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Space and Time Part II: Black Holes by Mark Lawrence 
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In Part I, we saw that a gravitational field slows down time. We learned that on a piece of paper we measure distance the way Pythagoras taught us back in about 500 BC. The distance between two points is the square root of X^{2} + Y^{2}. Unfortunately, we don't live on a flat piece of paper. We live in a four dimensional curved space time. If we're very far away from all matter, the distance rule for our universe is the square root of T^{2}  ( X ^{2} + Y^{2} + Z^{2} ). Except for the pesky minus sign, this looks a lot like Pythagoras' formula. If we're near a large mass, like the earth or the sun, the distance rule gets more complicated. The distance rule near the Earth is square root of (1  2M/r)T^{2}  ( X^{2} + Y^{2} + Z^{2} ) / (1  2M/r) , where M is the mass of the earth and r is the radius of the earth in appropriate units. At the surface of the Earth, 2M / r is about one billionth. That means here on the surface of the Earth, a clock would be slow by about one second every 32 years compared to an identical clock which was far away from all galaxies. This is why Newton never noticed this: you have to be able to measure things to within about one part in ten million to notice this at all, and then you still need a reference that's very far away from you.
At the surface of the earth, gravity slows time by about 1 second every billion seconds, about 1 second every 32 years. Suppose the gravity field were a billion times stronger, either because the Earth weighed a billion times more or it was shrunk to a billionth of its current size. It seems like time would slow in this case to a stop  our clocks would lose a second every second. The radius of the Earth is about 3940 miles. There's 5280 feet in a mile, so that's about 21 million feet. There's 12 inches in a foot, so that's about 250 million inches, about 1/4 billion inches. So, if the Earth were shrunk to a ball ¼ inch in radius, about ½ inch across, then on the surface of the Earth time would stop. What does it mean to shrink the Earth to the size of a marble? What does it mean to have an object that stops time? We have a name for this, such an object is called a Black Hole.
Our Sun weighs about 333,000 times as much as the Earth, so if we shrunk the Sun into a black hole apparently it would be about 333,000 times as big as the Earth at black hole size. This is about 166,000 inches, which is about 3 miles. So, if we shrunk the Sun into a ball 3 miles across, it would be a black hole. The Sun is actually about 860,000 miles across, so that's a lot of squeezing.
Another thing we notice is that as we get close to the surface of a black hole there is more and more space, until right at the surface of the black hole there seems to be an infinite amount of space. Right at the surface of the black hole we are dividing our spacial measurements by zero, and that makes them infinite. What does this mean? This means something quite interesting. If we were floating around in space near a black hole and we tossed a rock into the black hole, as the rock moved towards the black hole it would speed up, just as a rock dropped near the earth speeds up as it falls. However, when the rock got quite close to the black hole, it would appear to slow down. Starting at about 10 times the radius of the black hole, it would appear to quite suddenly hit the brakes. In fact, it would appear to go slower and slower, and just barely short of the surface of the black hole, it would appear to us to stop completely. We would never see the rock hit the black hole  it takes the rock an infinite amount of our time to hit the black hole. The rock, by the way, has its own sense of time and space, and things look different to the rock, but that's not important to us. What's important to us is that from our perspective nothing from our universe can ever enter a black hole, because it takes an infinite amount of time to get there.
Let's imagine that there's a collapsing star. This star has some mass, and that mass has a black hole radius r = 2M (in the appropriate units). Suppose all the mass of the star has crossed over the black hole boundary save for one last atom. Since we're 1 atom short of black hole mass, this is not yet a black hole and the equations don't quite ever get to zero. Now, what's going to happen to that last atom? It will attempt to enter the black hole, but as it gets close to r = 2M time starts going very slow and space starts getting very big  very doesn't quite mean infinite here, but very very large. This last atom seems to us to take billions of billions of years to reach the black hole radius. So, we see that there actually aren't any black holes in our universe  they take a long, long time to form.
The idea of a collapsed star forming a black hole is relatively new  John Wheeler invented the name "black hole" in the 1950s. However, the idea of a black hole is surprisingly old. In 1784, the English astronomer John Michels used Newton's equation for escape velocity v = square root( 2GM / r ) to predict them. He set v equal to the speed of light, c, and deduced that at a radius r = 2GM / c^{2} light could not escape from a mass. This is the correct formula for the Schwarzschild radius of a black hole.
Although we've just concluded that there are no actual black holes in our universe, there are objects which are almost just like a black hole. It's annoying to write "almost just like" all the time, so I'm going to call these almost objects "black holes," and we'll just have to remember that they're really "almost black holes."
Recently people have become very interested in some quite esoteric questions about quantum probabilities and black holes. Some people assert that information that drops into a black hole is lost forever from our universe; other people say that information leaving our universe like this violates the rules of probability and quantum mechanics. Then, a big "how many angels fit on the head of a pin" type of discussion starts. We see here that no particles from our universe actually ever make it all the way into a black hole  indeed, no real, live actual black holes can ever form in our universe. We can get things which are gravitationally very very close to a black hole, and we can get particles very very close to entering these things that are very very close to a black hole, but the "very very close" words mean that the laws of quantum probability continue to be observed in our universe. Information that's in our universe stays here, and the zero time and infinite space things never quite happen. I offer Lawrence's 1st axiom of physics: Our universe doesn't have infinities in it. People who worry about "but what if it did?", well, that's a fun topic for the pub but it doesn't actually come up in real life.
If we squeeze matter into a small enough space, it becomes something a lot like a black hole. What does this mean?
In our everyday lives, matter is made up of atoms. Atoms are a collection of protons, neutrons, and electrons. The protons and neutrons are very little objects that sit in the middle of the atom, in the nucleus. The electrons are very light objects that swarm around the nucleus. All atoms are roughly the same size, about 1 angstrom across. This means that if you line up 10 billion atoms in a row, that makes about a meter, or a yard if you prefer.
The nucleus is much smaller. An atom is about 10,000 times as big as the nucleus. So, if you wanted to measure off a meter using only nucleuses, you would need about 100 trillion of them all lined up.
Electrons like their space. There's a rule in our universe, that two electrons can't be in the same place at the same time  this is called the Pauli exclusion principle, named after Wolfgang Pauli, who first figured it out. So, when you push really hard on matter, it pushes back. We all know this from our ordinary lives: if you push really hard on steel or concrete, nothing much happens. But gravity can pull matter together much harder than we can push.
If you push matter really hard, much much harder than anything ever achieved on Earth, you can squeeze the electrons into the nucleus. At this point, something unusual can happen: the electrons can pair up with the protons and make neutrons. Sometimes when a star collapses, it turns into what we call a neutron star  it's a star composed of almost pure neutrons. Inside the star, the neutrons are all jammed together. This material is sometimes called neutronium. All the mass of the atom is still there, but it's been crowded into an area that's 1/10,000 times as big across. So, since volume is size cubed, the density of this matter is 10,000 * 10,000 * 10,000 times the density of normal atoms. Roughly, this stuff weighs about a billion tons per cubic centimeter, about a billion tons per marble sized object. Is this a black hole? Nope. We learned that a black hole the size of a marble weighs as much as the Earth, which is about 10 thousand billion billion tons.
What happens when you squeeze the neutrons? They also obey the Pauli exclusion principle, and they also push back. But gravity is not to be denied. If you make a clump of neutronium which weighs about 3 times as much as the Sun, this matter will collapse and try to form a black hole. When the atoms collapsed, the electrons combined with the protons to make neutrons, and then everything got real small. What combines when the neutrons collapse? We have no idea. Not a clue. This is so far beyond anything we can achieve with our billion dollar accelerators that we can't even speculate without feeling stupid. But, the neutrons collapse.
How do we know the neutrons will collapse? We have no idea what happens to matter when neutrons collapse, so how can we be so certain? The answer is that gravity is caused by energy, not mass. Pressure is a form of energy. So, as the gravity puts more and more pressure on the neutrons, and the neutrons resist harder and harder, the pressure starts build up to some really very impressive levels. Pretty soon, the gravity from the pressure in a massive neutron star is greater than the mass. At this point the neutrons have no hope: the harder they resist, the higher the pressure, and the higher the gravity. There is no limit to the gravity, so far as we know. We believe there must be some limit to the pressure. At that point, whatever that pressure point is, however they do it, the neutrons will collapse.
So, we don't know what's in a black hole, all we know for certain is that it will be very small and very dense, much denser than neutrons or anything else we understand.
What's the density of a black hole? Our instant answer is, um, well, billions of billions of tons per cc. Is this correct? We have our equation for black holes, r = 2M. This is interesting: if the mass gets twice as big, the black hole gets twice as big across. But twice as big across is eight times the volume. So a black hole twice as massive is eight times the volume and one eighth the density, apparently. The bigger the black hole, the lower the density. Maybe when black holes get really big and massive, they're not packed quite so densely.
We learned in part I that when something is very close to a black hole, the last few atoms take almost forever to enter. We think the threshold for making a black hole is about three solar masses. Less than three solar masses and the neutrons are strong enough to hold the star up and stop the collapse. So, imagine we have just more than three solar masses. This object will contract to just barely short of black hole size  about 10 miles across. After it gets to be almost a black hole, it takes almost forever to finish. This means if you keep throwing mass towards a black hole, the mass will not enter the black hole, but will get stuck just short of the black hole radius. If you throw a lot of mass at a black hole, say 10 solar masses, the entire 10 solar masses will be stacked up neutron by neutron just short of the black hole radius. But the black hole radius for 10 solar masses is 10 times as large as for 1 solar mass. So this stacked up matter is not getting squeezed nearly as much as the matter in the middle of this thing. Each new neutron you toss in sees not only the core black hole, but also the gravity from all the previous stacked up neutrons. So each neutron you throw in increases the size of the slowtime region. Thus we see that a black hole is a bit like an onion  there are layers of matter stacked up, each layer covering up the previous layer. As you the black hole gets larger and larger, the layers get less and less dense.
We're going to calculate some numbers now, so we need to talk a little bit about units. Traditionally, people measure time in seconds, distance in inches or centimeters, and mass in pounds or kilograms. These are very inconvenient units when you're talking about black holes. If you want to keep using these units, then your formulas are all full of strange constants like c, the speed of light, and G, Newton's gravitational constant. Frankly, I'm too dumb to keep track of all these constants  I always get them screwed up. Fundamental physics is normally done in what we call Natural Units. In natural units, we measure distance and time in units called Plancks. Plancks are very small. There are 2.5e32 plancks in a centimeter, and 7.5e40 plancks in a second. Stones, our unit of mass, are also very small. There are 18,350 stones in a gram. A single grain of salt weighs about a stone. These are very odd units, very remote from anything in our normal lives. However, in these units the speed of light is 1 and Newton's gravitational constant is also 1. This means we never have to worry about getting the constants right and in the right place, because multiplying by a couple fewer or a couple extra ones just doesn't matter.. It also means we're going to have to convert back to normal units to understand our answers.
We can calculate the density of this stacked up matter. We know that M = r/2, so if you change the mass a little bit, call it dM, then you change the radius a little bit, call it dr/2. Each time the black hole gets a little bigger, it holds a little more mass. Interestingly, each time the black hole radius increases by one planck, the black hole mass increases by 1/2 stone  no matter how big or small the black hole is, the amount of mass needed to increase the radius by one planck is the same. Add one grain of salt to any black hole and the black hole radius grows by one planck, by 4e33 centimeters. Each layer in a black hole weighs about the same as a grain of salt, and is 4e33 centimeters thick.
When you change the radius a little bit, you add to the volume of the black hole. The volume added is the area of the black hole, 4πr^{2}, times the height of the new shell, dr, so the volume of the new shell is 4 πr^{2} dr. The density is dM / 4πr^{2} dr, but dM = dr / 2. So the density is 1 / 8πr^{2}. Remember, we're using plancks and stones here, not grams and centimeters. There are 5.45e5 grams in a stone, and 4.05e33 centimeters in a planck, so if we multiply the density function by these conversion factors, we'll get answers in grams per cc which we understand better. 1 / 8πr^{2 } * 5.45e5 / 4.05e33 is 1.35e28 / 8πr^{2}, which is 5.4e26 / r^{2} grams per cc. Now let's plug in some numbers. For the Earth, the black hole radius r is about 1 centimeter, so the density would be 5.4e26 grams per cc at the edge of black hole. Frankly, I can't even imagine what this matter is like. The Sun's black hole radius is about 3 kilometers, about 300,000 cm. The density at the edge of the black hole radius is 5.4e26 / 9e10, about 6e15 grams per cc. This is about the density of neutronium, so if the Sun were compressed into a black hole, at the edge it would be neutronium.
M74, the Spiral Galaxy in Pieces
A galaxy a lot like ours with about 100 billion stars and
about 100,000 light years across. In the very center of this
galaxy is a black hole that weighs as much as 100 million suns.
We believe there's a black hole at the center of our galaxy which weighs about 100 million solar masses. So, this black hole would have a radius of 100,000,000 times the sun's black hole radius, about 300,000,000 kilometers. This is about the size of the orbit of the Mars, so we're talking about 100 million suns packed into a volume as big as the orbit of Mars. What's the density of this matter at the edge? It's 5.4e26 / (300,000,000 * 1000 * 100)^{2}, which is 5.4e26 / (3e13)^{2}, which is about .6. The black hole at the center of our galaxy weighs 100 million times as much as the sun, it's packed into a volume about as big as the orbit of Mars, and the density of matter at the edge of the black hole is less than the density of water. You could sail into this black hole in a submarine. People watching you would say it took you all the rest of eternity to sail in, but you could do it.
I know this all seems very counterintuitive. Remember, we have very little personal experience around black holes, so we must consider that our intuition means little here. Richard Feynman, a nobel prize physicist, used to say, "Your intuition is just the misunderstandings and misconceptions crammed into your empty childhood mind by your ignorant parents." Well, that's perhaps a bit harsh, none the less your parents also spent very little time hanging around black holes.
So, how can this be, that a black hole has only the density of water? Remember, there actually is no black hole. There's an object at the center which weighs about three solar masses, jammed into a volume about 10 miles across  a radius of about 9 km. This object is nearly a black hole. The density of matter at its edge is 5.4e26 / (9 * 1000 * 100)^{2}, about 6e14 g/cc. So, this object has a layer of neutronium around it trying to enter, and taking forever to do so. When more matter tries to enter the black hole, it gets stacked up outside in the area where time gets really slow. Each layer of new matter adds to the total mass of the black hole, and as the mass increases, so does the black hole radius and the slow time area. So, there's always more room for another layer of material stacked up in the slow time zone. As the mass continues to increase, the size of the black hole increases. We've seen that the mass required to increase the black hole radius is always the same, but the volume added to the black hole gets bigger and bigger. So the density drops off as we get to the edge of the black hole. Super massive black holes, like the one at the center of our galaxy, are so big that the matter density at the outer edge is not much different than the density of water.
M87 is a very large elliptical galaxy in Virgo.
The Hubble telescope took this picture of the accretion disk of its black hole.
M87 is a very large elliptical galaxy in Virgo, and has an entire cluster of other galaxies that orbit around it. At the center of this galaxy we believe there is a black hole that weight 3 billion times as much as the sun, 30 times as big as the black hole at the center of our galaxy. The Hubble telescope has even managed to get a picture of this black hole, or more precisely a picture of the matter that's falling into the black hole. Let's work out the details for this black hole. The size is 3 billion times the size of the Sun's Schwarzchild radius, 3 billion times 3 km. 9 billion km is 60 AU, 60 times the orbit size of the Earth. Pluto orbits at 39 AU, so this black hole is about half again wider than our entire solar system. The density of matter at the edge of the black hole is 5.4e26 / (9 billion * 1000 * 100)^{2}, which is 5.4e26 / (9e14)^{2}, which is about 1/1500 g/cc. The density of air is about 1/800 g/cc, so the density of matter at the edge of this black hole is half the density of air at sea level, about the same as the density of air at 15,000 feet. According to the FAA, you need an oxygen mask up there. Below is a graph of the density of the outermost layer of a black hole versus the mass of the black hole in number of suns. I made the line at a density of 10 thick and silver, because this is the density of silver. The next line down is bluegreen, this is the density of water. Two lines down is blue, this is the density of air. We see in the graph that a black hole of mass about 25 million suns has an outermost layer which has about the density of silver. About 80 million suns in a black hole makes the outermost layer have a density of water. About 2 billion suns in a black hole makes the outermost layer have a density of about air.
We're throwing this "r" variable around an awful lot, maybe we should talk about what it means. We learned from the metric that when you get really close to a black hole, there's more and more space, until right at the edge of a real black hole there's an infinite amount of space. So what does "r" mean when there's an infinite amount of space between you and the black hole? Imagine you are standing still near the black hole. To do this you need a rocket and a lot of fuel, but that's an engineering problem.. Measure the length of a circle around the black hole, perhaps by laying a lot of yardsticks end to end. Divide the length of this circle by 2π. We'll call the orbital circumference divided by 2π "r." Although from far away it looks like there's an infinite amount of space just barely outside of the black hole, the circumference of the circles around the black hole are always well behaved. So this is our definition of "r." We have to be careful about this  this definition of "r" doesn't work for someone who is orbiting  he's moving at a high velocity, and will see the yardsticks as Lorentz contracted.
Black holes generate a lot of gravity, so sometimes people think this means it's dangerous to get near them. But if we try to enter a black hole, we'll be in free fall  just as the astronauts orbiting the earth have no sensation of weight, neither will we. As they say, it's not speed that kills, it's the sudden stop. But we're not going to hit any sudden stops.
There is another issue, however. We're used to thinking of the Earth's gravity field as uniform, but it's not. This effect is even more apparent when near a black hole. Suppose you're in an elevator falling near the Earth. Can you tell you're falling, as opposed to just floating in space somewhere far from all matter? Yes, here's how.
Below is a picture of four tennis balls floating in an elevator. The elevator is about 12 feet across and 12 feet tall. The elevator and the four tennis balls are all falling towards a black hole, which is also about 12 feet across. This black hole is 12 feet * 12 inches per foot * 2 times as big as a black hole formed from the Earth, so evidently it weighs about 300 times as much as the Earth  about as much as Jupiter.
Left: Four balls in an elevator, falling towards a small black hole
Right: The same elevator, with the average force subtracted away.
Do the four tennis balls all fall at the same rate and in the same direction? No. As we can see in the picture above, the tennis balls at the sides of the elevator are falling towards the center of the black hole, not straight down. The arrows drawn in the picture on the left are what we would see as a distant external observer. What about a guy in the elevator? He is falling along with everything else. However, the elevator has no windows, so he has no sense of falling. He sees the four balls all dropping at the same speed as him, but with small additional forces. The two balls on his left and right appear to him to be attracted to each other; the two balls above and below him appear to be repelled by each other. When we subtract away the average force vector, we see the same thing.
Gravity drops off as 1/r^{2 }, and the tennis ball at the top of the elevator is about twice as far from the center of the black hole as the tennis ball at the floor. So, the tennis ball at the floor of the elevator feels 4 times as much force as the tennis ball at the ceiling. Now, if you imagine standing in this elevator, you can see that you would feel squeezed from the sides, and your head and feet would be pulled apart. How much? A certain amount of pulling and squeezing is no problem, but these black hole things sometimes crush matter into neutronium and beyond, so we had better be worried about getting crushed ourselves.
Earth with Tidal Bulges, being pulled towards the moon.
This pushing and pulling and squeezing is called the tidal force. It's called the tidal force because this is how the moon makes tides on the Earth. The Earth is pulled towards the moon with an average force, shown by the arrow at the center of the Earth. However, the water in the ocean that's nearer to the moon is pulled a bit harder than the Earth average, because it's closer to the moon. Since it's pulled a bit harder towards the moon, this water bulges out away from the Earth towards the moon. Similarly, the water at the far side of the Earth is pulled a bit less because it's further from the moon. Since it's pulled a little less than the Earth, the Earth pulls away from this water and it bulges out away from the Earth and away from the moon. The Earth spins on its axis between these two bulges. This is why there are two high tides each day  one when the moon is directly overhead, and the second when the moon is completely hidden behind the Earth. The north and south poles of the Earth are also being squeezed together by the tidal force from the moon, but we almost never think about this. That's because these forces are squeezing on rock, not pulling on water, so you need lasers and satellites to measure this tidal effect. If we try to enter a black hole, we're going to get stretched and squeezed by a tidal force. The moon is relatively small and light and far away. A black hole is big and heavy and we're interested in getting up close and personal, so inquiring minds want to know  will this tidal force hurt us?
How big is this tidal force? Before we calculate anything, we'd best understand what we're calculating. The Earth is about 13000 kilometers across, and about 384000 kilometers from the moon. So, the far side of the Earth is about 3.3% farther from the moon than the near side. This means the gravitational acceleration, which goes as M/r^{2}, is about 7% larger on the side of the Earth near the moon than on the far side.
What's the acceleration on the Earth due to the moon? The acceleration of gravity that we're used to is the Earth's gravity on the surface of the Earth, about 10 meters per second per second. This means if you drop a rock, each second it accelerates 10 meters per second, about 20 miles per hour. But the moon is much smaller than the Earth, and also much farther away. The force from the moon is M/r^{2} in natural units. We'll need a bunch of constants from our table of units . The mass of the moon is 1.343e30 stone, and the distance to the earth is 9.5e42 planck. M/r^{2} is 1.48e56. This is an acceleration, so we use our conversion factor of 2.27e50 g / stone to get 3.3 microgs. The gravitational pull on you from the moon is 3 millionths of the gravitation pull on you from the Earth. We calculated above that the tidal pull on the water is about 7% of this number, spread over 13000 kilometers. So the tidal force from the moon is 3.3e6 * .07 / 13,000,000 g/meter = 1.8e14 g / meter.
The tidal force is going to be the difference in acceleration between two points. The idea is that the acceleration on the surface of the Earth due to the moon is different at different places on the Earth. How different? We measure gravity in terms of acceleration, so we'll measure tides in terms of acceleration per meter. The time delay near a mass is M/r. The acceleration of gravity is the derivative of this, M/r^{2}. The tidal force is the derivative of this, 2M/r^{3}. So, the number we're looking for is the 2 * mass of the moon / (orbital radius of moon)^{3} * conversion to g/meter, which is 2*1.343e30 / (9.5e42)^{3} * 5.62e84. This multiplies out to 1.8e14, just as above.
Now, let's think about flying into a black hole. The moon is stretching the Earth as on a rack from the Spanish Inquisition, and the tides are a sign of this. What if we're flying into a black hole and we get a tidal force big enough to rip us apart? Imagine we're falling feet first into the black hole. When we get really close, our feet will be pulled much more strongly than the rest of us, and our head will be pulled much less strongly than the rest of us. The black hole will be trying to tear our head and feet off. We'll arbitrarily decide we can handle 1g / meter. Since a person is very roughly 1 meter tall, this force is about like hanging from a parallel bar in a playground. We're quite confident this won't tear us apart. But 100 gs per meter means a person who weighs 150 pounds would have a force of about 15,000 pounds trying to rip them apart. We're pretty confident we can't survive 100 gs per meter.
Let's work this out for a black hole that masses the same as the Sun. We want 2M/r^{3} times our conversion factor, which is 2*3.646e37 / (7.292e37)^{3} * 5.62e84. This is just about exactly 1 billion gs per meter. We try to fly into this thing, and all they will find is a couple of red splotches where parts of us hit the floor and ceiling.
Let's see what black hole we can enter. At the Schwarzchild radius, r = 2M, so the tidal force is 2M / 8M^{3} which is
1 / 4M^{2}. We want this to be 1 g/meter,
so 1 / 4M^{2} * 5.62e84 = 1. M = 1.2e42. We'll divide this by
the mass of the Sun, 3.646e37, to get 32,500. A black hole which weighs about 30,000 times as much as the Sun has a tidal force
of about 1g / meter, and we can fly into it with only mild discomfort. The black hole at the center of our galaxy has a mass of
100 million Suns, so at the Schwarzchild radius the tidal force is 5.62e84 / 4 (1e8 * 3.646e37)^{2} = 1e7 g/meter. We
can't feel this at all. The massive black hole in the Virgo cluster weights 3 billion times what the sun does, 30 times what our
galaxy's black hole weighs. This would have 1/900 of the tidal force, which is about a billionth of a g / meter. The Virgo black
hole has a density at the surface of half sea level air density, and a tidal force which we can't even measure. Really doesn't
sound very threatening, does it?