How we know we're dealing with white dwarfs and neutron stars		Today I'm going to talk about black holes! That will be exciting! But first, I forgot to mention some of the ways we know neutron stars are real. X-ray bursters have a blackbody spectrum. The spectrum tells us the size of the star, and that comes out to a 10 km radius or so--what we expect for a neutron star. X-ray pulsars pulse on and off because the star is spinning. (Same with radio pulsars, which are neutron stars off alone instad of in a binary.) Well, there's a limit to how fast a white dwarf can rotate, so these stars can't be white dwarf stars! Why is there a limit? Because when a white dwarf star is rotating, matter at the surface is moving pretty fast. If the star goes faster than once a second, the surface ends up going faster than the escape velocity, and the star falls apart! A white dwarf is limited to below a mass of 1.44 times the Sun's mass (named the Chandrasekhar limit, after Subrahmanyan Chandrasekhar.) But is there a limit to how massive a neutron star can be? Yes! We don't know exactly what that limit is, but we're pretty sure anything with a mass more than 3 times the Sun's mass couldn't be a neutron star--it would have to be a black hole!
Black Holes -- formation		When a degenerate iron core has more than 3 times the Sun's mass, what can halt its collapse? Nothing! Because even if matter when squeezed so tight has some kind of pressure that pushes back--say some kind of pressure that our physics theories don't yet know about--that pressure itself will help to pull the matter into a black hole! Why? Pressure itself is energy. If matter acts like tiny springs that push back when you push them in, that means that the matter has potential energy that increases when squeezed. But Einstein's theory of gravity says that not only mass, but also energy can be a source for gravity. (Remember, E=m c2!) So the pressure doesn't help much! Because the pressure pushing out is partially offset by the extra gravity pulling inwards, caused by the pressure energy!
Black Holes -- the Newtonian View		Even before relativity and E=mc2, people suspected there might be things like black holes. French physicist Laplace realized that if an object were so massive and compact that its escape velocity was the speed of light, then light would eventually have to fall back onto the object! Remember the escape velocity is vesc=SQRT(2 G M / R), where SQRT means square root. This means that if the mass M is greater, then you have to go faster in order to escape. Also, if you start closer to the center of the mass, with a smaller R, then it will also be harder to escape. Well, set vesc equal to the speed of light, c=2.99792x108 meters/second, and you'll find from the algebra that RS=2 G M / c2. This is called the Schwarzschild radius. It means that if you squeeze a mass M to within the size RS, it will be a black hole. If you point a flashlight outward at a distance closer than RS from the center of the black hole, according to Newton's theory, the light will eventually be pulled back. If the Sun were to become a black hole, it would have to be squeezed into a radius of 3 km. Einstein's theory makes a black hole more radical. It also places importance on the Schwarzschild radius. But now the Schwarzschild radius is the radius of the event horizon. From within the event horizon no information can ever get out! As of yet, we don't have definitive evidence that event horizons exist. Michael Garcia, Jeff McClintock, and Ramesh Narayan at Harvard has been trying to show that event horizons exist because, he and his team claim, dim accretion disks around black holes are not as bright as the dim accretion disks around neutron stars. Why? Because disks of gas swirling around neutron stars can release energy when they smash into the surface of the neutron star, but when the gas falls into an event horizon, it will never be seen again.
Einstein's Theories of Relativity		Check out his hair! In order to truly understand black holes, you have to understand something about Einstein's theories of relativity. I say theories plural, because he actually had two different versions of his theory. You may find it helpful to review some chapters that are not required: Chapter S2 (space and time--special relativity), Chapter S3 (spacetime--general relativity), and S4 (building blocks of the Universe--can help in understanding the Pauli exclusion principle that's responsible for the pressure of degenerate matter!) The first of Einstein's theories, which he came up with in 1905 when he was a 26-year old patent clerk, is called the Special Theory of Relativity. Its basic assumption was that all physical laws are the same, no matter what state of constant motion you describe them from. This might seem commonsensical. But the speed of light, c=300,000 km/s or so, plays an essential part in the laws of physics. Remember, light is an electromagnetic wave. This means that it's made of electrical and magnetic force fields. You can figure out the speed of light, according to theory, just by doing experiments with electricity and magnets! But according to the pre-relativity way of looking at things, light, as a wave, had to be a wave in something, just like ripples are waves in water and sound is a wave in air. So light was assumed to be a wave in a mysterious medium called the ether that was supposed to fill all of space. So this meant that light speed was always relative to this "ether." The speed of light, it was thought, is c=300,000 km/s if you're standing still relative to the ether, but you'd measure a different value if you moved through it. But that meant that electricity and magnetism experiments were related to how you moved relative to the ether, contradicting the assumption that all physical laws are the same, no matter what state of constant motion you describe them from. In fact physicists named Michelson and Morley did an experiment to find how fast the Earth was moving through the ether, and they found that there was no motion at all! Either the Earth didn't move--something that would surprise us astronomers!--or something was wrong with the ether theory. Einstein came up with the idea that our common sense ideas of space and time would have to be adapted so that everyone would always measure the same value for the speed of light in a vacuum. (Actually light can move slower through different materials--it gets slowed down in water or glass, and if something moves faster than light in water, it gives off Cerenkov radiation, which is like a sonic boom, but for light instead of sound. Also, Harvard physicist Lene Hau has slowed down light in her laboratory down to every-day speeds. It may be possible to build a "black hole" on your desk-top with special slow-moving light!) This assumption goes counter to common sense. According to common sense, if you went very fast towards a light beam, you would measure the speed of that light, relative to your motion, as greater than c. But according to Einstein, it's still c! And if you run away from a light beam, it doesn't seem slower, it just gets redshifted, but always at the speed c. The speed of light is the same relative to everyone, not relative to a special "ether" filling all of space. Einstein's theory said that the ether was no longer needed. Following logically from the assumption that you can't catch up or run away from light, Einstein showed that it is not always possible that all observers would agree on whether two events are simultaneous. There is no absolute measure of how much time passes between two events. Someone moving very fast relative to someone else sees that person's clock appear to run slowly. There are sub-atomic particles called muons that have a certain lifetime before they decay into other particles. And yet if these muons move very fast--as they do when they fall to the Earth as cosmic rays--they live a longer time! Their "clock" appears slowed down to us. Likewise, distances are also relative to which state of constant motion they are referred to. General Relativity was developed to make all motion relative, although how much it succeeded is open to debate. Einstein was influenced by a philosophical position called Mach's Principle, which said that even accelerated motion was relative to matter and not absolute. When you're in an accelerating car, you feel yourself pushed into your seat. When you all of a sudden brake hard, you feel yourself pushed to the front. You can't point to anything that causes these forces, except for the fact that you're accelerating. Only according to Mach's principle, the relative acceleration of the rest of the Universe must somehow cause that force! Einstein tried to account for these "forces" an accelerated body feels by a generalized theory of gravity--these forces would actually be gravitational forces. General relativity ended up describing gravity as the result of the curvature of space and time. Sometimes you'll see pictures of a black hole as a grid with a hole in it--that's meant to convey the curvature of space near the black hole. The curvature of space and time mean that normal Euclidean geometry no longer holds when gravity is very strong.
Falling into a black hole		What would it be like to fall into a black hole? Well, it depends on what kind of black hole! There is currently excellent astronomical evidence that black holes really do exist. They are the most conservative possibility in some cases. In other words, if black holes don't exist, something stranger would have to. The two main places in the Universe where we have good evidence for black holes is in stellar remnants and in the centers of galaxies. Stellar remnant black holes: When a star goes supernova, we think in some cases it may leave behind a black hole instead of a neutron star. The black hole can float through space alone, or as part of a binary system (if the star that went supernova had a companion star.) By looking at the Doppler shift of the visible star--knowing the period of the orbit and the velocity, we also know the semimajor axis--we can figure out the mass of the unseen star. One famous "black hole candidate" is called Cygnus X-1, the first X-ray star in the constellation of Cygnus to be discovered. The normal star is an O star, and the object that's probably a black hole has at least 6 times the mass of our Sun. Here is an artist's impression of Cygnus X-1: And here is a map showing where this star system is in the sky in relation to the constellation Cygnus: Cygnus X-1 is so famous that it was celebrated in song by the rock group Rush! This system is thought to have a black hole because (1) the mass of an unseen star in the system is at least 6 times the Sun's mass (we can tell this from the Doppler shift in the spectrum of the normal star)--and this is above the limit for how massive a neutron star can be, and (2) there are very bright X-rays that change brightness quickly. The X-rays can be given off when the gravity speeds up infalling gas, and then from friction it becomes hot and gives off radiation. The X-rays are not from within the event horizon, but from the region around the black hole. In order for X-rays to change brightness very quickly they must be coming from a small region. All this fits with the idea that Cygnus X-1 has a black hole. Another very exciting black hole candidate is called GRS 1915+105; it shoots out jets that appear, by an optical illusion, to be moving faster than light. In the links section, you can see a NASA animation of what may be happening in this system. Galactic center black holes: in the centers of galaxies are probably black holes with millions or billions of times the mass of our Sun. In short: falling in to a stellar mass black hole would kill you before you reached the event horizon, but if you fell into a supermassive black hole in the center of a galaxy, you'd notice nothing strange without looking out your window! If you fell into a stellar black hole, you'd be killed by the tidal forces before ever reaching the event horizon. Assuming you fell feet first into the black hole, gravity would put more strongly on your feet (closer to the black hole) than on your head. You'd stretch out until you snapped and died. Yuck. But the tidal forces are much stronger for a low mass black hole than for a high mass black hole! Why? Well, the force of gravity is G M1 M2/R2. For those of you who know some calculus, the derivative of this with respect to R tells you how much the force changes as distance changes. So to find the difference of force between your head and your feet, you multiply the derivative of G M1 M2/R2 by your height. This is proportional to M2/R3 (let M2 be the mass of the black hole.) Now, as you approach the event horizon, you're approaching the Schwarschild radius Rs=2 G M2/c2. So overall, the tidal force on your body is proportional to M2-2, or in other words, the greater the mass of the black hole (the greater M2), the weaker the tidal force! Even if you went through the event horizon of a high mass black hole where the tidal force is low, you'd still fall towards the very center, where there's a singularity--a point where the density becomes infinite. When you get too close to the singularity, at a point within the event horizon, then the tidal force will pull you apart! Right now we have no evidence for singularities. In fact, it's been theorized that "naked singularities" can never be seen, that every singularity is shielded by an event horizon, which prevents us from ever learning about it. Now, if you were smart enough to convince someone else to enter the black hole for you, you wouldn't have to get killed!
Spinning black holes		Spinning black holes differ in some ways from the garden-variety non-spinning black holes. The event horizon becomes ellipsoidal, for one. The singularity at the center of the black hole becomes a ring instead of a point. The black hole's spin can also drag matter along with it. Matter near a spinning black hole is forced to spin along with the black hole. This is called the "Lense-Thirring Effect" after the people who predicted it. Remember that one of the goals of General Relativity was to implement Mach's principle that all motion is relative. When you spin around, you feel a force pulling your arms away. What causes that force? According to Mach's principle, that force is caused by the relative motion of the rest of the Universe. Because the rest of the Universe is so massive, it "out-votes" you and decides what the state of "rest" is and what the state of "spinning" is. Near a black hole, its proximity and mass give it extra "voting power" to decide what's spinning and what's not! Spinning black holes may or may not be related to the jets seen coming off of stellar black holes as well as the black holes in the centers of galaxies.
Black holes have no hair		Unlike Albert Einstein, black holes have no hair! Well, what does that mean? There's a famous theorem, proved mathematically about black holes, that says that there are only 3 things that black holes can have to distinguish themselves: spin--a black hole can spin charge--a black hole can have electric charge mass--a black hole has mass Nothing else! Black holes don't have magnetic fields--they get cut off when the black hole forms. Black holes don't have bumps. The event horizons are spherical--if a collapsing black hole has a bump, the bump radiates gravitational waves until it disappears. Black holes have no hair! None!
Black holes evaporate: Hawking radiation		Stephen Hawking, one of today's foremost black hole theorists Stephen Hawking, building on work of Jacob Bekenstein, has shown that black holes can actually radiate away their mass! They lose energy as they give off light, until they've lost all their mass and nothing is left! This would take a very very long time for a black hole formed from a star to radiate away its mass--and its radiation would be very faint. Still, if somehow a very low mass black hole existed--say, formed not from a star, but in the Big Bang--it's possible we could detect its radiation. If detected, it would probably earn Stephen Hawking a Nobel Prize. Right now, the Hawking radiation is a daring theoretical idea that combines both Einstein's General Relativity with the theory of quantum mechanics. These two theories have never been melded in a general way. Why should black holes radiate? Well, it was noticed than when you merged two black holes, their total surface area always went up, never went down. (You can use the Schwarzschild formula to prove this!) What else in nature has the same property, that it can increase only and never decrease? A physical quantity called entropy. Entropy can be thought of as a physical system's ability to surprise you. It can also be thought of as a measure of randomness or disorder. For example, my apartment is a mess. You walk in one day, and there are papers on the floor, and another day they're on the chair! The room can surprise you, it's disordered. In a closed physical system, total disorder can only increase. You can work hard to create a little order here and there, but in your effort, you will release excess heat that will contribute to the total disorder of the Universe. So a black hole's area is like entropy. Bekenstein argued that the black hole's area really was its entropy. But that would mean that a black hole would have a temperature, and therefore that it would radiate. That didn't seem to make sense! Stephen Hawking came up with an explanation of how a black hole could radiate in terms of quantum mechanics. In the theory of quantum mechanical fields of matter and energy, even a vacuum, even empty space, isn't entirely empty. It's filled with "virtual particles" that can't be said to have definite existence individually, but whose possibility has a statistical effect on the outcomes of experiments. There is some "zero-point energy" in empty space--some people have looked into ways to tap this energy, although they're often thought of as over-eager by scientific peers. Anyway, this energy in empty space can turn itself into virtual particles. A virtual electron and virtual positron (the antimatter counterpart to an electron) can form out of vacuum energy as long as they soon collide with each other, annihilating themselves and leaving behind the energy of their mass (when matter and antimatter touch, that's what happens!) But if this happens near a black hole, one of the virtual particles can fall in, while the other can escape and become a real particle, at the expense of the energy of the black hole. This is what's responsible for the Hawking radiation! But why should a small black hole radiate so much more than a large one? Well, there's no way we're going to get into the detailed physics of Stephen Hawking's theories, but here's a vague kind of argument that physicists often use when they're in uncharted territory. Let's assume a black hole radiates like a blackbody. (Seems appropriate--it'll never reflect any radiation!) On what factors could its temperature depend? Well, a black hole has no hair. That means it could only depend on the black hole's spin (let's assume we have a garden-variety non-spinning black hole), electric charge (in most real cases electric charge would be 0), or mass. A black hole's mass is related to its radius through Rs=2 G M/c2. So let's say that the temperature will depend on the black hole's radius somehow. How can you relate a distance (like radius) to a temperature? Well, there's Wien's law relating the wavelength of light (a distance) to temperature. So we can make a guess that T(K) = 2.9x106 / RS (nm) (Wien's Law) Actually, if you do it right, you find the temperature depends on the circumference and not the diameter--so this is a factor of 2 pi too big. We're just being approximate here! But it shows that just considering what factors could go into the equation to give the right units, it appears the temperature of a black hole is inversely proportional to its radius. If the Sun were to become a black hole, its radius (given by RS=2 G M/c2) would be 3 km. This means its temperature would be about 10-7 K! Incredibly cold! Stephen Hawking's appearance on The Simpsons