## Equation [31 defines the effective potential Vr in its dependence on r by what we have to take away from the squared energy term to get the square of the radial velocity

Figure 8 Computer plot: Effective potential curve versus radius for an object orbiting a black hole for angular momentum L/m = 3.75M. Both effective potential and particle energy are measured along the vertical axis. Particle energy does not vary with radius—it is a constant of the motion. As a result the energy-versus-position plot is a horizontal line. Upper diagram: When the particle has energy corresponding to the minimum of the potential (point indicated by open circle), the particle remains at a constant radius and orbits the blade hole in a cirde. When it has a somewhat greater energy 0ine with double arrow), the particle oscillates back and forth in radius while orbiting around the center of attraction. Lower diagram: When the particle comes in with energy only slightly greater than the maximum of the potential at the left of the lower diagram, it slows its radial motion to nearly zero at the peak and orbits the black hole on a nearly arcular path before plunging, as in the terminal portion of the orbit drawn in Figure 4. For further analysis of Figure 4, see Figure 9.

Figure 8 Computer plot: Effective potential curve versus radius for an object orbiting a black hole for angular momentum L/m = 3.75M. Both effective potential and particle energy are measured along the vertical axis. Particle energy does not vary with radius—it is a constant of the motion. As a result the energy-versus-position plot is a horizontal line. Upper diagram: When the particle has energy corresponding to the minimum of the potential (point indicated by open circle), the particle remains at a constant radius and orbits the blade hole in a cirde. When it has a somewhat greater energy 0ine with double arrow), the particle oscillates back and forth in radius while orbiting around the center of attraction. Lower diagram: When the particle comes in with energy only slightly greater than the maximum of the potential at the left of the lower diagram, it slows its radial motion to nearly zero at the peak and orbits the black hole on a nearly arcular path before plunging, as in the terminal portion of the orbit drawn in Figure 4. For further analysis of Figure 4, see Figure 9.

### Define effective potential.

The squared effective potential is the product of two simple factors. The first depends entirely on the attracting mass M and the r-coordinate, not at all on the angular momentum. The second factor depends not at all on the attracting mass, but only on the angular momentum per unit mass L/m of the satellite and on its r-coordinate.

There are two obvious difficulties with equation [31 J.

First difficulty: Equation [31] is the difference of SQUARES of the total energy per unit mass and the effective potential per unit mass. Why squares? Why isn't it the simple difference between total energy per unit mass and the effective potential per unit mass, as in the case of the corresponding Newtonian equation [29]?

Second difficulty: For very large values of the radius r, the expression [32] for V(r)/m takes on the value unity. In contrast, the Newtonian effective potential V/m in equation [28] approaches the value zero at large r, as it must by the definitions used in Newtonian mechanics

Both these features of the effective potential are characteristic of relativity. To take the second objection first, the total energy of the particle at rest at infinity is E = m, so Elm = 1. Therefore the effective potential per unit mass must also have the value unity (Vim —> 1 as r—> so that the difference of their squares—equal to the square of drldt—remains zero in equation [31] for a particle at rest at infinity In contrast, the Newtonian effective potential is arbitrarily given the value zero at large separation In essence the relativistic formula provides the new value of unity for this previously arbitrary constant

Concerning the difference of squares in equation [31], we mentioned earlier that in general relativity one cannot separate different forms of energy. Rest energy, potential energy, and kinetic energy all combine into a larger unity in the term E Therefore we cannot claim that V(r) defined in equation [32) is the actual potential energy. Instead it is a quantity that helps us to visualize the radial component of the trajectories of particles—it is an "effective potential." This phrase does not explain the difference of squares in equation [31 ], but it shows that the interpretation of the separate terms is different from the Newtonian case

Plotting both V/m and E/m on the same graph, as in Figure 8, provides a measure of the radial speed, as predicted by equation [31]. The greater the vertical separation in this graph between the E/m line and the V/m curve for a given r, the greater is the radial component of the particle velocity at that r. (The vertical separation does not tell us whether the particle is moving inward or outward.) In particular, the radial motion is zero when the two graphs touch.

"Knife-edge" orbit predicted The decisive new feature of orbits around the black hole is shown in the lower diagram of Figure 8. Let the particle have energy slightly greater than the peak of the effective potential. Then its radial motion will slow to nearly zero as it creeps down in radius, spiraling in to the radius corresponding to this peak. Very slow radial motion corresponds to nearly

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