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Gravitational lensing
It may of course happen that photons from the same star will travel trajectories that pass on opposite sides of the deflecting star and intersect each other after deflection, as illustrated in Fig. 11.7. Rays 1 and 2 are essentially parallel if the star ( ) is far from the deflecting object (S). An observer at position B would then see two images of the star, coming from apparently different directions.
This is a very simple and special arrangement of the objects, but it illustrates the principle that gravitating bodies act as lenses. Lensing is essentially universal: no matter how weak the deflection, it would always be possible to place the observer B far enough away from S to see two rays from the same point on the source. We don’t get such “double vision” when we look at the heavens because the probability of ‘being in exactly the right spot’ B for any given star and lens is small, and because many sources are not pointlike like our star: if the angular separation of the images at B (which is of order the deflection angle) is small compared to the angular size of the object on the sky, then we are not likely to be able to tell the difference between the two rays.
But as astronomers have built larger and more powerful telescopes, able to see much greater distances into the universe, they have revealed a sky filled with lensed images. Of particular importance is lensing by clusters of galaxies. There are so many galaxies in the universe that, beyond any given relatively distant cluster, there is a high probability that there will be another group of galaxies located in just the right position to be lensed into multiple images in our telescope: the probability of ‘being in exactly the right spot’ has become reasonably large. What is more, the masses of galaxy clusters are huge, so the deflections are much bigger than the sizes of the more distant galaxy images, so separating them is not a problem. In fact, we often see multiple images of the same object, created by the irregularities of the lensing mass distribution. A good example is in Fig. 11.8.
Even more important than the creation of separate images can be the brightening of single images by the focusing of light from them. This is called magnification. The magnification of galaxy images by lensing makes it possible for astronomers to see galaxies at greater distances, and has helped studies of the very early universe. Lensing also helps us map the mass distribution of the lensing cluster, and this has shown that clusters have much more mass than can be associated with their luminous stars. Astronomers call this dark matter, and it will be an important subject in the next chapter, when we discuss cosmology.
Detailed modeling of observed lenses has shown that the dark matter is distributed more smoothly inside clusters than the stars, which clump into the individual galaxies. The mass of a cluster may be ten or more times the mass of its stars. The dark matter is presumably
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Figure 11.7 |
Deflection can produce multiple images. |
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11.1 Trajectories in the Schwarzschild spacetime
A picture taken with the Hubble Space Telescope of a cluster called 0024+1654 (fuzzy round galaxies) showing many images of a much more distant galaxy. The images are distorted, all of them stretched in the direction transverse to the line joining the image and the center of the cluster, and compressed along this line. Picture courtesy of W. N. Colley and E. Turner (Princeton University), J. A. Tyson (Bell Labs, Lucent Technologies) and NASA.
made of something that carries no electric charge, because light from the distant galaxies passes through it without absorption or scattering, and it seems to emit no electromagnetic radiation. Physicists do not know of any elementary particle that could serve as dark matter in this way: apart from neutrinos, all electrically neutral particles known at present (2008) are unstable and decay quickly. And neutrinos are thought to be too light to have been trapped in the gravitational fields of clusters. Dark matter is one of the biggest mysteries at the intersection of astronomy and particle physics today.
Studies of gravitational lensing have the potential to reveal any matter that clumps in some way. Lensing observations of individual stars in our galaxy (microlensing) have shown that there appears to be a population of objects with stellar masses that are also dark: they are observed only because they pass across more distant stars in our Galaxy and produce lensing. In this case, observers do not see two separate images. Instead, they see is brightening of the distant star, caused by the fact that the lens focuses more light on
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the telescope than would reach it if the lens were not there. This brightening is temporary because the lensing object itself moves, and so the lensing event is transitory. Optical studies of the region around the lensed stars have shown that the lensing object is not as bright as a normal white dwarf star but has a mass that is a good fraction of a solar mass. The nature of these lensing objects, which could represent a significant fraction of the mass of the Galaxy, is still (2008) a mystery.
11.2 N a t u re o f t h e s u r fa ce r = 2M
Coordinate singularities
It is clear that something funny goes wrong with the line element, Eq. (11.1) at r = 2M, but what is not clear is whether the problem is with the geometry or just with the coordinates. Coordinate singularities – places where the coordinates don’t describe the geometry properly – are not unknown in ordinary calculus. Consider spherical coordinates at the poles. The north pole on a sphere has coordinates θ = 0, 0 φ < 2π . That is, although φ can have any value for θ = 0, all values really correspond to a single point. We might draw a coordinate diagram of the sphere as follows (Fig. 11.9 – maps of the globe are sometimes drawn this way), in which it would not be at all obvious that all points at θ = 0 are really the same point. We could, however, convince ourselves of this by calculating the circumference of every circle of constant θ and verifying that these approached zero as θ → 0 and θ → π . That is, by asking questions that have an invariant geometrical meaning, we can tell if the coordinates are bad. For the sphere, the metric is positive-definite, so if two points have zero distance between them, they are the same point (e.g. θ = 0, φ = π and θ = 0, φ = 2π : see Exer. 18, § 11.7). In relativity, the situation is more subtle, since there are curves (null curves) where distinct points have zero invariant distance between them. In fact, the whole question of the nature of the surface r = 2M is so subtle that it was not answered satisfactorily until 1960. (This was just in time, too, since black holes began to
θπ
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Figure 11.9 One way of drawing as sphere on a flat piece of paper. Not only are φ 0 and φ 2π really the same lines, but the lines θ = 0 and θ = π are each really just one point. Spherical coordinates are therefore not faithful representations of the sphere everywhere.
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11.2 Nature of the surface r = 2M |
be of importance in astronomy within a decade as new technology made observations of quasars, pulsars, and X-ray sources possible.) We shall explore the problem by asking a few geometrical questions about the metric and then demonstrating a coordinate system which has no singularity at this surface.
Infal ling particles
Let a particle fall to the surface r = 2M from any finite radius R. How much proper time does that take? That is, how much time has elapsed on the particle’s clock? The simplest
= ˜ =
particle to discuss is the one that falls in radially. Since dφ 0, we have L 0 and, from
Eq. (11.11), |
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(the minus sign because the particle falls inward). It is clear that if E > 1 (unbound par-
˜ =
ticle), the integral of the right-hand side from R to 2M is finite. If E 1 (particle falling
from rest at ∞), the integral is simply |
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which is again finite. And if E < 1, there is again no problem since the particle cannot be
− ˜ 2 =
at larger r than where 1 E 2M/r (see Eq. (11.58)). So the answer is that any particle can reach the horizon in a finite amount of proper time. In fact there is nothing about the integral that prevents us placing the lower limit smaller than 2M, that is the other side of the surface r = 2M. The particle apparently can go inside r = 2M in a finite proper time.
We now ask how much coordinate time elapses as the particle falls in. For this we use
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Therefore we have |
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˜ =
For simplicity, we again consider the case E 1 and examine this near r the new variable
ε := r − 2M.
Then we get
dt = −(ε + 2M)3/2dε . (2M)1/2ε
(11.61)
2M by defining
(11.62)
