did a quick mental calculation to make sure I had the numbers right, and related to Roger that a solar system full of air would be a black hole. He thought about it for a moment and said, "Yes, that sounds about right."
And that, me droogs, was one of the coolest moments of my hole life. But thinking about it still makes my brain hurt.
Conclusion:
Well, what can I say? Black holes are weird.
As it so happens, there was a lot more that could be said about them, of course. What about wormholes? What about how they form? what about Hawking radiation? Can black holes totally evaporate?
You can find answers to these and other questions elsewhere on the web (and even on this very blog); I couldn’t cover everything in just ten sections! But I’ll note (shocker) that chapter 5 of my book Death from the Skies! talks in detail about how they form, and what they can do if you get too close to them. Later chapters also talk about the black hole in the core of the Milky Way, and what will happen to black holes a long time from now… literally, 1060, 1070, even a googol years from now.
But even then, that’s not the scariest thing about black holes. I almost didn’t put this in the post, it’s so over the top mind-numbingly horrifying. But I’m a scientist, and we’re skeptics here, so we can take it. So I present to you, the worst thing about black holes of all:
 |
| "If there’s any justice at all, the black hole will be your grave!" |
October 30th, 2008 at 8:54 pm
Phil, I’m disappointed. You took all day to write this, and you neglected the obvious “they suck” joke?
Tsk, tsk.
(Great article, btw. I’m about 2/3 of the way through DFTS and enjoying every minute of it.)
October 30th, 2008 at 9:04 pm
Hey, if the FSM’s noodly appendage doesnt save you, it’s because you are supposed to die that way.
And… Hi, Teddi!
October 30th, 2008 at 9:06 pm
My first astronomy professor told me to remember that black holes don’t suck, physics does.
October 30th, 2008 at 9:09 pm
I knew that.
Some of these can be added to the list of weird things in general. Like if you thrust to go forward in orbit, you slow down (your orbit gets higher, which is a slower orbit). And when a star runs out of fuel, it gets hotter (due to gravitational collapse).
Is this a more modern graphic for the Milky Way? (I need a new T-Shirt).
http://www.spitzer.caltech.edu/Media/releases/ssc2008-10/release.shtml
October 30th, 2008 at 9:16 pm
I actually love that film for some inexplicable reason - it’s one of my guiltiest pleasures…
October 30th, 2008 at 9:21 pm
I love the notion that we could all get sucked into a black hole and have out individual atoms mingling around with one another like socks and underwear in a clothes dryer.
So, you know, we’ll meet someday, Phil. I hope you’ll sign my book.
October 30th, 2008 at 9:23 pm
Any information about string theory and BHs? That we can understand?
October 30th, 2008 at 9:25 pm
The low density black hole thing blows my mind too! In intro to astrophysics last year I did a quick calculation and seemed to find that a black hole with the mass of the observable universe would be smaller than the observable universe…
October 30th, 2008 at 9:28 pm
Is it really accurate to consider the density of a black hole to be the total mass within the event horizon, since all of the matter exists in the singularity (save that which is in the process of getting sucked between the event horizon and the singularity). It’s a bit like, say, considering the density of the earth and the moon to be mass(earth) + mass(moon) / volume(sphere of 300,000 km radius). Except that the event horizon is perhaps an even more arbitrary distance than the orbit of the moon!
October 30th, 2008 at 9:29 pm
Point #10 was referred to in Heechee novels - where (IIRC) it was postulated the aliens jumped into a mega-large BH to escape the heat death of the universe. I have not checked the math ( hell, I probably can’t DO the math), but the gravity gradient might be slight enoutgh to avoid spaghettification as you entered….
October 30th, 2008 at 9:33 pm
Point 11…Your bank account. I KID!!!!
cool info.
October 30th, 2008 at 9:33 pm
The other wild thing about a solar system full of air is that it would have the mass of A BILLION SUNS! That’s 0.5% the mass of the GALAXY. We really do live in a nearly perfect vacuum.
October 30th, 2008 at 9:48 pm
With #1:
I sat through a talk by Lawrence Krauss recently at an undergrad physics conference titled “Our Miserable Future.” One point he brought up (I’m hoping I get this right), is that a black hole of the mass of this universe would oddly enough have the density and size of this universe! This means we could essentially be on the *inside* of the largest observable black hole. And that’s what really blows my mind.
October 30th, 2008 at 9:55 pm
Well, I guess you managed to put one thing I didn’t know about black holes, which was the last point. Does that still allow me to have bragging rights?
October 30th, 2008 at 10:00 pm
I knew a lot of this from your book of course, but a thought just occurred to me: So wait…If black holes rotate, and are spherical, then they have poles? I mean if it shrinks, to that itty-bitty size, then the spin must be HUGE, right? Interesting to think about.
As for never seeing past the event horizon, you may have heard- we’ve replicated it in the lab with lasers. I’m always hopeful we can pull off the mathematically improbable, even if there’s little to support that hope.
October 30th, 2008 at 10:06 pm
Black holes always leave me with more questions, the more answers I get, so maybe I’m too dense:
How would we know if we (right now on Earth) were already within the event horizon of some ultra-hyper-super-massive, yet extremely non-dense “black hole?”
Does the empty space within chemical bonds and the huge empty space between sub-atomic particles decrease as mass is ‘falling into’ a black hole?
October 30th, 2008 at 10:23 pm
In Michio Kaku’s Parallel Worlds, he makes the point that a black hole with the mass of the universe . . . has about the same density as the universe.
I had to put the book down.
October 30th, 2008 at 10:27 pm
Phil, you and I are roughly the same age - hence the loud laugh that came out of my mouth when I read the Folgers Crystals joke!
J. D.
October 30th, 2008 at 10:42 pm
My brain won’t stop going over the last point!
My worst thing about black holes: insomnia!!
October 30th, 2008 at 10:45 pm
I knew 1-9, and the basic behind 10, but I never thought of 10. This is cool. So the Universe is a black hole?
October 30th, 2008 at 10:47 pm
Point 10 leads to another interesting fact about black hole feeding habits.
As “Pete” points out, if you fall into a really big black hole, you don’t go through spaghettification (at least, not until you’re well inside the event horizon). But think about what this means for “bright” black holes.
For a relatively small black hole (less than a billion solar masses) you can’t really get a star into it without the tidal affects tearing it apart into a disk. It is the energy loss in these accretion disks that are slowly spiraling to the black hole that makes quasars so bright.
However, a big enough black hole doesn’t have to do that. It can start swallowing stars whole (hole?). As a result, a really big black hole may not be a luminous as a smaller one, just because it’s not ripping apart passing stars and making nice bright disks and jets out of them.
October 30th, 2008 at 10:51 pm
You mentioned the Plank length in the second point but referenced the Fermi length in the fourth point. I only hazily remember the Plank length, but haven’t heard of the Fermi length.
Could you clarify
October 30th, 2008 at 10:52 pm
Rich, nice catch. That’s a typo. I was thinking about the Fermi Paradox earlier today, and must have written that instead of Planck! I fixed it, thanks.
October 30th, 2008 at 10:55 pm
I came close to talking about SMBHs not having big tides, but ran out of topics. This was a tough one to write, and a tough one to narrow down to these topics. I wanted to cover stuff not necessarily in the book, or easy to find on the web.
To make it more fun, my calculator batteries died, so I was using one I downloaded on my Mac, which freaked out over numbers with big exponents. I was taking inverses and getting 0. Grrrr.
October 30th, 2008 at 11:05 pm
But Phil, as a loved one of yours pointed out to me once upon a time, air does not have a constant density so you cannot do that calculation
October 30th, 2008 at 11:08 pm
This is Science. This is Interesting.
October 30th, 2008 at 11:27 pm
Folger’s Crystals. Bwahahaha…hee.
And as for the Disney flick- Dangit, Phil! You’ve probably kicked off another round of the recurring nightmare that movie caused me to have…
October 30th, 2008 at 11:30 pm
You know the thing that makes me the most curious about this post is: What are the beers on the bookshelf in the background of the video from #1
October 30th, 2008 at 11:59 pm
[…] 10 things you don’t know about black holes […]
October 31st, 2008 at 12:00 am
“So volume goes up really quickly as you increase the size of a sphere. ”
Um, is the size of a sphere not measured by its volume? The volume increases very quickly with an increase in radius/diameter…
Pedantic perhaps…
October 31st, 2008 at 12:48 am
11) Black holes evaporate. See Hawking Radiation.
October 31st, 2008 at 12:51 am
I’ve read that black holes eventually evaporate (for the lack of a better word) but later saw a BBC documentary claim they can only get bigger.
October 31st, 2008 at 12:57 am
3) My first thought was — “The enemy’s door is down!”
Seriously this is the one thing that gets me in almost every space sci-fi movie, why people assume that there is some fixed point that should be up or down for everyone everywhere in space is beyond me. (and yes I do realize that you are talking about a 3-d object being assumed to exist in a 2-d space, but it is a similar mistake to assume everyone will always have the same reference frame and coordinate system; and one I believe contributes to this misunderstanding.)
October 31st, 2008 at 1:23 am
Umm, Phil, You’re pulling my leg(s) when you said this, right?
“Astronomers call this spaghettification. Ewwww.”
October 31st, 2008 at 1:31 am
re: Your point #1 - ouch!
While there is a certain consensus about that film sucking so hard it’s DVD might be called an accretion disk (SCNR), it was quite effective in giving me nightmares when I was a child. And ignoring the vapid science of it all, it works quite well as a “horror house” story. The scene with the book (+anthony-perkins-) shredding - urgh. To this day, I don’t know how that film managed to get its age rating - maybe someone didn’t watch further than “Disney” in the titles.
Of course, thanks to the wonders of modern digital distribution, we can marvel at the plainly visible “zero gravity” wire-fu of moviemaking days gone by…
October 31st, 2008 at 1:40 am
Okay, 2 questions.
1) What happens to the matter entering the region of the ergosphere where spatial warping exceeds the speed of light?
2) This one’s been bugging me for a while. What happens when the expansion of the universe meets the event horizon of a black hole?
October 31st, 2008 at 2:22 am
“Of course, we’d freeze to death. You can’t have everything.”
Wouldn’t we be fried by the radiation first?
October 31st, 2008 at 2:37 am
I’ve read that theories that a black hole can spin quickly enough that the event horizon forms torus shape with the event horizon still in the middle of the torus. In such a case, one not need to pass the event horizon to reach singularity. What up with that?
October 31st, 2008 at 3:47 am
I always thought the core of a black hole collapses down to Planck density, not length. Would be slightly more intuitive(as far as there can be anything intuitive about black holes), as collapsing down to Planck length would mean every black hole core is the exact same size, regardless of it’s mass. Even more weirder would be if you had a hypothetical black hole with mass equal to Planck mass, the size of the core in that case would be equal to the size of it’s event horizon.
Could you provide a reference to some math that shows why black hole core collapse is limited at Planck length instead of density?
October 31st, 2008 at 4:32 am
Ooh, 9/10!
October 31st, 2008 at 4:33 am
I was waiting for number 10 the whole time I was reading this. It was easily the craziest thing I remember learning about black holes back in college. Great list!
October 31st, 2008 at 5:00 am
The “hole life” pun made me giggle.
October 31st, 2008 at 5:03 am
I have quite a soft spot for The Black Hole, as I saw it at the cinema when I was 10. And, seriously, how trippy is that Reinhardt-trapped-in-Maximilian’s-body finale in (presumably) hell?
October 31st, 2008 at 5:10 am
[…] de detalles curiosos sobre los agujeros negros. Fuente original (ING): blogs.discovermagazine.com/badastronomy/2008/10/30/ten-things-you-dont/ sin comentarios en: cultura, divulgación karma: 19 etiquetas: agujeros negros, curiosidades, […]
October 31st, 2008 at 5:18 am
Wow, the detail in point 5. that space could be dragged around the black hole at more than the speed of light…quite amazing, and new to me.
But regarding point 10…for the “would be a black hole” calculation, you’re only considering whether the concentration of matter would have an event horizon. This ignores entirely what would happen inside the event horizon, which I think is relevant for a meaningful definition of “density”.
The first question is, is it possible to have something with sufficient gravity to have an event horizon outside its surface without at least part of what’s “left inside” inevitably collapsing into a singularity?
A quick intuitive guess would be maybe, if it’s sufficiently large. Has anyone done any calculations on how large that would be?
The second question is, if such a structure would be stable, could something like that actually form in nature, and if so, how?
October 31st, 2008 at 5:22 am
Phil, that was masterful! Thanks so much. (runs off to make Useful Notes)
Best! — DD
October 31st, 2008 at 5:30 am
You also forgot that referring to the term “Blackhole” in a council meeting is classed as being racially offensive, as happened earlier this year.
October 31st, 2008 at 6:02 am
The knowledge! IT BURNS!
Lots to think about there. Great article. I’m dead thick and that so pretty much all of this was news to me and I don’t pretened to understand all of it but damn, that was good readin’!
October 31st, 2008 at 6:04 am
Awesome article!
I knew a few, but most were new to me. The plank length thing and the density thing were the most insightful (read: confusing).
Might be worth pointing out pamela’s recent post http://www.starstryder.com/2008/10/20/black-holes-only-grow-so-big/ about black holes. That they have a theoretical limit to their size would make a good #11
October 31st, 2008 at 6:27 am
That low density bit is fairly cool. I didn’t know that. I’m annoyed at myself for applying what’s been ground into me in physics classes.
You see the thing derived for you. You see all the variables cancel out, and you still need to have rather obvious implications pointed out. I so don’t grok physics.
October 31st, 2008 at 6:33 am
That was a fantastic read, I feel like my brain capacity has doubled since I finished it
Can’t wait till I can afford your book, it’s next on my reading list.
October 31st, 2008 at 6:38 am
Regarding your last point, if you *really* want to bake your noodle then make it a double feature. First watch “The Black Hole” and then follow it up with “Event Horizon”. They’re fundamentally the same story but told in very different ways.
Which one is scarier? Well, one has people tearing each other apart with their bare hands and clawing their own eyes out after literally traveling to Hell, and the other has Ernest Borgnine. You make the call…
October 31st, 2008 at 6:39 am
Numbers one through nine I knew;
but number ten, ah that was new.
October 31st, 2008 at 6:44 am
Great article. For regular readers of your blog and books, I’m sure many had a grasp of most of these, however, I am making links to this post like crazy! This is great stuff, and more people need to be exposed to this! Thanks again Dr. Plait!
October 31st, 2008 at 6:58 am
This is an excellent article.
October 31st, 2008 at 7:03 am
Phil,
Would it be ok for me to run off a copy of that section for my kids to read in Physics when we cover gravity later on in the year, and for astronomy for next year?
Secondly, regarding the funnel analogy we use to describe warped space….Its a 2D model to explain something 3D (or 4D with time…). When I use this in class every year with a big piece of spandex representing space, a large mass in the middle for the sun, and marbles of various masses for planets. Every year a student will say: Ok, what is the planet is “over here” or “down here” (somewhere not on the plane of the spandex). I tell then it acts as though the “spandex” plane is at that spot as well. This got some students and to talking about this and thinking:
Is there any way to create a 3D model of all of these “funnels” around the mass in the center, and if you could what would it look like? In short, is there a way to create a 3D visual representation of curved space? I don’t have the expertise to know if it could be done, but I am willing to bet that the person who created that 3D nearby stars program could. Thoughts on this?
October 31st, 2008 at 7:05 am
Something that’s been bothering me about black holes lately is “how long does it take stuff to reach the middle of a black hole?” Clearly the answer requires a “that depends” the size of the moon.
Yes, from the perspective of the falling thing the answer is not long at all. From the point of view of those of us on the outside time slows as it gets closer and closer until it pretty much just stops right there on the event horizon, right? The object passes through and keeps falling. How long does it take to get from there to the core? Does the gravitational well of the black hole become lopsided? Would there be layers of attracted matter falling inwards for eternity? I can see how there could be billions of tons of low density matter slowly falling toward a high density core for eternity.
But I could be visualizing it all wrong.
October 31st, 2008 at 7:07 am
Sorry for all the typos in my post there, my coffee hasn’t kicked in this morning: Eliminate When in the second paragraph, change to OK, what if the planet if “over here…, and change to I tell them…..
I need a spell check on this thing….
October 31st, 2008 at 7:34 am
Phil, don’t put Disney’s movie down too much. The concept was good. The exdcution was terrible. I was lucky enough to have read the book before watching the film. It was fantastic! I think Disney was attempting to dumb it down to the kiddie level and sadly, they succeeded. Read the book. You’ll love it!
October 31st, 2008 at 8:19 am
Ibid: rom the perspective of the falling thing the answer is not long at all
No, from the perspective of the falling thing you will see the event horizon to glow from hawking radiation and shrink faster than you can fall towards it till it implodes just before your eyes, and you will find yourself in distant ( like 10^1500 years ) future without having to fall in any singularity at all
October 31st, 2008 at 8:21 am
Your “hole life”, Phil?
We don’t need to live in holes anymore….
Well, welcome back from the stone-age
But: This was an awesome article, even if I knew many of these things AND that you already mentioned some of the points in the very first (in the video…). Thanks a lot!!
October 31st, 2008 at 8:37 am
re: The Black Hole
I just started as a lowly intern at the Disney studio when that film was wrapping. Everyone on the production was unhappy with it. The studio head (son in law of Walt Disney, Ron Miller) was still pissed because he had passed on Star Wars. The director, Gary Nelson, thought the special effects guys had taken over. The special effects guys (great old-timer Peter Ellenshaw, among many others) thought the story sucked worse than a black hole and that their effects were the only thing anyone would be interested in. Actors didn’t like it because those zero-g wire rigs were probably making them sterile (”ball busters,” in the words of Anthony Perkins…not that he’d care about being sterile). And the hard-core Disneyphiles were upset that it was going to get a PG rating…the first in Disney history.
Nonetheless, the publicity department did a huge campaign on the film, only to see it land with a big ol’ clunk at the box office.
One more BTW…the guy who designed the robots, including the most annoying mechanical man in the history of the world, Old Bob, was a Disney imagineer whom I worked with later, and who was also a serious Christian fundie. I mean, serious. Who knows what he thought about that “heaven and hell” ending!
October 31st, 2008 at 8:40 am
If black holes evaportate… then will the universe also evaporate?
October 31st, 2008 at 8:55 am
If black holes emit so much energy (point 7), if we replace the sun with a black hole, wouldn’t we all fry instead of freezing to death?
also, I was always told that the universe sucks, not blows (I guess it’s all relative, though, right?)
October 31st, 2008 at 8:55 am
check that, Blows not sucks
…I blow (or do I suck?) for typos
October 31st, 2008 at 8:58 am
James: Try a Klein bottle, ie, a 2D representation of a 3D bottle with an entry but no exit,,,
,,,always loved those mathematical inventions,,,
There are a number of characterizations of black holes that produce paradox.
The only identifying characteristics of a black hole we can know are the mass, charge and radius. All other qualities disappear, therefore,,,
1) Charge: how can it exhibit a charge, if the force transfer particle(photon) cannot exceed light speed?
2) How can we even feel a G-field from gravitons(the force carrying particle) since C is the escape velocity of the hole and presumably everything must succumb to that force??? Note: we have yet to detect gravitons,,,
Me thinks we must either rework our standard model of physics, which proposes force carrying particles,(to include force carrying particles that exceed C) or accept that those forces (electric field, gravity field) generate their effects without the mediation of force transfer particles at all,,,
Black holes: one paradox piled atop another
If space/time can expand, it must also be able to contract. If the space/time in which our matter is imbedded contracts, we would not know we were falling into a singularity, because as with Zenos paradox, we just keep getting closer, but the space/time between us and it keeps getting smaller and the particles of which we’re made also get smaller.
Woo-Hoo,,,and the ride goes on forever,,,
GAry 7
October 31st, 2008 at 9:02 am
Thanks for the BH posting, Phil.
Been wanting to get those paradoxes off my chest for a loooong time,,,
GAry 7
October 31st, 2008 at 9:07 am
One question, if two black holes collide and unify their event horizons - do the actual singularities unify too? Or does the preservation of momentum mean that they circle each other for eternity?
October 31st, 2008 at 9:09 am
Phil, I always love the “ten things you don’t know” posts. Keep ‘em up!.
October 31st, 2008 at 9:36 am
Phil, this is a great article. I particularly like #10, which I didn’t know at all. The others were all pretty familiar.
Oh, and, uh… I like The Black Hole! It was one of those movies I saw as a little kid that made me go: SPACE EXPLORATION IS AWESOME. I know it has lots of wrong stuff in it, but just like all the movie mistakes in Total Recall, somehow they just help me enjoy the movie all the more. (This is not usually the case with me: normally I’m the first one to suffer intense pain when confronted with Treknobabble or ScriptScience.) And, really, Roddy McDowall as a wise-cracking robot? Priceless.
“Vincent, were you programmed to bug me?”
“No, sir; to educate you.”
October 31st, 2008 at 9:43 am
if black holes eat eat each other why not 1 big black hole eats everything in the iniverse and then it is to big to hold all the matter and it blows up like the first big bang and we start all over again
October 31st, 2008 at 9:45 am
Ibid said:
If I understand correctly, an observer outside the event horizon would see an infalling object gradually slow down, then stop at the event horizon, where its light gradually redshifts and fades into invisibility (how long this takes depends on the gradient in the gravitational field).
An infalling observer does not notice the event horizon (assuming a large BH and hence a relatively gentle gradient in the local graviational field at that point). However, as the infalling observer gets closer to the singularity, the gravitational gradient increases, to the point where tidal forces (or, OK, Phil, differential gravitational forces) cause spaghettification. How long it takes to reach the singularity depends on the radius of the event horizon and the strength of the gravitational acceleration. And, potentially, on the local density and orbital velocities of other infalling matter.
October 31st, 2008 at 9:51 am
So beyond cool. My second favorite post from Phil (I also agree the “10 Things you Don’t Know” ones are some of my faves).
If you like that one he wrote this one is fun to read (or re-read) as well.
-blogs.discovermagazine.com/badastronomy/2008/09/08/ten-things-you-dont-know-about-the-earth/
October 31st, 2008 at 9:55 am
Great posts,
You guys give me a headache (and in a good way).
My heads full, for the day, but I will come back and re-read some of these posts again.
Thanks,
rod
October 31st, 2008 at 10:02 am
Re: #10
Okay, but wouldn’t a solar system full of air (I’m presuming you assumed sea level density) quickly collapse due to the mutual gravitation of all that mass?
Which begs the next question: What would the pressure of all that air have to be to prevent collapse? (Probably not a trivial calculation–I’m getting Lane-Emden equation flashbacks right about now…)
Followed by: What would the temperature (assuming the ideal gas law applies) be that corresponds to that pressure? I’m guessing (based on nothing more than gut intuition) that the temperature would have to be ginormous, as in “particles moving relativistically” big. And once that happens, the particles act like they have more mass ala Lorentz, so I’ll bet it collapses anyway.
Fun to think about, though.
October 31st, 2008 at 10:05 am
Re: #2
Isn’t it likely that once we have a theory of quantum gravity, it might very well describe a state of matter that the black hole becomes, and stops collapsing at, well before it reaches the Planck scale? (I recall talk of “quark bags” some years ago…)
October 31st, 2008 at 10:09 am
Re: #5
Related to this is the fact that at the photon sphere, if you’re traveling in a circular orbit, as you thrust the engines to go faster, centrifugal force (yeah, I know, it’s a pseudo-force) actually shoves you _inward_, not outward because spacetime is sort of turned inside-out. (Scientific American had a nice article about this some years back.)
October 31st, 2008 at 10:21 am
Phil Plait:
Not necessarily, there is still geothermal energy that we could rely on, and let’s not forget all the hot air from politicians that could be channeled into gas turbines to drive electrical generators.
@ kuhnigget
You had better watch your language, kuhnigget, because I’ve already been told off twice by his Lordship, Phil Plait, for using “bad language”.
Phil Plait: “… coolest moments of my hole life.”
October 31st, 2008 at 10:30 am
Re: #6
Personally, my favorite way a black hole can kill you is this:
Even if you find a huge, low density black hole (like the monster in M87) and fall in, knowing that the tidal forces wont spaghettify you (Kip Thorne writes about this in “Black Holes & Time Warps), you will die in a spectacular way. As you approach the event horizon, time in the universe will appear to speed up due to relativistic effects. Just before you cross the event horizon, you’ll see stars winking out, going supernova, etc. Sounds like a great show, except that the photons’ “clocks” will be running hyperfast too (thanks to relativistic blue-shifting). Meaning that the frequency of every photon falling on you will be shifted to the shortest possible wavelengths (ultra-high energy gamma rays) and incinerate you. Ouch.
Okay, I’ll shut up now.
October 31st, 2008 at 10:54 am
@BA “See where I’m going here? If you were to rope off the solar system out past Neptune, enclose it in a giant sphere, and fill it with air, it would be a black hole!”
Hmmm, not according to my calculations. The basic equation is this:
Rs = 2 * G * M / c^2
Rs is the Schwarzschild radius in meters
G is the gravitational constant (6.67E-11 m^3 per kg per sec^2)
M is the mass within the event horizon in kilograms
c is the speed of light (2.998E8 meters per second)
So M = c^2 * Rs / (2*G)
rho = M / V = M / ((4/3) * pi * Rs^3)
rho is the average density in kilograms per cubic meter
V is the volume in cubic meters
pi is 3.14…
So we have rho = 3 * c^2 / (8 * pi * G * Rs^2)
And then Rs = c * Sqrt(3 / (8 * pi * G * rho)
Plugging in the density of air at sea level for the standard atmosphere which is 1.225 grams per cubic centimeter = 1,225 kilograms per cubic meter we get:
Rs = 3.62E11 meters = 2.4 A.U.s
This is only out to the asteroid belt, not the entire solar system.
If you wanted the entire solar system out to the orbit of Neptune (30.1 A.U.s) then the density should be 7.9 kilograms per cubic meter which is the density of air at an altitude of about 40 kilometers, not at sea level.
October 31st, 2008 at 11:03 am
@BA “Black holes can eat other objects, including other black holes, so they can grow. We think that early on in the Universe, when galaxies were just forming, matter collecting in the center of the nascent galaxy can collapse to form a very massive black hole. As more matter falls in, the hole greedily consumes it, and grows.”
If you drop any object straight down into a black hole from and outside observer’s perspective it takes an infinite amount of time for the object to reach the event horizon. That should apply to other black holes falling towards the event horizon. Thus, from our perspective no merger of black holes should have ever occurred since they would take infinitely long to happen. This seems to be a contradiction in terms of the supposed supermassive black hole at the center of the Milky Way galaxy. Theoretically it should have taken an infinite amount of time for it to form from component black holes and yet we observe that it’s there. That seems to be a pretty big paradox that I’m trying to get my mind around, … unsuccessfully.
October 31st, 2008 at 11:07 am
@Tom Marking
Ahhh! Mathematics!
*crickets*
October 31st, 2008 at 11:21 am
@BA “Worse, magnetic and other forces can focus two beams of energy that go plowing out of the poles of the disk. The beams start just outside the black hole, but can be seen for millions or even billions of light years distant.”
Yeah, I’ve never quite understood the two jets coming out from quasars and how they relate to the black hole at the center. A couple of points:
1.) Magnetic field lines CANNOT cross the event horizon boundary.
2.) Therefore if the jets are aligned with the magnetic field then the magnetic field exists in the gas disk and has nothing to do with the black hole itself. If the black hole has a magnetic field inside the event horizon it has nothing to do with the magnetic field outside the event horizon.
3.) The axis of the jets does NOT have to be the same as the angular momentum axis of the black hole’s rotation. They may be totally separate.
October 31st, 2008 at 11:27 am
@Todd “Ahhh! Mathematics!”
Run away! Run away!
ARTHUR: Right. How many did we lose?
KNIGHT: Gawain.
KNIGHT: Hector.
ARTHUR: And Boris. That’s five.
GALAHAD: Three, sir.
ARTHUR: Three. Three. And we’d better not risk another frontal assault, that rabbit’s dynamite.
ROBIN: Would it help to confuse it if we run away more?
ARTHUR: Oh, shut up and go and change your armor.
.
.
.
October 31st, 2008 at 11:34 am
I got one that’s really weird. The other day, I was playing around with the equations and decided to see how big a black hole it would make if you squished all the known matter in the universe together.
Accounting for all the dark matter as well, I found out that the black hole produced is actually LARGER than the co-moving distance to the cosmological horizon. In other words, all the matter currently known to exist is currently inside its own event horizon, meaning that it should immediately form a black hole. (Without accounting for dark matter, the size of the black hole is “only” about 20 billion light-years in diameter).
Now dark energy/the expansion of the universe probably negates this in some way, but it is still kind of spooky.
Also, a comment on #6. Interestingly, supermassive black holes don’t do this as much. They’re billions of kilometers across and the human body is only a very tiny fraction of the whole distance. You could orbit quite close without being shredded by tidal forces (though it will happen if you get sufficiently close to ANY black hole).
October 31st, 2008 at 11:37 am
@Tom Marking
Phil’s statement was correct: The Solar System filled with air would be a black hole. You just calculated the _minimum_ density needed, which turns out to be less.
That being said, your calculation uses the formula V = 4/3 * pi * r^3, which only works in flat (Euclidean) spacetime, not the case with a black hole! All that warped space would increase (significantly) the volume contained within a given radius, bringing Phil’s air-filled solar system closer to the critical density. How much closer, I couldn’t say.
October 31st, 2008 at 11:42 am
sometimes it’s the little things that count.
Exactly what I told my wife.
October 31st, 2008 at 11:45 am
@Myself “Rs = 3.62E11 meters = 2.4 A.U.s”
Arrrggghhhh!!! Egad!!!! The equation is right but I plugged in the wrong number for rho. It should be just 1.225 kg/m^3 not 1225 kg/m^3. (Damn that conversion from grams per cubic centimeter to kilograms per cubic meter!!!) This makes the Schwarzschild radius for an average density of air equal to 76.6 A.U.s., not the 2.4 A.U. I previously stated. So it’s actually 2.5 times the size of Neptune’s orbit that has to be filled with air to make a black hole. Surprised no one caught that mistake before I did. But then, I guess they had already run away by that time.
* crickets *
October 31st, 2008 at 11:45 am
Hi all, first time posting here, though I’ve been reading for years.
Anyway, I just wanted to point out something minor about #2. Typically (in general relativity) we call the event horizon the black hole, and the r=0 place the singularity. This is because it is possible to have a spacetime that has an event horizon with no singularity, they are weird and probably can’t exist in reality, but they do exist mathematically. The other way around, a singularity with no event horizon is impossible, cosmic censorship and all that.
To summarize, Phil you win that argument with your boss, at least according to most any general relativist.
October 31st, 2008 at 11:55 am
It’s so nice to come here after reading the latest logical atrocities being perpetrated in the political world and read a science discussion…
I remember sitting through that awful Black Hole movie and smirking every few minutes as another law of science went crashing down in flames…
For some reason that movie was particularly egregious about it. Almost as bad as the “Kesla run in 12 parsecs” line from Star Wars IV.
October 31st, 2008 at 11:56 am
Oh, and about those beers behind Phil’s head… I wonder if he’s ever thought of brewing his own? Black Hole Brew maybe?
October 31st, 2008 at 11:59 am
Re: #10
Since the force of gravity by distance is reduced by the inverse square law, wouldn’t the mass of the black hole would have to increase by a factor of 4 to double the radius of the event horizon? The volume would still decrease as the black hole got larger, but only by a linear factor, not by a square.
October 31st, 2008 at 12:07 pm
Contrary to the title, I knew all ten already. I can has cookie?
October 31st, 2008 at 12:17 pm
Disney’s “Black Hole” had the worst robot in high budget sci-fi, and that’s saying something.
I think I understand the current thinking on the formation and effects of black holes, but nobody seems to answer my question: How can matter or energy pass through the event horizon from the greater universe. As matter falls toward the event horizon, it goes faster and faster, time for the matter passes more and more slowly, and the matter becomes shorter and shorter in the direction of travel. As the matter approaches the event horizon it is also approaching the speed of light, and relativistic effects become more and more pronounced. Taken to the speed of light wouldn’t an object becomes two dimensional in the direction of travel and virtually detached from the universe in time. The matter would never cross the event horizon.
Now I’ve got a headache.
October 31st, 2008 at 12:25 pm
Ok, you got me on # 10…low density. Who’d ‘a’ thunk it, being fed all this stuff about neutron stars and a spoonful weighing a google of grams. But how does that jive with #2? Where did all that mass go?
Can you answer # 11 for me? What the temperature of a black hole?
If the temp is above 0 K, how can all that matter collapse to a Planck length, since Brownian motion keeps things apart?
October 31st, 2008 at 12:27 pm
Bert,
Didn’t that happen on an episode of Stargate SG-1?
October 31st, 2008 at 12:29 pm
I remember hearing this line (or a very similar one) on either The Daily Show or The Colbert Report. So either great minds think alike, or else your friend is a certain director of the Hayden
overhead projectorplanetarium and he stole your line.October 31st, 2008 at 12:30 pm
@ccpetersen
I think I’ve seen the Star Wars Faithful ™ argue Han Solo’s apparent conflation of time and distance units this way: By navigating along the shortest path, Solo could get to his destination faster, with his 12 parsec path taking him dangerously close to objects normally avoided by more cautious pilots choosing longer (and thus slower) routes.
October 31st, 2008 at 12:50 pm
I suppose someone should also mention Greg Egan’s Java applet that shows what the view looks like as you get close to a black hole.
October 31st, 2008 at 1:04 pm
Re. #9 –
In the event that the black hole at the center of our solar system did start “feeding” and sucking stuff in, how long would it take us to find out?
October 31st, 2008 at 1:05 pm
Cuz, it could have happened already. Just sayin’. Which would explain a lot about the current election and the last few years of high fashion.
October 31st, 2008 at 1:15 pm
BTW, the “Astronomy Fail” photo is of the Black Hole, a laboratory surplus store in Los Alamos, so called because “everything goes in and nothing comes out.” So the name does make some sense…
October 31st, 2008 at 1:42 pm
[…] Skies! just came out, and there
October 31st, 2008 at 1:42 pm
Dammit Dr. P, now I have to listen to Muse all day. I was really enjoying Lisa Hannigan, but now I have a hankering for Supermassive Black Hole.
October 31st, 2008 at 1:46 pm
@Chris “All that warped space would increase (significantly) the volume contained within a given radius”
I believe it goes the other way for positively warped space - the volume is less than you expect. For example, consider the following 2D case. You have a circle of radius S. The expected Euclidean area contained in the circle is pi * S^2.
Now suppose S is really a curve on a sphere going half-way around the sphere so that S = pi * R where R is the radius of the sphere. Now the surface area enclosed by the “circle” is 4 * pi * R^2 or (4/pi) * S^2. This i