100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

The Physical Basis of Black Hole Astrophysics 13512V(r)0.90.81.8531.60.70 5 10 15r/r gFig. 2. Effective potential for motion in the Schwarzschild spacetime. Curves are labeledwith the values of L/r gc. The dashed line at V = 0.95 and the dotted line at V = 0.998illustrate turning points.Fig. 2. Since V (r) plays the role of a potential in Eq. (15), we can infer thenature of particle orbits from Fig. 2. For r/r g ≫ 1 the orbit is determinedby the usual effects of gravitational attraction (−r g /r), and centrifugal replulsion(+L 2 /c 2 r 2 ). The extra non-Newtonian term at the end of Eq. (16)represents the inescapably strong pull of gravity on a particle close to thehole.Consider, for example, a particle with E 2 = 0.95. The dashed line inFig. 2 shows that the particle will have turning points approximately atr/r g = 3.4 and 15.0, which turn out to be not very different from theNewtonian turning points. This particle would then be in a very eccentricbound orbit which, when combined with the equation for dφ/dτ in Eq. (11),can be shown not to be a closed orbit, but rather has the nature of aprecessing ellipse. The turning point at r/r g = 1.56 is a new, completelynon-Newtonian feature. It is, of course, not a feature of the eccentric orbitbetween the turning points at r/r g = 3.4 and 15.0. Rather it applies to aparticle that is created very close to the hole, moving outward with E = 0.95and L/r g c = 2. This particle will reach a maximum of r equal to 1.56 r gwhere it turns, and begins its spiral into the hole.The effect of the non-Newtonian peaks of the potential has an importantconsequence for the shape of orbits. For a particle with L/r g c = 2, the peakof the potential is at V = 1. If this particle happens to have E 2 = 0.998