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General Relativity

Gravitational Lenses

Light Bending

A gravitating object inevitably bends the trajectory of light under the equivalence principle; the only issue in a post-Newtonian theory of gravity is the amount of bending that occurs as light passes by an object. In general relativity, an object with a weak gravitational field—an object such as a main-sequence star or aplanet,in contrast to a neutron star or a black hole—bends light by the angle α, which is given by the following formula:

α = 4 GM/c2b

In this equation, G is the gravitational constant, M is the mass of the object, b is the minimum distance between the light path and the object, and c is the speed of light. Because the light path bends by an angle that is inversely proportional to the closest approach, light just skimming the surface of an object experiences the largest angle of deflection.

Light-rays are bent at the limb of a star like the Sun by very small amounts; Light passing by the limb of the Sun is deflected by 1.75 arc seconds. From Earth, this bending is seen as a slight displacement of a star's position on the sky towards the Sun. This effect is subtle at Earth, but if we move back from the Sun, so that the size of the Sun on the sky is smaller than 1.75," we see an interesting and pronounced effect: the region around the Sun acts as a lens that magnifies the light from more distant objects.

Gravitational Lens

To see why the gravitational field around a star can act as a lens, let us consider a distant source behind the Sun and on a direct line with the Sun and Earth, a star for instance. Compared to the distance of the Sun from the Earth, a star is effectively at an infinite distance from the Sun. The light traveling from the star directly to Earth is blocked by the the Sun. The light from the star that passes the limb of the Sun is deflected by 1.75,"which is too small of an angle to send it to Earth; to reach Earth, the starlight must be deflected by an angle equal to the ratio of the Sun's radius to Earth's distance from the Sun, which is 15' 59.6." If we pull back from Earth, however, the angle that the starlight must be deflected to reach us decreases; pulling back to 570 AU, the required angle of deflection equals the deflection angle at the limb of the Sun, and we suddenly see the star. But where do we see it? The light bends around the whole limb of the Sun, so we see the star as a ring around the limb of the Sun; this ring is iconically called an Einstein ring. Like a piece of glass, the gravitational field of the Sun distorts the image of the star.

Like any lens, we can define a focal length for a gravitational lens (The focal length of a lens is the distance from the lens to the point where an image forms of an infinitely distant object). But unlike a well-made optical lens, a gravitational lens has an infinite number of focal lengths, each associated with a different minimum distance of the light path from the object creating the lens. Each focal length has it own Einstein ring with a radius on the sky that is given by the following equation:

α = 2 ( G M/c2 D )1/2

In this equation , D is the distance from us to the lens source. The minimum focal length occurs for the light path passing over the limb of the lens source. We have already found the minimum focal length for the Sun: 570 AU. If we pulled farther back from the Sun, we would continue to see the ring of quasar light. The ring's diameter on the sky would become smaller as we pull back, decreasing as D-1/2, but the ratio of the ring's diameter to the Sun's diameter on the sky would increase as D1/2. This means that for us every star within our Galaxy is a gravitational lens.

Time Delays

Light passing to us through a gravitational lens takes longer to reach us than if it were traveling directly to us. This is caused by the longer distance traveled by light passing through a lens. For a light source at infinity, the time delay is independent of the distance of the lens from us; the mass of the object creating the lens is the only parameter that sets the magnitude of the time delay.

tdelay = 2 GM/c3

The time delay is only 10 microseconds for a solar mass star, but it is 0.3 years for a galaxy of 1012 solar masses.

Types of Lenses

Every gravitating object creates a lens, but not all of these lenses are observable. The issue is the probability of seeing an object behind a lens. This becomes an issue of distance to the lens and the density of lenses. Distance sets the area on the sky covered by the lens. This area is proportional to M/D, which favors galaxies and clusters of galaxies over nearby star; the larger mass of a galaxy over a star more than compensates for the galaxy's greater distance. For instance, the gravitational lens of a 1012 solar mass galaxy at 1 Gpc cover 1000 times more area on the sky as the lens of a 1 solar mass star at 1 parsec. A large galaxy cluster can produce a lens that covers 106 times the area of a stellar gravitational lens.

As expected from their greater area on the sky, the lenses we find most easily in astrophysics are those of galaxy clusters. The gravitational lens of a galaxy cluster has a structure that depends on the distribution of the galaxies within the cluster. Because the mass is not all concentrated at the center of the cluster, and because the mass can be distributed anisotropically, the gravitational lens of a cluster can produce complex multiple images. The lens is large enough on the sky that we can resolve theses images with a telescope. If these images are of highly-variable objects such as quasars, we see the time delays introduced by the lens.

The point gravitational lenses of stars are also observed, but these lenses are rare, and they are only found by systematically observing changes in the brightness of a million very distant stars every night. The image produced by a stellar lens is too small to resolve with a telescope, and the time delay associated with a stellar lens is too short to measure. The only observable effect of a stellar lens is the increase in apparent magnitude of stars seen through the lens.

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