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General Relativity

In Astrophysics

The equivalence principle, an extension of the mass-independent acceleration of Newtonian gravity, provides general relativity with its fundamental effects on light propagation. In addition to the effects associated with the equivalence principle, general relativity adds three more features not found in Newtonian gravity: under the principle that gravity appears as the curvature of space-time, the radius-volume and radius-circumference relationships of Euclidean geometry no longer hold in a gravitational field; gravitational fields in general relativity can propagate as waves moving at the speed of light; and the static solution of general relativity is a black hole, meaning that it has a static event horizon. Usually these traits of general relativity are invisible in astrophysics; for most problem, Newtonian gravity is entirely adequate. There are only six cases in astrophysics where the effects of general relativity appear. Three of these are generally weak effects that are observable within the Solar System: the drift of a planet's perihelion, the gravitational bending and focusing of light, and the gravitational Doppler shift of light. The remaining cases are on the edge of detection: gravitational waves, black holes, and the curvature of space-time within our expanding universe.

Gravitational Doppler Shift

In special relativity, an observer accelerating at a constant rate sees light falling to him as blue-shifted in frequency, and he sees light rising to him as red-shifted in frequency. The equivalence principle says that acceleration in a gravitational field produces the same effect, so an observer standing on Earth should see light falling from above as blue-shifted. This test was first performed in 1960, and the results are as expected under the equivalence principle.

In astronomy, the gravitational redshift is a mild effect that is usually lost in the Doppler shift from the motion of the gas emitting the light and from the thermal motion of the electrons and ions producing the light. The gravitational redshift has been seen in some white dwarfs, although the effect is a redshift of less than a factor of 10-4 in frequency of the spectrum. The redshift expected for radiation from neutron stars is larger, about 15% in frequency. For neutron stars in binary systems, this redshift is mixed with the red-shift from orbital and free-fall motion, but for isolated neutron stars, the gravitational redshift should be untainted. So far, lines have not been seen from the surface emission of isolated neutron stars, but the hope persists.

Gravitational Bending of Light

An early test of general relativity was the bending of light rays by the Sun. From the equivalence principle we know that light must travel on a curved path by a gravitating body, but the magnitude of the effect depends in part on the curvature of space around that body. The Sun is massive enough to produce an observable bending of starlight; under general relativity, starlight passing the limb of the Sun is deflected 1.75." Astronomers sought to see this effect by measuring the positions of stars around the Sun during a solar eclipse. This effect was measured by several teams of astronomers during the May 29, 1919 eclipse, although the accuracy of that experiment is now questioned. Today the same effect can be done by measuring the path followed by radio waves reflected off of Venus or emitted from a spacecraft on the far side of the Sun.

The weak gravitational fields of stars, planets, and galaxies bend light by very small angles. Under most circumstances, we cannot see this effect, but for very distant objects, where this small angle is comparable to the angles separating objects on the sky, we see this effect with regularity. The two instances where the gravitational bending of light is important is in the appearance of the most distant galaxies, which can be affected by galaxies and clusters of galaxies between us and them, and in the appearance of stars in nearby galaxies, which can be affected by intervening brown dwarf stars in our own galaxy.

If a cluster of galaxies lies between us and a high-redshift galaxy, that cluster will act as a lens, splitting the image of the distant galaxy into multiple images. The number of images depends on how the mass in the cluster is distributed. For example, a spherically-symmetric galaxy cluster can produce three images if the the cluster is slightly off the line running from the lensed galaxy to us. One image will come from the light that passes almost undeflected through the center of the cluster, a second image will come from light deflected towards us as it passes to the right of the cluster, and a third image will come from light deflected towards us as it passes to the left of the cluster. Many multiple-image galaxies are known. By studying these galaxies, we can gather information about the structure of the intervening galaxy cluster.

Dwarf stars within our own galaxy can also cause multiple images, but they are too close to the lensing star to separate with a telescope. But gravity also can cause the images to appear more luminous. A distant star can therefore appear to grow luminous as an dim intervening star passes between it and ourselves. This effect has been observed by several groups, and it is being used to estimate the density of dim stars in our galaxy.

Perihelion Drift

A second early test of general relativity is influence of the curvature of space on the orbit of a planet. The curvature of space by gravity causes a breakdown in the relationship between area and circumference. This effect changes the orbit of a planet from the closed ellipse of Newtonian gravity to an orbit that does not close on itself. In effect, in the time it takes a planet to travel 360° around the Sun, it does not complete the full excursion from aphelion to perihelion and back.

The strength of this effect depends on distance from the Sun; the closer a planet is to the Sun, the stronger the perihelion drift. Mercury, the planet deepest in the Sun's gravitational potential, exhibits this effect most strongly. With every orbit around the Sun, Mercury's perihelion shifts by 0.1038". This effect is so strong that it was observed before general relativity was developed. Venus and Earth also have perihelion shifts that are strong enough to observe, predicted at 0.058" and 0.038" per orbit, as does the asteroid Icarus, with a perihelion sift of 0.115" in general relativity. All of these perihelion shifts are observed.

Gravitational Waves

When a gravitating body is accelerated in general relativity, its gravitational field must change throughout space. But the change to the field can only propagate at the speed of light. This is a property that general relativity shares with electromagnetism, and as with the electromagnetic field, the change in the gravitational field propagates outward across space as a wave.

There is one place in this universe where we clearly see the radiation of gravitational wave: compact binary star systems. As a two stars orbit each other, the binary system emits gravitational waves that carry energy way from the system. Over time, the energy loss causes the stars to spiral closer together. While not the only mechanism for losing energy, gravitational radiation is the dominant mechanism in very compact systems containing two neutron stars. In binary pulsar systems, this energy loss has been measured and shown to be consistent with the losses expected from gravitational radaition.

The other side of the gravitational wave problem, the detection of gravitational waves at Earth, has yet to be accomplished. Several machines of various design are currently attempting to detect these waves, and several new machines of greater sensitivity are under development. (Continue on the Gravitational Waves survey path.)

Black Holes

The event horizon is not a consequence of general relativity, but of special relativity. If I accelerate at a constant rate in a rocket, an event horizon forms behind me. The event horizon is simply an abstract boundary that separates light that can reach me from light that cannot; no special physics occurs there.

Inevitably in any theory of gravity, we must have a static, spherically-symmetric gravitational field. After all, that is what we have here on Earth. For an observer sitting at a fixed point in such a field, where his length is small enough to make the tidal force negligible, the acceleration he experiences is identical to the constant acceleration he would experience in special relativity. But in special relativity, an observer has an event horizon below him; must this event horizon also exist in general relativity? If the distance to the event horizon is much shorter than the distance to the center of the gravitational field, so that the distance to the event horizon is much shorter than the distance that the tidal force is strong, there must be a static event horizon. This occurs in general relativity, giving the theory a static solution that is a black hole.

An event horizon implies a second feature not found in Newtonian gravity: a radius of last stable orbit. In special relativity, when an observer is accelerating at a constant rate, the light he emits inevitably falls to the event horizon. In particular, if he shines light parallel to the event horizon, it will bend and fall onto the event horizon. But far from the event horizon, where the gravitational field resembles the Newtonian gravitational field, light will travel in a nearly straight line. These two limits imply that at some radius, light emitted perpendicular to the event horizon orbits bends just enough as it travels to keep a constant distance from the center of the black hole. Such a radius exists around the black hole of general relativity, and it is called the last stable orbit; at larger radii, objects with mass can orbit the black hole, but inside this radius, light objects with mass fall to the event horizon.

The most striking feature of a black hole is the infinite number of images that they create of all the objects surrounding them. The reason for this is that light from a point in space can orbit the black hole many times before escaping to the observer. Depending on the angle of emission, light from an object can orbit the black hole once before escaping, it can orbit twice before escaping, or it can orbit a dozen times before escaping. Each photon path will appear as a distinct image to the observer. The images created after many orbits are quite dim, so most of the light that reaches the observer comes from only several images.

Black holes are the ultimate end in general relativity for the most massive stars. Black hole candidates are found in compact stellar binaries, were they are several times the mass of the sun, and on a much larger scale at the centers of galaxies, where they can be a billion times the mass of the sun. These objects are massive and compact, and assuming that general relativity is correct, they must be black holes. But are they black holes? The difficulty in proving the existence of black holes is in developing a diagnostic for the event horizon. The problem is that nothing happens physically at the event horizon. The only physics that is unique to the black hole is at the last stable orbit and the creation of an infinite number of images. The last stable orbit will have an orbital period associated with it that is dependent on the mass of the black hole. The multiple images created when light is bent by the black hole, is more a complication that must be accounted for than a diagnostic.

Cosmology

Over short distances within our expanding universe, we do not see the effects of general relativity. We can describe the expansion of the universe near our galaxy with Newtonian gravity alone. As we look farther away from our galaxy, however, we begin to see the curvature of space-time caused by the mass within the universe.

At its most extreme, gravity provided by the matter in the universe can cause the universe to stop expanding and than collapse upon itself. The curvature of space-time in such a universe is so severe that the volume of space in the universe is finite. At the opposite extreme, a universe with virtually no mass, the universe expands forever. In such a universe, there is no curvature of space-time, so the universe expands to infinity, and the volume within a fixed radius is set by Euclidean geometry. Between these two extremes is the universe with just enough mass to slow the expansion of the universe for all time without causing collapse. The galaxies moving outward from us in such a universe are similar to a spacecraft leaving Earth at precisely the escape velocity: as the galaxy moves to infinity, its velocity away from us goes to zero. This universe with the closure density does not have a volume at a fixed radius that obeys Euclidean geometry.

The curvature of space-time within our universe can be observed if we are able to see far enough out, where the gravity of the universe is sufficiently strong to decelerate the galaxies. This effect is seen in the curvature of space in two ways: in the number of galaxies out at high redshift, which is a proxy measurement of the volume versus radius, and in the apparent diameter of a galaxy at a given redshift, which is a proxy measurement of circumference versus radius. These effects are difficult to separate from other characteristics displayed by galaxies, such as the changes in their character with age of the universe, so they are more effects that must be included in an analysis of galaxies than diagnostics of general relativity.

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