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General Relativity

The Schwarzschild Black Hole

The gravitational field of a Schwarzschild black hole depends on only one constant: a mass. Around this central mass, which is assumed to exist at a single point in space, the gravitational field is isotropic, so it varies only with the distance from the central mass.

The principal features of a Schwarzschild black hole—time dilation, a gravitational Doppler shift, the bending of light, and the presence of an event horizon—arise because of the equivalence principle. All of these effects are seen under special relativity by a traveler accelerating at a constant rate; the difference is that around a black hole, an accelerating observer can remain motionless relative to the black hole. Over distances much smaller than the observer's distance from the black hole, the observer sees physics that is identical to that seen by an accelerating traveler in special relativity. An observer hovering over a black hole sees the radiation arriving from above Doppler shifted to higher frequencies, and he sees the radiation arriving from below Doppler shifted to lower frequencies. He sees the clocks of observers hovering farther way from the black hole's center as moving more quickly than his own, and he sees the clock of those closer the the black hole's center as moving more slowly. He sees the path followed by light bend, so that objects at the same distance from the black hole as himself appear to be above him. The only thing missing is the event horizon.

In special relativity, the distance from an accelerating traveler to the event horizon is inversely proportional to the strength of the observer's acceleration. Far from a black hole, the acceleration of an hovering observer is inversely proportional to the square of the distance to the center of the black hole. We expect the event horizon for a black hole to appear at approximately the place where the acceleration experienced by the hovering observer produces under special relativity an event horizon distance equal to the distance from the black hole center. This approximation in fact holds, for the precise radius of a black hole's event horizon is r = 2GM/c2, where the radius r is defined by the circumference of the event horizon, G is the gravitational constant, M is the black hole's mass, and c is the speed of light. This radius is called the Schwarzschild radius. For a solar mass black hole, the event horizon has a radius of only 3 km. For a one billion solar mass black hole, which is about the maximum mass expected for a black hole at the center of a galaxy, the radius is 20 AU, or about the distance of Uranus from the Sun. In other words, black holes are small.

You must have noted the strange phrasing for the radius of the black hole's event horizon. In general relativity, space ceases to obey Euclidean geometry. This means that if I took a ruler, measured out a radius, and then measured the circumference of a circle having that radius, I would not get the Euclidean relationship C = 2 π r, but some other relationship that depends both on my acceleration and on the variation of the local gravitational field. The value of the radius from the black hole's center is normally defined from the circumference of a circle centered on the black hole. With this definition of radius, if one moves from one radius to another, one cannot derive the distance traveled by simply taking the difference of the two numbers. The actual formula for the distance traveled depends on the mass of the black hole.

In nature, objects orbit black holes; they don't hover over them. When an object orbits a star in a circular orbit, its speed increases as the radius of the orbit decreases. Where does this speed equal the speed of light? For a black hole, it equals the speed of light at a radius of 1.5 times the Schwarzschild radius. The circular orbit at this radius is called the last stable orbit of the black hole. Outside this orbit, matter orbits the black hole at a speed less than speed of light. At this orbit, only light can orbit the black hole; matter place in this orbit falls into the black hole. Inside this orbit, even light falls to the event horizon.

The existence of the last stable orbit implies something very unusual about black holes: light passing by a black hole can orbit it several times before escaping to a distant observer. This is a much more dramatic effect than the gravitational lens associated with stars and galaxies. The bending of light by a star can only produce two images of a more distant star, but a black hole can produce an infinite number of images. These images are created in pairs, with the first pair corresponding to the images produced by a star. The next pair of images is created when the light passing on either side of the black hole completes a single orbit around the black hole before traveling to the observer. A third pair is created when the light completes two orbits of the black hole. This progression continues to infinity, with each image pair smaller on the sky than the previous pair. Because the pairs of images become smaller with each orbit, the amount of light reaching the observer from a star becomes smaller with each image, which keeps finite the total amount of light contained in this infinity of images. Perhaps, rather than as “black holes,” these objects should be known as “black crystals,” for their affect on distant stars is similar to the multitude of images created by a piece of cut crystal.

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