Effective Potential for Light

Orbits of light at a glance!

How to define effective ls there some way to set up an effective potential for light in order to visu-

potentiaP alize its trajectory simply and directly, the way we did in Chapter 4 for particles with mass? What is an effective potential, anyway? To define an effective potential for a particle with mass, we earlier made use of the equation for the radial motion (equations [30] and [31], pages 4-15 and 4-16):

Equation [23] gives us a recipe for the effective potential. On the left side of equation is a measure of the radial velocity of the particle. On the right side is the algebraic difference between two terms: The first term is a constant of the motion, independent of the position of the particle. The second term depends on the radius.

Try to find a similar equation for the motion of a light flash. The analogous measure of radial motion of such a pulse is given by equation [14]:

Equation [14] does not meet the requirement that the first term on the right be a constant independent of the radial position of the particle. Therefore this equation as it stands cannot be used to define an effective potential. However, if we look instead at equation [17], we do see a constant first term on the right-hand side:

Ur ^2 shell

In this equation, the "constant of the motion" is the impact parameter b. Put b into the first term on the right by dividing both sides of the equation by b2:

dr shell dt shell,

The left-hand side of this equation is a (rather strange!) measure of the radial velocity of the particle. The first term on the right-hand side depends, through b, on the choice of orbit but not on the Schwarzschild geometry. The second term on the right depends on the Schwarzschild geometry but not on the choice of orbit. This second term acts like the square of an effective potential:

^effective potential^2 < for a light flash J

Shell radial equation leads to effective potential

Effective potential for light

Only one effective potential for light of ALL frequencies

Qualitative predictions are easy, even with strange measure of radial motion

Different kinds of light trajectories for different values of b

Light orbits on a knife edge at r = 3M.

The expression [25] for effective potential makes no reference to the energy of the light or its impact parameter b. Therefore it applies to light of all wavelengths. Only one effective potential is needed to analyze the motion of all light (including radio waves, radar pulses, and gamma rays)! A plot of this square of the effective potential is shown in Figure 5.

Using this effective potential we can simply and quickly predict the major features of light motion around a nonspinning spherically symmetric center of attraction.

Equation [24] has one obvious drawback: The left-hand side expresses radial velocity in shell coordinates (with an extra coefficient 1/b2), rather than the accustomed Schwarzschild bookkeeper's coordinates r, <|>, and t. The good news is that shell velocity is "real," the possible result of a local measurement. The bad news is that shell coordinates are only local coordinates, usable only over a small range of radial coordinates r. Question: Of what use are local shell measures of velocity? Answer: Equation [24] and the resulting graphical plots help us to gain a qualitative understanding of the motion of light even on a global level. They allow us to make statements such as the following: "Initially the light pulse moves to smaller radius." "At a particular radius r the radial component of velocity of the flash goes to zero, so the flash moves tangentially, perpendicular to the radial direction.""Finally the light pulse moves to larger radius again." Once we have such a qualitative understanding of the motion, we can use computations based on equations [21] and [22] to assemble a wider-reaching Schwarzschild bookkeeper's accounting of the trajectory expressed in coordinates r, <|>, and t.

Figure 5 shows the square of the effective potential for light. It has a maximum at r = 3M and a value 1/(27 M2) at this maximum. Horizontal lines represent various possible values of 1/b2, where b is the impact parameter. According to equation [24], if the light beam has a value of 1 /¿r greater than the peak of the effective potential (small enough impact parameter b), the light is captured by the black hole. In contrast, if 1/ir has a value less than the peak of the effective potential (large enough impact parameter b), then the inward component of the light velocity goes to zero at the radius r for which the value of 1/b2 is equal to the effective potential. In this case the light subsequently moves outward again and flees the black hole. Figure 6 presents these results in another form.

Finally, if 1/&2 is just equal to the peak of the effective potential—in other words, if b = bcritical = (27)1/2 M = 5.20 M—then the light pulse stops its radial motion for some time at a coordinate radius r = 3M. But the tangential motion does not stop; the light moves for a while in a circular orbit. The light flash may stay at this radius for a fraction of an orbit or for many orbits, teetering on a knife edge before making a choice: return outward to a great distance or plunge on into the black hole. Which way it goes may seem random, because the choice is extremely sensitive to details of the way the light flash arrived in this orbit—similar to the question of which way a pencil balanced on its point will fall.

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