Schwarzschild Maps of the Motion of Light

The larger view that no observer observes!

A light flash moving under the influence of a spherically symmetric center of attraction of given mass M has an orbit whose size and shape, praise be, depends on only a single quantity, the impact parameter b.

The trajectory of a light flash near a black hole lends itself to a simple description using the effective potential. For example, Figure 6 is what we call a Schwarzschild map of the orbits of light for three sample values of the impact parameter b. The Schwarzschild map shows three light trajectories as a function of Schwarzschild bookkeeper coordinates r and <|>. Figure 5 traces the radial motions along three such trajectories using the effective potential.

From these figures we can derive a qualitative description of the trajectory of any light pulse, no matter what the value of its impact parameter b. The more formal—and accurate—Schwarzschild map of the trajectory comes from integrating equations [14] and [15] themselves.

Size and shape of orbit depends only on b

Describe orbits with "Schwarzschild map "

Magnetic Reconnectin Black Hole
Figure 6 Schematic Schwarzschild map of light trajectories around a black hole for the three values of the impact parameter b shown in Figure 5, page 5-13.

1 Light is captured for b less than the critical value 27u2 M

2. For critical impact parameter, light teeters on unstable circular orbit atr- 3M. Eventually the light will plunge into the black hole—or escape to infinity, as shown

3 For larger impact parameters, the trajectory is deflected but light is not captured

SAMPLE PROBLEM 2 Escaping Light Flash?

Does the laser pulse described in Sample Problem 1, page 5-10, escape the black hole? SOLUTION

That pulse had an impact parameter b = 5.59 M. This value is greater than the critical impact parameter 6crit,cai = (27)1/2 M = 5 20 M So that pulse escapes from the black hole

The simplicity and power of the effective potential is witnessed by the brevity of this sample problem, which is the shortest in the book'

Schwarzschild map does not Figures 5 and 6 do not tell us what we would see if we stood on a spherical predict what shell observer shell near a black hole nor what color light we would perceive. Those fig-

sees ures focus on a Schwarzschild map, a plot artificially constructed from the accounting entries of the Schwarzschild bookkeeper, using coordinates r, <|>, and t. The shell observer does not agree with the Schwarzschild bookkeeper about the direction of motion of these light beams. He does not even agree with the bookkeeper on the value of the speed of light! What we actually see as we stand on a shell is the subject of the following section. Here we explore Schwarzschild maps of some additional trajectories of light as plotted by the far-away bookkeeper.

As a specific example, think of light beams from different directions converging on a point at r-coordinate r = 3M, symbolized by a small open circle in Figure 7. Beams from a distant star located along the horizontal line move straight in from the right. Beams coming from stars at the left, directly behind the black hole, arrive at this point from both above and below at an angle of 90 degrees to the outward radial direction (radial component of velocity equal to zero where the 1/b2 horizontal line grazes the top of the effective potential, as shown in Figure 5). Beams from stars farther forward in angle arrive in directions less than 90 degrees to the outward radial direction, again from both above and below—and indeed from all transverse directions obtained by rotating Figure 7 around its horizontal axis.

Orbtas Png
Figure 7 Schematic Schwarzschild map of trajectories of light that converge on and are absorbed at a point (small open circle) on the shell at r = 3M,

Shell observer's view of the black hole

"Halo" around the black hole

The "light sphere"

Bookkeeper and shell observer disagree on direction of motion of flash.

What about light arriving from angles greater than 90 degrees from the radially outward direction? One sample light beam is shown leaving the given point at angle greater than 90 degrees and plunging into the black hole. No light can come the other way along the same trajectory because that light would have to come out of the black hole, which light—or anything else—cannot do! Hence a point at r = 3M receives no light from angles greater than 90 degrees from the outward direction. In Section 9 we show that the viewer standing on the spherical shell at r = 3M sees the edge of the black hole in this direction, at 90 degrees from the radially outward direction.

The effective potential for light allows us to predict another effect unique to general relativity. Figures 5 and 6 show that the r-coordinate r = 3M is the radius for the knife-edge orbit for a light beam approaching the black hole at the critical impact parameter bcrjtiCal=

M. But some light along the wave front from every star approaches the black hole at this critical impact parameter. Therefore every visible star contributes to a spherical shell of light circling the black hole at the radius r = 3M, each beam bound at least temporarily. These beams arrive at that radius from all sides at an angle of 90 degrees in the Schwarzschild map. In the following section we show that the viewer standing on this shell also sees these beams arriving from opposite directions at angles of ±90 degrees from the straight-outward direction. In other words, a viewer stationed at r = 3M sees additional images of all the stars in the sky scattered on a narrow bright ring that extends all around him, transverse to the radially outward direction. This bright ring forms a "halo" around the image of the black hole. In Section 11 of this chapter we meet a similar image, called there an "awesome ring bisecting the sky."

These results apply to a reception point at r = 3M on the so-called light sphere. For points on other spherical shells, the black hole will also be surrounded by a halo, but light from given stars will arrive at different angles than for r = 3M, and the view of the sky can be quite different. For example, Figure 8 describes light from a single star arriving from several directions at a fixed point on the shell, indicated by the small open circle in the figure. Looking around him, the shell observer sees multiple images of this single star in different directions.

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