According to Einstein's general theory of relativity, light rays are deflected as they pass by a massive object like the sun. The trajectory of a ray influenced by a central, spherically symmetric object of mass \(M\) lies in a plane with coordinates \(r\) and \(\theta\) (so-called Schwarzschild coordinates); the trajectory must be a solution of the differential equation

where \(u=1 / r, G\) is Newton's gravitational constant, and \(c\) is the constant speed of light.

(a) The right-hand side of this equation is ordinarily small. In fact, the ratio of the righthand side to the second term on the left is \(3 G M / r c^{2}\). Find the numerical value of this ratio at the surface of the sun. The sun's mass is \(2.0 \times 10^{30} \mathrm{~kg}\) and its radius is \(7 \times 10^{5} \mathrm{~km}\).

(b) If the right-hand side of the equation is neglected, show that the trajectory is a straight line.

(c) The effects of the term on the right-hand side have been observed. It is known that light bends slightly as it passes by the Sun and that the observed deflection agrees with the value calculated from the equation. Near a black hole, which may have a mass comparable to that of the sun but a much smaller radius, the right-hand side becomes very important, and there can be large deflections. In fact, show that there is a single radius at which the trajectory of light is a circle orbiting the black hole, and find the radius \(r\) of this circle.

Transcribed Image Text:

d-u 3GM +u= -u d02 c