II) Black Holes

One of the most exciting subjects in the study of gravity, and in particular General Relativity, is the topic of black holes. It is interesting to note that the concept of a black hole was not first introduced by General Relativity.

As long ago as 1784, Rev. John Michell (remember he built the device that Cavendish used) suggested in a paper that if light was a particle, it could not escape from a sufficiently massive star. Twelve years later, in 1796, the French mathematician Laplace conjectured that a sufficiently massive object might have an escape velocity greater than that of light:

"A luminous star, of the same density of Earth, and whose diameter should be two hundred and fifty times larger than that of the Sun would not, in consequence of its attraction, allow any of its rays to arrive at us; it is therefore possible that the largest luminous bodies in the Universe may, through this cause, be invisible."

These are objects so massive and so dense that nothing can escape! Their escape velocities are so high that even light, moving at the absolute speed limit, cannot escape the gravitational pull of a black hole. This is where a black hole gets its name: if light cannot escape, then it is truly black! Black holes and General Relativity are strongly linked concepts. They are usually spoken in the same breath. Nevertheless, we can understand many of the important ideas about black holes just with a straightforward application of Newtonian gravity.

To get a feeling for how gravity works around a black hole, let's start with an example about the Earth. We can calculate the escape velocity of Earth as 11.2 km/s (for a detailed calculation of this escape velocity see later in the section on Gravitational Potential Energy). What would happen to the escape velocity if we compressed the Earth to half its present size? Substituting known values into the formula for escape velocity we get

In the case of the Earth the result we get is impossibly small. No force in the Universe is going to collapse the Earth into a ball less than an inch across! But what about other objects? What is the Schwarzschild radius of the Sun?

During its life, a star is held up against the inward force of gravity by the high pressure pushing outward from its center. This pressure is caused by the release of energy from nuclear fusion. Without this energy source, the star would collapse. For a very massive star, at the end of its life, this energy source fails spectacularly when the fusion in the star's core produces iron instead of light elements, such as helium. Fusion cannot produce elements heavier than iron without energy input, and so no new energy can be released. Eventually (in a fraction of a second), the pressure is not sufficient to support the star against gravity, and the star collapses. During this collapse, the density increases dramatically. For intermediate mass stars, it is possible that the collapse can be halted by neutron pressure. If this happens, the star becomes a neutron star. If the star is massive enough, however, even neutron pressure is not sufficient to stop the collapse, and the star shrinks indefinitely until a black hole is formed.

A) Spaghettification

Imagine that an astronaut decides to visit near a black hole. The black hole is 3 km in radius, so to be super cautious the astronaut decides not to venture closer than a hundred times this, or 300 km. Let's calculate a few things. Using the equation for Schwarzchild radius, the mass of a 3 km radius black hole is M= 2x10³⁰kg. At a distance of 300 km, the gravitational acceleration is

This is 1000m/s² larger than the acceleration experienced by the astronaut's head. So even if the astronaut could balance the accelerations on the rest of his body by orbiting, his feet would be pulled by the equivalent of 100 earth gravities! The unfortunate astronaut would be ripped apart! For obvious reasons, this process is called spaghettification...

This differential force is called the tidal force and occurs everywhere in nature, not just near black holes. When two objects orbit each other, the gravitational force is stronger on the sides facing each other than on the far sides. But the gravitational force is balanced by the orbital motion (i.e. the motion of the objects revolving around each other) only precisely at the center of the objects. Thus, the forces are unbalanced both on the near sides and the far sides. In the case of the Earth and the Moon the result on Earth is the ocean tides. Water on the Earth piles up on the sides close to and far away from the Moon (but only by a little bit, a couple of feet typically; see figure below).

But the tidal force can have much more major consequences. Just as in the case of the astronaut, there exists a critical distance between two massive objects where this tide-raising force is sufficient to tear the smaller object apart. This distance is known as the Roche Limit, after its discoverer Edouard Roche (1850), a French mathematician. A beautiful example of this was the disruption of comet Shoemaker-Levy 9 as it rounded Jupiter for the first time. Before the tidal forces from Jupiter acted, the comet was all in one piece. But when it approached the Roche Limit of Jupiter, it broke apart, as seen in this image taken by the Hubble Space Telescope.

B) Time Dilation

So far, we have simply described a place where the force of gravity is very, very strong. What's so special about black holes? Why are these objects so mysterious?

In 1916, Einstein explored the behavior of time near very massive objects. He was able to show that the more massive the object, the greater the effect on time. Close proximity to the intense gravity of a black hole is the perfect place to warp time itself. Let's follow an astronaut falling into a stellar mass black hole (stellar mass black holes are less massive and more abundant than supermassive black holes). Falling from a thousand kilometers away would take only perhaps a hundredth of a second... or would it? One would expect that the huge gravity of the black hole would drag the astronaut into the black hole in the twinkling of an eye. And this is exactly what happens, from the viewpoint of the astronaut. Before she can even blink, she is pulled into the black hole past the event horizon and falls to the singularity (or center). But the very strange thing is that this is not what a observer at a safe distance would see at all! To someone far away (observing with a telescope perhaps), the astronaut would take an infinite time to fall into the black hole!! As the astronaut falls closer and closer to the event horizon she would appear to slow down, and finally halt just before falling beyond the event horizon. She would be frozen in time for all eternity! At the same time, however, the astronaut's image would get fainter and fainter and redder and redder until, just before reaching the event horizon, they would fade from view entirely!

How is this possible? Basically, it is possible because "clocks run slow" in a gravitational field. We don't mean just physical clocks, but really all physical processes. That is, time itself slows down deep in gravitational fields. This effect has actually been measured here on Earth! Of course, on Earth gravity is not nearly as strong, as in a black hole, and so the effect is incredibly tiny. But near a black hole, the effect is huge and, in fact, infinite at the event horizon. This slowing down of time also explains why the astronaut's image gets faint and red. The astronaut is only emitting a finite number of photons every second. But as the time gets slowed down the number of photons per second that we receive decreases because one second for the astronaut is perhaps a hundred (or more) seconds for us. As the astronaut gets closer to the event horizon, the photons get spread out over a longer and longer time for the outside observers. Light also has a frequency, or color. Let us imagine that the astronaut is running slow by a factor of 100 times. Then a photon of frequency say 7.5 x 10¹⁴ Hz (blue), would be seen far from the black hole as a photon of frequency 7.5 x 10¹² Hz. That's far into the infrared!

C) Singularities and Event Horizons

While black holes are very weird objects, they are perhaps not the strangest consequence of General Relativity. Stranger still are the singularities hidden deep in the heart of every black hole. Singularities are regions of space where the intense gravity has crushed the star into a single point. The density of matter becomes infinite, and the very concepts of matter, space and time lose their meaning at this point. They are formed where matter is crushed literally "out of existence" by its own gravitational field. In their vicinity, time travel becomes possible and the laws of physics break down completely. Luckily for us, surrounding the singularity is the event horizon. This can loosely be thought of as the surface of the black hole. But it isn't a solid surface that one could land on. The event horizon is the point of no return. If one is above it one can still escape the black hole. If one falls below it then return to the outside world is impossible. This is actually a very good thing because otherwise a brave enough astronaut could enter a black hole, travel in time around the singularity, and then return to cause trouble here on Earth. As it is, once the astronaut enters the black hole he can travel in time but can never get out of the black hole again.

So, even if I have fallen beyond the event horizon, can't I stop my fall to the singularity, perhaps just hovering above the singularity forever? No. The reason has to do with the nature of space and time. Remember that we said that gravity warps space. This is technically wrong. Gravity warps spacetime, not space alone. This is the reason for the slowdown of time as we approach a black hole.

As we cross over the event horizon time and space swap roles. The spatial direction towards the singularity becomes timelike. That is to say that we are swept along it. The singularity becomes part of our future. Just as it is impossible to avoid one's 40th birthday, one can't avoid the singularity of a black hole. Just as time inexorably marches on, so too would one fall into the singularity.

D) The Existence of Black Holes

Black holes are intriguing theoretical constructs, but do any actually exist? Yes, we have found a handful of objects in the Milky Way galaxy that are very likely black holes. We think they are black holes for two reasons. First, they are very massive, and yet they don't emit light like a normal star. This alone is suggestive. Equally important, however, is that high energy x-rays have been detected coming from these objects. X-rays are generally associated with very hot energetic objects. What could be massive, very hot and yet emit no light? Certainly not a normal star. We believe that these objects are black holes that are sucking in matter. As the matter falls into the star it gets squeezed into a smaller and smaller space. The friction between all this matter, along with the compression, heats it up to tremendous temperatures, high enough to emit x-rays. We refer to these types of black holes as stellar black holes, because they are created as the result of the death of some very massive stars.

Much more massive black holes, millions and even billions of times the mass of the Sun, have been found lurking in the hearts of galaxies. These supermassive black holes are not well understood but are thought to be involved in the formation of galaxies many billions of years ago.

Proceed to Content Background III) Mass, Weight and Gravity at Earth's Surface: Key Ideas