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General Relativity

The Kerr Black Hole

In astrophysics, everything within the universe—planet, star, and galaxy—spins. This is true of the black hole as well. A star that collapses to form a black hole give part of its angular momentum to the black hole. A black hole in a binary star system acquires angular momentum as it pulls the atmosphere from its companion star onto itself. The massive black hole candidates we see at the centers of galaxies are surrounded by accretion disks that dump both matter and angular momentum onto the black holes.

A “spinning” black hole is called a Kerr black hole; it is described by only two physical properties: mass and angular momentum. The Schwarzschild black hole, which has zero angular momentum, is a special case of the Kerr black hole. The amount of angular momentum a black hole may carry is limited by the black hole's mass; the maximum magnitude of the angular momentum is proportional to the mass.

A Kerr black hole's gravitational field has an axis of symmetry that can be thought of as an axis of rotation. The angular momentum vector is in the direction of this axis. The gravitational field is also mirror-symmetric about an equatorial plane that contains the black hole's center. The angular momentum vector is perpendicular to the equatorial plane. As the angular momentum goes to zero, the gravitational field becomes spherically-symmetric.

Like the Schwarzschild black hole, the Kerr black hole has an event horizon. The radius of the event horizon is dependent both on the mass and the angular momentum of the black hole. For a given mass, the circumference of the event horizon is at its maximum for zero angular momentum. The circumference falls as the magnitude of the angular momentum rises. At the maximum angular momentum, a minimum circumferences of half the maximum circumference is found.

The striking feature of a spinning black hole is that the gravitational field pulls objects around the black hole's axis of rotation. This effect, called frame dragging in the jargon of general relativity, prevents an accelerating observer close to the black hole's event horizon from holding a fixed position relative to the stars. Regardless how much he accelerates, the observer is incapable of stopping his motion around the black hole, although he can keep a fixed distance above the event horizon. The boundary surrounding the black hole that separates the space where an accelerating observer can remain static with the distant stars from the space where no amount of acceleration can keep an observer at a static location is called the static limit. This boundary touches the event horizon at the poles, but it extends much farther out than the event horizon away from the poles, reaching its maximum radius at the black hole's equatorial plane. The volume enclosed between the event horizon and the static limit is called the ergosphere.

In astrophysics the effect of a black hole's angular momentum would be seen on an object falling onto the black hole or in orbit around the black hole. If we drop an object onto a Kerr black hole, it would spiral around the black hole's spin axis as it falls. The only way to fall in a straight line into the black hole is to fall along the axis of rotation. Traveling along any other line, an object spirals down a cone whose apex is at the black hole's center and whose axis is the black hole's axis of rotation. The direction of motion is set by the direction of the angular momentum vector; a dropped object always travels counter-clockwise around the angular momentum vector. As an object approaches the event horizon, its radial motion slows to a crawl, but its orbital motion goes to a constant rate, so that the object appears to be stuck to a rotating event horizon.

The counterclockwise rotation of an object free-falling from rest at infinity onto a Kerr black hole is matched by the same counterclockwise rotation of an object thrown way from the black hole is such a way that it comes to rest at infinity. This counterclockwise spiral for both infalling and outgoing objects means that when an object falls onto a Kerr black hole, we see it spiral inward, because the light that escapes from the object back to us spirals outward with the same handedness.

Orbits around Kerr black holes are generally complex. As with the Schwarzschild black hole, an orbit around a Kerr black hole is not closed. Unlike an orbit around the Schwarzschild black hole, a Kerr black hole orbit is not generally confined to a plane. The only orbits confined to a plane are those on the equatorial plane. Orbits out of the equatorial plane move in three dimensions. These orbits are confined to a volume that is limited by a maximum and minimum radius and by a maximum angle away from the equatorial plane.

Circular orbits exist around Kerr black holes, but they are confined to the equatorial plane. These orbits are very different from the circular orbits around a Schwarzschild black hole, because, while there are an infinite number of circular orbits around a Schwarzschild black hole of a given radius, all with the same orbital period, there are only two circular orbits of a given radius around a Kerr black hole, one clockwise and the other counterclockwise around the black hole's axis of rotation, with the period of the counterclockwise orbit shorter than that of the clockwise orbit.

One consequence of the split in clockwise and counterclockwise orbital periods is that the radii of the last stable orbit is different for the clockwise and counterclockwise orbits. As with the Schwarzschild black holes, the last stable orbits for a Kerr black hole are the circular orbits closest to the event horizon. Objects in orbit inside of the last stable orbit fall onto the event horizon. The last stable counterclockwise orbit is closer to the event horizon than the last stable clockwise orbit.

As with the Schwarzschild black hole, the Kerr black hole is the vacuum gravitational field far from a spinning body. This means that the frame dragging that is so striking close to the black hole should also appear around any body that rotates, such as Earth. The effect can in principle be seen with a gyroscope, because the frame dragging causes the orientation of a gyroscope to change. As with measurements of the deflection of light passing the Sun, measurements of frame dragging around the Earth or Sun only test general relativity in the limit of weak gravitational fields. The effect, however, is so small that it is extremely difficult to measure. So far no instrument has measured the effect. A NASA satellite called Gravity Probe B, which is designed to measure the effect, just completed gathering data in the first quarter of 2006 after more than 17 months in space. Project scientists expect to know if they have successfully measured the frame dragging caused by Earth's rotation in a year's time.

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