Orbits or Trajectories
How can we obtain the orbit of a planet around the sun, in general relativity? Taylor and Wheeler changes this question into a smilar question about the orbit of a satellite of a black hole. You should go through their kind and interesting exposition, but the essence of their exposition may be conveniently summarized in the Figure 2. It makes use of "effective potential", V(r) depending on r-coordinate, which in essense determines the orbit or trajectory. V is defined by the following two equations:
(dr/dĄ)(dr/dĄ) = (E/m)(E/m) - (V/m)(V/m)
(V(r)/m)(V(r)/m) = (1 - 2M/r)(1 + (L/m)(L/m)/rr).
E/m is the total energy per unit mass, and L is the angular momentum. Remember that if the satellite has a different amount of angular momentum (see the Figure 1), the curve of effective potential is different.
Figure 1
You may imagine the whole figure (Figure 2, center) is rotating around the vertical axis (the center of the black hole) with a given angular momentum (L/m, per unit mass); thus the satellite's motion is a compound motion of this rotation and and the motion along the radial direction. Given a curve (of effective potential), the level of total energy per unit mass, E/m, determines the orbit. For details, see Taylor & Wheeler (2000), chapter 4. The following figure is adapted from their Figure 11, 4-23. The effective potential curve is mostly similar to the Newtonian curve, except for the smaller values of r-coordinate; the relativistic curve has a characteristic "knife" edge, because of the curvature factor.
Figure 2
Everything in this figure is calculated from Schwarzschild metric. Take the case of "elliptical orbit", the case 2 (orange orbit). The effective potential curve determines the length of radius (r-coordinate) at each instant; the case is quite analogical to a ball rolling up and down in a bowl (Wheeler 1999, ch. 10); when it reaches a maximal height, it turns around and goes inward. The only difference is that the satellite goes around the black hole (along a geodesic), but the length of radius (from the black hole) changes exactly as the distance of a ball from a specified center.
Figure 3
The important difference from the Newtonian mechanics emerges because, according to general relativity (i.e. Schwarzschild geometry, in this case), the satellite takes extra "dwell time" in closer part to the black hole; this makes a small difference of angle when it begins new cycle, and this is in essence the cause of the precession of the "perihelion of Mercury" (for an actual calculation, see Project C of Taylor & Wheeler 2000; see how physicists make use of approximations). Einstein made a similar calculation in November 1915, and found the result which exactly agrees with the observed data; this made him very happy, and then, this was the only experimental support of his theory.
Thus we know the orbit of a satellite. Then what about the trajectory of light? How does a light ray travel around a black hole? Taylor and Wheeler answer this question in chapter 5. Since light differs from a mass point in that it has 0 mass, the the preceding effective potential is not applicable. However, we can define the effective potential for light in the following way. The radial motion of light depends on what is called the impact parameter b: how far away its trajectory is from the light going straight to the center of the black hole (figure adapted from Taylor & Wheeler 2000, Fig. 2, 5-6).
Then the radial motion of light is related to b as in the following formula (derived from Schwarzschild metric).
(1/bb)(dr_shell/dt_shell)(dr_shell/dt_shell) = (1/bb) - (1 - 2M/r)/rr
From this, we can define the square of effective potential for light:
square of effective potential = (1 - 2M/r)/rr.
Thus, we can know the behavior of light, from the relation of this (squared) potential and 1/bb, just as we could know the orbit of a satellite from its effective potential. For details see Taylor and Wheeler, chapter 5. The upshot is the following figure (figure adapted from Taylor & Wheeler 2000, Fig. 5 and Fig 6, 5-13, 14). You may notice that the deflection of starlight around the sun is a simple application of this.
References
Taylor and Wheeler (2000) Exploring Black Holes, chapters 4 and 5, Addison Wesley Longman.
Wheeler (1999) A Journey into Gravity and Spacetime, chapter 10, Scientific American Library.
Last modified Dec. 4, 2004.(c) Soshichi Uchii webmaster
