## Schwarzschild Map vs Shell View

Different maps; different directions

The observer on a given shell and the Schwarzschild bookkeeper both track the path of a light flash between two events, A and B, that lie near the shell observer. The shell observer tracks this light flash moving past him at an angle 0shell with respect to the radially outward direction (Figure 9). At what angle %chw does the Schwarzschild bookkeeper record the light beam to be traveling? Because of the way we define the reduced circumference r (Section 4 of Chapter 2), the two observers agree on the tangential displacement component rdiJ) of motion as the light moves outward. However, they disagree about the radial separation between these shells.

Figure 8 Schematic (not computed!) Schwarzschild map showing the formation of multiple images of a single star. Light from a single distant star approaches the black hole along effectively parallel paths (shown here coming from the upper left) These beams can arrive at a given visual observer (open circle) along alternative trajectories around the black hole Of all possible parallel light paths from a given star, four labeled with circled numbers are selected here for examination Ray 1 is the most direct path arriving at the observer A second ray from the distant star follows path 2, skirting the black hole on the opposite side from the first ray but closer to the blade hole and therefore bent more. Incoming ray 3 circles the black hole once or many times clockwise near the radius 3M. Most of the light in this ray eventually falls into the black hole or escapes outward to infinity along various trajectories. 8ut some small fraction of the light in ray 3 escapes along a trajectory that arrives at the observer. Likewise, incoming ray 4 ardes the black hole counterclockwise and makes a fourth image of the distant star for the observer. In brief, the shell observer sees multiple images of the same star in several directions (see Section 8). Double images of distant galaxies corresponding to paths 1 and 2 in the map above have been observed from Earth. In each such case the attractive gravitational center between us and the imaged quasars is believed to be a low-luminosity galaxy or cluster of galaxies (See the Einstein ring in Figure 14).

The brightness of each of these different star images seen by the observer depends on the focussing properties of space near each trajectory Do nearby rays converge or diverge along each path? Answering such questions is beyond the scope of this book

The Schwarzschild bookkeeper reckons the angle to be given by the equation (from Figure 9):

The shell observer claims that the radial separation between shells is not dr but rather drs^e\\. So for the shell observer the tangent of the angle is shell rz Figure 9 Angle of light motion for bookkeeper and shell observer compared Light flash moves from event A to event B. Each observer measures an angle of travel 8 with respect to the radially outward direction. They agree that the tangential displacement is rdty during this travel between adjacent shells. However, they disagree on the radial distance between these shells The radial distance is greater as measured by the shell observer. As a result, the shell observer measures a smaller angle of motion 6 with respect to the radially outward direction

Schwarzschild (bookkeeper) map

### Shell observer map

Figure 9 Angle of light motion for bookkeeper and shell observer compared Light flash moves from event A to event B. Each observer measures an angle of travel 8 with respect to the radially outward direction. They agree that the tangential displacement is rdty during this travel between adjacent shells. However, they disagree on the radial distance between these shells The radial distance is greater as measured by the shell observer. As a result, the shell observer measures a smaller angle of motion 6 with respect to the radially outward direction

Angles between light trajectories for bookkeeper and shell observer

But we know that the relationship between the two measures of radial distances is given by equation [D] in Selected Formulas at the end of the book:

dr shell

Therefore the relation between the two tangents is ry/2

shell 