7. BLACK HOLES

If you want to learn more about this fantastic subject, read Gravity's Fatal Attraction (1996: Scientific American Library), by Mitchell Begelman (who is a professor in our Department) and Sir Martin Rees of Cambridge University.

The Relativity of Time: of all the aspects of Einstein's Theory of Relativity, the most wonderful is the realization that time is relative. The very rate at which time elapses depends on the relationship between the observer and the observed phenomenon.

Einstein's 1911 theory of Special Relativity is a theory of force, energy and motion that extends Newton's theory of motion. Newton's theory is great to account for most kinds of motions, and engineers and scientists use it all the time to describe the motion of automobiles, airplanes, rockets, and stars. But Newton's theory of motion becomes inaccurate when we try to describe how objects move at very high speeds -- say, a few percent of the speed of light.

To formulate a theory of motion that included the motion of light itself, Einstein had to abandon the concept of absolute time. According to the theory of Special Relativity, time, as measured by a stationary observer, runs slower for a rapidly moving object. For example, suppose you have a twin brother who takes a round-trip to a star 10 light years away, travelling at 90% of the speed of light and stopping for one year to visit a planet there before he returns. You and your brother synchronize your calendar watches upon departure. You stay on Earth. When your brother returns you check your watches again. You find that you have aged 23.2 years since your brother departed. But, what does your space-travelling brother read on his watch? Using simple formulas from Einstein's theory of Special Relativity, I calculate that he would have aged by only 10.7 years! His watch is not running slow, his time has slowed down. His biological processes have run slower. Your hair has become gray but his has not. Of course, we can't yet test this theory by sending people on rocket trips at 90% of the speed of light. But physicists have tested it by accelerating unstable subatomic particles to speeds very close to the speed of light, and they can see that the rate at which the particles decay into other particles slows down exactly according to Einstein's theory.

In 1915, Einstein took a step further and proposed his theory of General Relativity to extend Newton's theory of universal gravitation. When he proposed his theory, he suggested some observations to test it. One was to detect a slight deviation of the orbit of Mercury. Another was a slight bending of starlight by the Sun, that might be tested by very accurate observations of the positions of stars during a total eclipse of the Sun. Within a few years, scientists made these observations and verified that this theory was correct. (Arthur S. Eddington made the observations during the solar eclipse.)

General Relativity predicts a new effect: gravity makes time run slower

For example, our time on Earth is slowed down compared to time as measured by an observer in outer space where the effect of Earth's gravity is not as strong. For Earth's gravity, the effect is very weak: when we put an atomic clock in space and compare it with one on Earth, the Earth clock seems to be running slow by about 2 parts in a billion compared to the one in space. But atomic clocks are now so accurate that we can measure this difference and see that it is exactly what Einstein's theory predicts. Indeed, there are two atomic clock standards in this country: one is located in Washington, DC, and another one at NIST here in Boulder. But because Boulder is about 1 mile above the sea level, and thus above Washington, DC, the gravity here is a tiny bit weaker than in Washington, and the atomic clock at NIST goes slightly faster than the Washington clock. The difference is really small, the NIST clock will run ahead by 1 second in 160,000 years! But our modern atomic clocks are so accurate, that this difference is noticeable, and scientists at NIST and in Washington must take it into account to keep the USA standard time accurate! General Relativity predicts many fascinating phenomena, the most fascinating of which is a black hole! What is a black hole? Let's imagine that you stand here on Earth and throw a rock straight up. As you well know, the rock will not go into orbit and become a satellite, but rather will fall back on your head. Ouch! But if you mount a rocket engine under your rock however, you can send it into open space. It could not only become the Earth satellite, but might even leave the Earth altogether. To do this though, your rock has to go really fast, more than 11 km/s. This special speed is called the escape speed. This is the speed an object must have in order to overcome the gravitational pull of a celestial body (a planet, a star, a galaxy, etc) and escape into space. Different objects have different escape speeds. For example, for the Sun the escape speed of about 70 km/s, which is 6 times larger than that of the Earth. For a neutron star, whose gravity is incredible strong, the escape speed is enormous, about 60% of the speed of light! Now, you are ready to make a guess what a black hole is... Suppose that a neutron star had a mass equal to 1.4 solar masses and a radius of 4 km rather than 10 km (its actual radius). Then the escape speed of such a neutron star will then be equal to the speed of light! - this means that even light itself will not be able to leave this object! We call such an object a black hole (The name "black hole" was initially invented by an astronomer who did not believe in black holes, just to make fun of the idea). General Relativity (GR) predicts that black holes have really weird properties. Inside the black hole nothing, even light, can move outwards. That means that no pressure can hold such an object from collapsing upon itself - in order for atoms in a black hole to stand still, they would have to move faster than the speed of light. General Relativity then predicts that anything inside a black hole will fall to its center and will get compressed to infinite density and zero size. Well, we know that there are no infinities in the real world, everything is finite. That means that General Relativity itself becomes invalid in this state of "infinite density" - we need a new, even more general theory of gravity that includes General Relativity as a subset, just like GR includes mechanics of Newton as a subset. We don't have such a theory yet. Many scientists are currently working on it, they even invented a name for it: "Quantum Gravity", but, frankly speaking, they have not moved much beyond inventing a name. Even if we don't know what happens at the very center of a black hole, we know that everything will be squeezed there to very high density, the gravity becomes unimaginably strong, and any object will be torn into pieces by such a strong gravity. Even atoms will be squashed and destroyed at the black hole center. We call such a place a singularity. Again, at least we have a name for it. Because nothing can stand still inside a black hole, the black hole is actually empty (well, it is a hole, after all!). Everything that ever falls into it, goes right away into a singularity. And the surface that surrounds the black hole, its "surface", is equally empty. It has a special name, it is called an event horizon. Imagine that you fall into a black hole. As you cross the event horizon, you will notice nothing, but once you crossed it, no matter what you do, there is no way back - your fate is predetermined, you will die at the singularity squashed by gravity into something smaller than the smallest atom.

General Relativity predicts even stranger things too. The greater the gravity, the more time slows down. This time dilation causes the frequencies of photons emitted by atoms on the surface of a white dwarf stars to be reduced by about 1 part in 1000. We observe this as a gravitational redshift of the spectral lines. The gravitational redshift on the surface of a neutron star is much greater -- say, 40%. But at the event horizon the redshift is 100% - any photon, be it visible light, a gamma-ray, or radio, gets redshifted to zero frequency, and thus becomes undetectable. And this is not the end of the story! For someone outside a black hole time appears to stop at the event horizon (well, it only appears, it does not actually stop of course). Imagine that you fall into a black hole, and your friend orbits above you in a spaceship. You will quickly fall into the singularity and die, but your friend will see a very different picture... For her, you never cross the event horizon but appear to hover just above it for the rest of eternity! Again, this is just an illusion, you will indeed fall through and die, but your image - the light that reflects off your body right before you fall through the event horizon - will take an infinite amount of time to get from the horizon to your friend in the spaceship. This is a spooky consequence of the fact that light will have to fight against the incredibly strong gravity of a black hole. Black holes are a very fascinating subject, but in this course we can only meet them for a very short time. If you want to learn more about black holes, the APS department has a special course on them - ASTR2020. Historically, the radius of the event horizon (often called "the size of the black hole") is named after Karl Schwarzschild, who found the solution of Einstein's equations describing a black hole (Schwarzschild himself did not use this term, he called such an object a "collapsed star"). The Schwarzschild radius is given by the simple formula R_S = 3 km (M/M_Sun). Thus, for a collapsed star with M = 16 solar masses, the black hole would have a radius of 48 km. If M = 50 million Suns, the black hole radius would be equal to 1 AU (the Earth's distance from the Sun).

What kinds of stars end up as black holes? They are the natural consequence of the evolution of massive stars. As we discussed at the beginning of this lesson, neutron stars have an upper mass limit of 2 - 3 solar masses. A collapsed object of greater mass will continue to collapse indefinitely, making a black hole. Evidently, if a star is massive enough to begin with, this will happen. Either the core of the star weighs more than about 3 solar masses when it begins to collapse, in which case a black hole will form directly; or enough additional matter falls onto a newborn neutron star so that its mass increases above the maximum limit. How massive must a star be to become a black hole rather than a neutron star? We're not sure, but we suspect that the star must have initial mass greater than about 30 Suns. Does a supernova explosion accompany the formation of a black hole? We're not sure. Perhaps the collapse to form a black hole is a rather quiet event compared to the formation of a neutron star. On the other hand, some astrophysicists suspect that the formation of a black hole will give rise to a gamma ray burst.

Here's a brief summary of the final fates of stars:

How do we detect black holes? Isolated black holes, the result of the collapse of a single massive star, are too small to detect with any telescope that we can conceivably build with present technology. There are probably millions of black holes in the Milky Way, weighing between 3 and 30 solar masses, but we don't know how to detect them.

But, if a black hole is orbiting another star in a binary system, the companion star might be swelling up and pouring gas into the black hole. As the gas falls toward the black hole, it swirls into a rapidly rotating accretion disk, as illustrated here: Artist's conception of Cygnus X-1. The inner part of the disk heats up to 10⁷ - 10⁸ K, hot enough to radiate X-rays. Most of the X-rays come from the part of the disk that is a few times the Schwarzschild radius away from the black hole. (Radiation might be emitted by the gas after it falls within the Schwarzschild radius, but it cannot be seen from outside.)

Today, astronomers have detected several binary X-ray systems containing black holes. We are confident that these objects are black holes because we can determine their masses from the orbital motions of their companion stars (just as we inferred the masses of extra-solar planets from periodic Doppler shifts of their companion stars). We have found several such systems, as described in the table below. We can tell that the X-rays come from a region smaller than the Earth because they vary on timescales shorter than the time it takes light to cross the Earth's diameter (0.04 seconds). The massive object must be smaller than that. There is no known force that can prevent such an object from collapsing. It will form a black hole in less than 0.1 seconds.

Distortion of light around a black hole: The gravity near a black hole is so great that it can deflect light by large angles, resulting in bizarre optical effects. For example, we can see stars that are actually behind a black hole because their light is bent around the hole, as illustrated here:

Notice that we would see each star as a double image, one on each side of the black hole. The net result is that the black hole seems to push the stars aside, as illustrated at the top of this page. This is an example of gravitational lensing. Although we don't have the technology to see this effect around stellar mass black holes, we will see more evidence of this phenomenon when we discuss distant galaxies.

We might, however, observe the effects of black hole lensing indirectly through observations of the spectrum of X-rays from the inner accretion disk. Just as the light from stars is deflected by the black hole, so is the X-ray radiation from the accretion disk. If we see the accretion disk nearly edge-on, the image of the side of the disk on the far side of the black hole will be doubled, as illustrated in this Animation of accretion disk around a black hole. (source).

Falling into a black hole: If you were falling into a black hole, you would experience quite a bizarre trip. The details of what happens are fascinating but complicated. If you want to learn more, you can find a good introduction and some excellent movies in Approaching the black hole by Prof. Andrew Hamilton of our Department. See also Virtual Trips to Black Holes and Neutron Stars by Robert Nemiroff.

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Last modified March 7, 2002
Copyright by Richard McCray