0 10 20 30 40 50 60 70 80 90 9 (degrees): Shell viewing angle (8 = 0 radially outward)

Figure 13 Computer plot of the direction $ to a distant star (vertical scale) as a function of the viewing angle 9 = 90 ^ (horizontal scale) at which a spectator sees the star when perched on a shell at radius r^ See Figures Wand 11. Both angles are measured from the radially outward direction. Example (dashed lines): You stand on the shell at r0 = 3M and see a star at 65 degrees from the radially outward direction (dot on horizontal axis) If there were no center of gravitational attraction behind you, this star would appear at 132 degrees from the radially outward direction (dot on vertical axis)—a bit behind you. For each value of r0, the graph also shows the angle at which the edge of the black hole appears to this shell viewer.

Light blue shift In addition to changing direction, light from a distant star also changes energy as seen by a shell inhabitant. Energy of starlight is upshifted (the so-called gravitational blue shift) as seen by the shell spectator. The formula for this energy change is equation [27], page 3-17.

Any astronomical body can act as a distorting lens, as Einstein himself recognized. However, Einstein doubted that we would ever see such an image, because it requires that the imaged star and the "lens star" line up almost perfectly. This alignment can be less perfect if both the distant object and the imaging object are galaxies. Figure 14 shows such an image, an Einstein ring, created by radio waves rather than light. The "ring" is actually the distorted image of a distant galaxy focused by an intermediate galaxy lying on a line between the imaged galaxy and our observation point, Earth. Figure 14 is the first Einstein ring ever observed; since then we have seen a number of them (and a large number of ring segments) at various wavelengths of electromagnetic radiation. For more on this subject see Project D, Einstein Rings.

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