Relativity Lite

56 | Relativity Lite Isaac Newton showed in 1687 that the gravitational acceleration would be the same if all of a planet’s mass were concentrated at its center. Einstein said that acceleration due to thrusters and acceleration due to gravitation must be indistinguishable. So someone in a spaceship in orbit around a point-planet or collapsed star (figure 9) encounters the same bending of light as in figure 1a. Figure 9. Gravitational acceleration would be the same if all of a planet’s mass were concentrated at its center, so the bending of light it causes would be the same. We wish to know how the bending of light, and the consequent time dilation, depends on closeness to a star whose mass is all contained within a point. Since gravitational acceler- ation increases as one gets closer, the bending, too, will increase, with a picture something like figure 10. ∞ ← r s c τ ct r → r Figure 10. Consider a sequence of versions of figure 1c, in which the acceleration is small at the top of the present figure (where r → ∞) and increasingly larger as one approaches the Schwarzschild radius r s , the distance from the singularity at which clocks appear to stop according to an outside observer. The length of the arc of the bent light gets longer and longer, for this sequence of versions of figure 1c, the closer one approaches the bottom of the figure, near the Schwarzschild radius r s . There is no bending when one is far from the star ( r → ∞ at the top of figure 10), and there is an infinite bending at the bottom as one approaches some, as yet undetermined, distance from the point in space where all the mass is concentrated. To get an intuitive sense that this distance is not zero requires some thought.