One can consider the motion of planets or of light in the gravitational field of the...

Transcribed Image Text:

One can consider the motion of planets or of light in the gravitational field of the Sun (i.e., of test bodies in Schwarzschild space-time). This motion is given by the geodesic equation where 10 = 1 = 33 m r(r - 2m) where a = r²=const. ds² == sin cos 0, 3 23 do² r ď² 1 do² r I'₁ r Then the equation for planets and light can be written as ď² 1 de" de vp ds ds + IH -T₁, T2=-(r-2m), 33 11, + + = cot 0, m = = 1 mc² 3m r 3m r 5 0, light, GM planets -(r-2m) sin 0, (27) (28) Hint In Newtonian dynamics, the gravitational force is just an inertial force. (29) (30) Figure 3: Coordinate system for binary stars in circular orbits, The origin is at the center of mass and m1a1 m₂a2. 1. Show that the radius at which light travels in a circular path around a body of mass M is r = 3m 2. Show that this radius is r = m in the Newtonian theory One can consider the motion of planets or of light in the gravitational field of the Sun (i.e., of test bodies in Schwarzschild space-time). This motion is given by the geodesic equation where 10 = 1 = 33 m r(r - 2m) where a = r²=const. ds² == sin cos 0, 3 23 do² r ď² 1 do² r I'₁ r Then the equation for planets and light can be written as ď² 1 de" de vp ds ds + IH -T₁, T2=-(r-2m), 33 11, + + = cot 0, m = = 1 mc² 3m r 3m r 5 0, light, GM planets -(r-2m) sin 0, (27) (28) Hint In Newtonian dynamics, the gravitational force is just an inertial force. (29) (30) Figure 3: Coordinate system for binary stars in circular orbits, The origin is at the center of mass and m1a1 m₂a2. 1. Show that the radius at which light travels in a circular path around a body of mass M is r = 3m 2. Show that this radius is r = m in the Newtonian theory